traj_id large_string | problem large_string | gold_answer large_string | teacher_answer large_string | is_correct bool | difficulty large_string | length_bin large_string | n_tokens int64 | thinking_paragraphs list | final_response large_string | nodes list | method large_string | exploration_share float64 | teacher large_string | annotator large_string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
deepmath-53165_3 | Determine the minimal polynomial of \(\sqrt[10]{5}\) over \(\mathbb{Q}(e^{2\pi i/10})\). | x^5 - \sqrt{5} | x^5-\sqrt{5} | true | hard | mid | 5,878 | [
"We need to find the minimal polynomial of α = 5^{1/10} over K = Q(ζ_10) where ζ_10 = e^{2πi/10}.",
"Let me think about this carefully.\n\nFirst, ζ_10 = e^{2πi/10} = cos(π/5) + i sin(π/5). This is a primitive 10th root of unity.",
"The field Q(ζ_10) is the 10th cyclotomic field. Its degree over Q is φ(10) = φ(2... | # Minimal Polynomial of $\sqrt[10]{5}$ over $\mathbb{Q}(e^{2\pi i/10})$
## Setting Up the Fields
Let $\alpha = 5^{1/10}$ and $\zeta_{10} = e^{2\pi i/10}$. I need to work over $K = \mathbb{Q}(\zeta_{10})$.
**Key identification:** Since $\zeta_5 = \zeta_{10}^2$, we have $\mathbb{Q}(\zeta_5) \subseteq \mathbb{Q}(\zeta_... | [
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"bran... | Tower law shows 5|[K(α):K] and α^5=√5∈K gives [K(α):K]≤5, so degree is 5 and the minimal polynomial is x^5−√5. | 0.2569 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-54201_2 | Find an irreducible polynomial in \(\mathbb{Z}[x]\) that is reducible modulo 2, 3, and 5. | x^4 + 1 | x^4+1 | true | hard | mid | 3,570 | [
"We need find irreducible polynomial in Z[x] that is reducible mod 2, 3, 5. Need reason step by step. Final boxed. Need solve.",
"We need polynomial integer coefficients irreducible over Q (primitive, irreducible in Z[x]) but reducible modulo 2,3,5. We can choose simple polynomial. Need ensure reducible modulo ea... | Consider the polynomial
\[
f(x)=x^4+1 \in \mathbb{Z}[x].
\]
We show it is irreducible over \(\mathbb{Z}\), but reducible modulo \(2\), \(3\), and \(5\).
---
### 1. Irreducibility over \(\mathbb{Z}\)
Use the shift \(x \mapsto x+1\). We have
\[
f(x+1)=(x+1)^4+1=x^4+4x^3+6x^2+4x+2.
\]
Now apply Eisenstein’s criteri... | [
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"bran... | Propose x⁴+1, verify reducibility mod 2, 3, 5 by explicit factorization, prove irreducibility over ℤ via quadratic factorization argument. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55540_7 | Evaluate the integral: $$ \int \cos x \cdot \cos^2(2x) \cdot \cos^3(3x) \cdot \cos^4(4x) \cdot \ldots \cdot \cos^{2002}(2002x) \, dx $$ | 0 | 0 | true | hard | mid | 4,023 | [
"Let me analyze this integral carefully.\n\nThe integrand is:\n$$\\cos x \\cdot \\cos^2(2x) \\cdot \\cos^3(3x) \\cdot \\cos^4(4x) \\cdot \\ldots \\cdot \\cos^{2002}(2002x)$$",
"The general term is $\\cos^k(kx)$ for $k = 1, 2, 3, \\ldots, 2002$.",
"So the integrand is:\n$$f(x) = \\prod_{k=1}^{2002} \\cos^k(kx)$$... | Looking at this integral, I need to identify the structure of the integrand:
$$f(x) = \prod_{k=1}^{2002} \cos^k(kx)$$
## Step 1: Establish the Anti-Periodicity of $f(x)$
I'll check what happens when I shift $x$ by $\pi$:
$$\cos(k(x+\pi)) = \cos(kx + k\pi) = (-1)^k\cos(kx)$$
Therefore:
$$\cos^k(k(x+\pi)) = \left[(... | [
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"bran... | Uses anti-periodicity f(x+π)=-f(x) to split the integral over [0,2π] into two cancelling halves, yielding 0. | 0.5336 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-59826_5 | Find a function $f \in L^p(\mathbb{R})$ such that for a given non-negative function $g \in L^1(\mathbb{R})$, the convolution $f * g = ||g||_1 \cdot f$, where $*$ denotes convolution. | 0 | f = 0 | true | hard | mid | 4,621 | [
"We need to find f in L^p(R) such that f * g = ||g||_1 * f, where g is a given non-negative function in L^1(R). This is a convolution equation.",
"Let me think about this carefully.\n\nThe convolution is defined as:\n(f * g)(x) = ∫ f(y) g(x-y) dy = ∫ f(x-y) g(y) dy",
"We want: f * g = ||g||_1 * f",
"So: ∫ f(y... | # Finding $f \in L^p(\mathbb{R})$ such that $f * g = \|g\|_1 \cdot f$
## Setting Up the Equation
We need to solve the convolution equation:
$$\int_{\mathbb{R}} f(y)\, g(x-y)\, dy = \|g\|_1 \cdot f(x) \quad \text{for all } x.$$
## Fourier Transform Analysis
Applying the Fourier transform $\hat{f}(\xi) = \int f(x)\, ... | [
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"bra... | Fourier transform reduces equation to f̂(ĝ−‖g‖₁)=0; triangle inequality shows ĝ=‖g‖₁ only at ξ=0, forcing f̂=0 a.e., hence f=0. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-57027_6 | Consider a compact connected Riemannian 2-manifold $M$ with positive curvature and diameter 1. Determine the supremum of the average distance between two randomly chosen points on $M$, distributed according to the volume form on $M$. Assume $M$ is diffeomorphic to a 2-sphere. What is this supremum? | \dfrac{1}{2} | \dfrac{1}{2} | true | hard | mid | 8,698 | [
"Let me think carefully about this problem. We have a compact connected Riemannian 2-manifold $M$ with positive curvature and diameter 1, diffeomorphic to $S^2$. We need to find the supremum of the average distance between two randomly chosen points, distributed according to the volume form.",
"First, let me clar... | Looking at this problem, I need to find the supremum of the average distance between two random points over all positively curved Riemannian metrics on $S^2$ with diameter 1.
## Setting Up the Average Distance
For a Riemannian manifold $(M,g)$ with area $A$, the average distance is:
$$\bar{d} = \frac{1}{A^2}\int_M \i... | [
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"bran... | Compute average distance on round sphere (1/2), then argue via shape comparisons that no positively curved deformation can exceed this value. | 0.5338 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-106289_6 | Find all real numbers \(\lambda\) such that the polynomial equation \[ x^{n} + \lambda\left(x^{n-1} + \cdots + x^{3} + W x^{2} + A B x + 1\right) = 0 \] has only real roots, where \( n \geq 4 \), and \(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n} \) are two sets of real numbers s... | 0 | \lambda = 0 | true | hard | mid | 5,481 | [
"Let me carefully analyze this problem. We have the polynomial equation:\n$$x^n + \\lambda(x^{n-1} + x^{n-2} + \\cdots + x^3 + Wx^2 + ABx + 1) = 0$$",
"where $n \\geq 4$, and $\\alpha_1, \\ldots, \\alpha_n$ and $\\beta_1, \\ldots, \\beta_n$ are two sets of real numbers with $\\sum \\alpha_j^2 < 1$ and $\\sum \\be... | # Finding All Real $\lambda$ for Which the Polynomial Has Only Real Roots
## Setting Up Vieta's Formulas
The polynomial is:
$$P(x) = x^n + \lambda x^{n-1} + \lambda x^{n-2} + \cdots + \lambda x^3 + \lambda W x^2 + \lambda AB\, x + \lambda$$
If all roots $r_1, \ldots, r_n$ are real, then by Vieta's formulas:
- $e_1 =... | [
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"bran... | Vieta's formulas express elementary symmetric polynomials in λ; Newton's inequality for k=2 forces λ²≤0 since the coefficient exceeds 1 for n≥4, yielding λ=0. | 0.1581 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101325_5 | Determine the value of \( c \in \mathbb{R} \) such that the function \( \kappa(t) = ct^4(1-t)^4 \) can be the signed curvature of a regular, closed, and convex curve \( \alpha: [0,1] \rightarrow \mathbb{R}^2 \). | 1260\pi | 1260\pi | true | hard | mid | 9,969 | [
"We need to find c such that κ(t) = c t^4 (1-t)^4 is the signed curvature of a regular, closed, and convex curve α: [0,1] → R^2.",
"First, let me think about what conditions are needed for a function κ(t) to be the signed curvature of a regular closed convex curve.",
"A convex curve has positive signed curvatur... | Looking at this problem, I need to find $c$ such that $\kappa(t) = ct^4(1-t)^4$ can serve as the signed curvature of a regular, closed, convex plane curve.
## Setting Up the Key Conditions
**Convexity:** A convex curve has signed curvature of constant sign. Since $t^4(1-t)^4 > 0$ for $t \in (0,1)$, I need $c > 0$ (co... | [
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"bran... | Apply turning tangent theorem (total curvature = 2π) with arc length parametrization: c·∫₀¹ t⁴(1-t)⁴ dt = 2π, giving c = 1260π. | 0.5085 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55262_1 | Evaluate the complex integral \( \int _{ C } \frac { \cos(iz) }{ { z }^{ 2 }({ z }^{ 2 }+2i) } dz \) where \( C \) is the rectangle with vertices at -3, 3, -3i, and i. | 0 | 0 | true | hard | mid | 8,318 | [
"We need to evaluate ∫_C cos(iz) / (z^2 (z^2 + 2i)) dz where C is the rectangle with vertices at -3, 3, -3i, and i.",
"First, let me understand the contour. The vertices are -3, 3, -3i, and i. Let me plot these:\n- -3 is at (-3, 0)\n- 3 is at (3, 0)\n- -3i is at (0, -3)\n- i is at (0, 1)",
"So the rectangle has... | Looking at this problem, I need to evaluate the integral of $f(z) = \frac{\cos(iz)}{z^2(z^2+2i)}$ over a rectangular contour.
## Setting Up the Problem
**Singularities:** The denominator $z^2(z^2+2i) = 0$ gives:
- $z = 0$ (pole of order 2)
- $z^2 = -2i \Rightarrow z = \pm(1-i)$, i.e., $z = 1-i$ and $z = -1+i$ (simple... | [
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"bra... | Residue theorem on a square contour; the two simple-pole residues cancel by symmetry and the double-pole residue vanishes, giving integral = 0. | 0.2885 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-51840_7 | Consider the set \(\mathcal{S}_X\) of trigonometric polynomials \(f(t) = \sum_{|k| \leq X} c_k e^{2\pi i kt}\) on the circle \(\mathbb{T} = \mathbb{R}/\mathbb{Z}\) with degree \(\leq X\), such that \(f(0) = 1\) and \(c_0 = 0\). Define
\[ M_X(f) = \sup_{\mathbb{T} \setminus [-\frac{1}{X},\frac{1}{X}]} |f|. \]
Let \(B_X ... | 0 | 0 | true | hard | mid | 3,367 | [
"Let me carefully analyze this problem about trigonometric polynomials.\n\nWe have the set $\\mathcal{S}_X$ of trigonometric polynomials $f(t) = \\sum_{|k| \\leq X} c_k e^{2\\pi i kt}$ on $\\mathbb{T} = \\mathbb{R}/\\mathbb{Z}$ with:\n- Degree $\\leq X$ (meaning $|k| \\leq X$)\n- $f(0) = 1$\n- $c_0 = 0$",
"Wait, ... | Looking at this problem, I need to determine the behavior of $B_X = \inf_{f \in \mathcal{S}_X} M_X(f)$ as $X \to \infty$, where $\mathcal{S}_X$ consists of degree-$X$ trigonometric polynomials with $c_0 = 0$ and $f(0) = 1$.
## Setting Up the Constraints
The conditions $c_0 = 0$ and $f(0) = 1$ mean:
- The average of $... | [
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"b... | Construct f=(Fejér kernel − 1)/X to satisfy constraints, then bound |f| away from 0 by O(1/X)→0. | 0.1629 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-103523_5 | Determine all real numbers \( C \) such that for any integer \( n \geq 2 \), there exists a "naughty set" in an \( n \times n \) grid containing at most \( Cn^2 \) small squares. | \dfrac{1}{2} | \dfrac{1}{2} | true | hard | mid | 3,084 | [
"Let me think about what a \"naughty set\" is in the context of an $n \\times n$ grid.",
"A \"naughty set\" in an $n \\times n$ grid is a set of cells such that every row and every column contains at least one cell from the set. This is essentially a \"hitting set\" for all rows and columns.",
"Wait, let me rec... | # Finding All Valid Values of $C$ for Naughty Sets
## Understanding the Problem
A **naughty set** in an $n \times n$ grid is a set of cells such that **every row and every column contains at least one cell** from the set. I need to find all $C$ such that for every $n \geq 2$, a naughty set with at most $Cn^2$ cells e... | [
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"bran... | Define naughty set as row/column hitting set, find minimum size n, derive C≥1/n for all n≥2, binding constraint at n=2 gives C≥1/2. | 0.4582 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-103876_3 | In quadrilateral \(ABCD\), the diagonals intersect at point \(M\) with an angle \(\alpha\) between them. Let \(O_{1}, O_{2}, O_{3}, O_{4}\) be the centers of the circumcircles of triangles \(ABM, BCM, CDM,\) and \(DAM\), respectively. Find the ratio of the areas of quadrilateral \(ABCD\) to quadrilateral \(O_{1} O_{2} ... | 2 \sin^2 \alpha | 2\sin^2\alpha | true | hard | mid | 6,734 | [
"Let me work through this problem carefully.\n\nWe have a quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at point $M$ with angle $\\alpha$ between them.",
"The four triangles are:\n- $ABM$ (circumcircle center $O_1$)\n- $BCM$ (circumcircle center $O_2$)\n- $CDM$ (circumcircle center $O_3$)\n- $DAM... | # Finding the Ratio of Areas of $ABCD$ to $O_1O_2O_3O_4$
## Setting Up Coordinates
I place $M$ at the origin with diagonal $AC$ along the x-axis and diagonal $BD$ at angle $\alpha$:
$$A = (-a,\, 0),\quad B = (b\cos\alpha,\, b\sin\alpha),\quad C = (c,\, 0),\quad D = (-d\cos\alpha,\, -d\sin\alpha)$$
where $a, b, c, d... | [
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"bran... | Coordinate geometry with M at origin; compute four circumcenters via perpendicular bisectors, find both quadrilateral areas via shoelace, take ratio. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-52595_1 | Find the value of \(a \in \mathbb{R}\) such that the solution set of the linear system \[ \begin{cases} -x+2y+az=3 \\ y-z=a \\ x+y-z=-9 \end{cases} \] represents a line perpendicular to the plane \(K\) given by \[ \begin{cases} x=t+1 \\ y=-t+2s-1 \\ z=s-1 \end{cases} \]. | -2 | -2 | true | hard | mid | 8,685 | [
"Let me work through this problem step by step.\n\nFirst, I need to find the solution set of the linear system and determine when it represents a line.",
"The system is:\n$$\\begin{cases} -x+2y+az=3 \\\\ y-z=a \\\\ x+y-z=-9 \\end{cases}$$",
"Let me write the augmented matrix:\n$$\\begin{pmatrix} -1 & 2 & a & | ... | # Finding $a$ for a Line Perpendicular to Plane $K$
## Step 1: Set Up the Augmented Matrix
The system is:
$$\begin{cases} -x+2y+az=3 \\ y-z=a \\ x+y-z=-9 \end{cases}$$
$$\left(\begin{array}{ccc|c} -1 & 2 & a & 3 \\ 0 & 1 & -1 & a \\ 1 & 1 & -1 & -9 \end{array}\right)$$
## Step 2: Determine When the Solution Set is ... | [
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"bran... | Compute determinant to find a = -2 as unique value yielding a line; verify consistency; accept as answer despite perpendicularity check failing. | 0.6349 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-100704_2 | Calculate the Euler characteristic of the surface given by the presentation \( \langle a,b,c \mid abcb^{-1}a^{-1}b^{-1}\rangle \) and determine the type of surface it represents. | -1 | -1 | true | hard | mid | 8,894 | [
"Let me work through this step by step.\n\nWe have a surface given by the presentation $\\langle a, b, c \\mid abcb^{-1}a^{-1}b^{-1} \\rangle$.",
"First, let me understand what this presentation means. This is a single-relator presentation with three generators and one relator. The relator is $abcb^{-1}a^{-1}b^{-... | Looking at this problem, I need to analyze the presentation $\langle a, b, c \mid abcb^{-1}a^{-1}b^{-1} \rangle$ to find the Euler characteristic and identify the surface.
## Step 1: Simplify the Presentation
The relator $abcb^{-1}a^{-1}b^{-1} = 1$ allows me to solve for one generator. I'll conjugate $c$ by $b$: set ... | [
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"bran... | Solve the relator algebraically to show the group is F₂, identify it as the fundamental group of a pair of pants (genus 0, 3 boundaries), compute χ = 2−0−3 = −1. | 0.2392 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55325_5 | Find the function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the equation \( f(x) = x f\left(\frac{1}{x}\right) = 1 + f(x+y) - f(y) \) for all \( x, y \in \mathbb{R} \setminus \{0\} \). | x + 1 | f(x) = x + 1 | true | hard | mid | 8,287 | [
"Let me carefully parse this problem. We need to find $f: \\mathbb{R} \\to \\mathbb{R}$ such that for all $x, y \\in \\mathbb{R} \\setminus \\{0\\}$:",
"$$f(x) = x f\\left(\\frac{1}{x}\\right) = 1 + f(x+y) - f(y)$$",
"Wait, I need to be more careful about the domain restrictions. The equation says \"for all $x,... | Looking at this problem, I need to find $f: \mathbb{R} \to \mathbb{R}$ satisfying both conditions simultaneously for all $x, y \neq 0$:
1. $f(x) = x\,f\!\left(\tfrac{1}{x}\right)$
2. $f(x) = 1 + f(x+y) - f(y)$
## Step 1: Transform the second equation
Rearranging equation (2): $f(x+y) = f(x) + f(y) - 1$
Substituting... | [
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"bran... | Substituting g=f−1 yields restricted Cauchy equation; proving g(0)=0 and Q-linearity extends to full Cauchy; first equation forces linearity; h=g−id argument rules out wild solutions. | 0.3811 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-108077_3 | Given the sequence \( \{a_{n}\} \) defined by \( a_{1} = 0 \), \( a_{2} = 1 \), and for \( n \geq 3 \), \( a_{n} = \frac{1}{2} n a_{n-1} + \frac{1}{2} n(n-1) a_{n-2} + (-1)^{n} \left(1-\frac{n}{2}\right) \), find the simplest expression for \( f_{n} = \sum_{k=0}^{n-1} (k+1) C_{n}^{k} a_{n-k} \). | 2n! - n - 1 | f_n = 2n! - n - 1 | true | hard | mid | 9,227 | [
"Let me work through this problem step by step.\n\nWe have a sequence $\\{a_n\\}$ defined by:\n- $a_1 = 0$\n- $a_2 = 1$\n- For $n \\geq 3$: $a_n = \\frac{1}{2} n a_{n-1} + \\frac{1}{2} n(n-1) a_{n-2} + (-1)^n \\left(1 - \\frac{n}{2}\\right)$",
"We need to find $f_n = \\sum_{k=0}^{n-1} (k+1) \\binom{n}{k} a_{n-k}$... | # Finding the Simplest Expression for $f_n$
## Setting Up the Exponential Generating Function
Let me define the EGFs:
$$A(x) = \sum_{n=1}^{\infty} a_n \frac{x^n}{n!}, \qquad E(x) = \sum_{n=0}^{\infty} f_n \frac{x^n}{n!}$$
**Expressing $f_n$ in terms of $A(x)$:** With the substitution $j = n-k$:
$$f_n = \sum_{j=1}^{n... | [
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"bran... | Exponential generating functions convert the recurrence and binomial convolution into algebraic equations; coefficient extraction yields the closed form. | 0.3869 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-54814_3 | Given the power series \( f(z)=\sum^{\infty}_{n=0}a_n(z-i)^n \) and \( g(z)=\sum^{\infty}_{n=0}b_n(z+i)^n \), both with a radius of convergence of \( \frac{3}{2} \), determine the radius of convergence for the series \( h(z)=f(z)+g(z) \) centered at 0. | \dfrac{1}{2} | \frac{1}{2} | true | hard | mid | 3,226 | [
"Let me work through this problem carefully.\n\nWe have:\n- $f(z) = \\sum_{n=0}^{\\infty} a_n (z-i)^n$ with radius of convergence $R_f = \\frac{3}{2}$ (centered at $z = i$)\n- $g(z) = \\sum_{n=0}^{\\infty} b_n (z+i)^n$ with radius of convergence $R_g = \\frac{3}{2}$ (centered at $z = -i$)",
"We need to find the r... | # Finding the Radius of Convergence of $h(z) = f(z) + g(z)$ Centered at 0
## Setting Up the Convergence Regions
- $f(z) = \sum a_n(z-i)^n$ converges in the disk: $|z - i| < \frac{3}{2}$ (centered at $i$)
- $g(z) = \sum b_n(z+i)^n$ converges in the disk: $|z + i| < \frac{3}{2}$ (centered at $-i$)
The function $h(z) =... | [
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"bran... | Compute the largest disk centered at 0 contained in both convergence disks via triangle inequality, yielding radius 1/2. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-56328_2 | Find \( \lim_{n \rightarrow \infty} \mathbb{P}(X_1 + \ldots + X_n \leq 0) \), where the independent binary random variables \(X_k\) take values \(\pm 1\) with probabilities \((1 \pm k^{-1/2})/2\) for \(k = 1, 2, \ldots\). | \Phi(-2) | \Phi(-2) | true | hard | mid | 3,527 | [
"Let me carefully parse the problem. We have independent random variables $X_k$ for $k = 1, 2, 3, \\ldots$ where $X_k$ takes values $+1$ and $-1$ with probabilities.",
"The values are $\\pm 1$ with probabilities $(1 \\pm k^{-1/2})/2$.",
"So for each $k$:\n- $P(X_k = +1) = \\frac{1 + k^{-1/2}}{2}$\n- $P(X_k = -1... | Looking at this problem, I need to analyze the sum $S_n = X_1 + \cdots + X_n$ where $X_k \in \{+1, -1\}$ with $\Pr(X_k = +1) = \frac{1+k^{-1/2}}{2}$ and $\Pr(X_k = -1) = \frac{1-k^{-1/2}}{2}$.
## Computing Mean and Variance
**Mean of $X_k$:**
$$E[X_k] = (+1)\cdot\frac{1+k^{-1/2}}{2} + (-1)\cdot\frac{1-k^{-1/2}}{2} = ... | [
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"bran... | Compute mean and variance of S_n, apply Lindeberg CLT, show the standardized threshold converges to -2, yielding Φ(-2). | 0.0191 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-100364_6 | Evaluate the limit: $$\lim\limits_{x \to -\infty} \left(\frac{\left((x+3)^2+x\cos(x)\right)(x)!}{(x-1)!+2^x}\right)$$ | 0 | 0 | true | hard | mid | 7,895 | [
"We need to evaluate the limit as x approaches -∞ of:",
"$$\\lim_{x \\to -\\infty} \\frac{((x+3)^2 + x\\cos(x)) \\cdot (x)!}{(x-1)! + 2^x}$$",
"First, I need to think about what the factorial function means for non-integer and negative values. The factorial function is typically defined for non-negative integer... | # Evaluating the Limit
## Setting Up the Factorial Ratio
I'll use the Gamma function extension $x! = \Gamma(x+1)$, which satisfies $\Gamma(x+1) = x\,\Gamma(x)$, i.e., $x! = x\cdot(x-1)!$.
Substituting this into the expression:
$$\frac{\bigl((x+3)^2 + x\cos x\bigr)\, x!}{(x-1)! + 2^x} = \frac{\bigl((x+3)^2 + x\cos x... | [
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"bran... | Factor (x-1)! from denominator, show Γ(x) decays super-exponentially while 2^x decays only exponentially, so denominator ≈ 2^x, making the whole expression super-exponentially small, dominating polynomial growth. | 0.4532 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-103457_0 | Find the constant \(c\) to which the sequence \(p_n\) converges, where \(p_n\) is the probability that a randomly-chosen \(2n\)-digit number is special. A number is special if its digits can be split into two sets of size \(n\) with equal sums. Assume leading zeros are allowed in \(2n\)-digit numbers. | \dfrac{1}{2} | \dfrac{1}{2} | true | hard | mid | 8,654 | [
"Let me think about this problem carefully.\n\nWe have a $2n$-digit number where each digit is from 0 to 9. A number is \"special\" if its $2n$ digits can be split into two sets of size $n$ each with equal sums.",
"Let the digits be $d_1, d_2, \\ldots, d_{2n}$ where each $d_i \\in \\{0, 1, \\ldots, 9\\}$.",
"A ... | Looking at this problem, I need to find the probability that a random $2n$-digit number has its digits splittable into two groups of size $n$ with equal sums, then find the limit as $n \to \infty$.
## Setting Up the Problem
A number with digits $d_1, d_2, \ldots, d_{2n}$ (each in $\{0,\ldots,9\}$) is special if there... | [
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"bra... | Show S must be even (prob 1/2), then argue that given S even, the exponential number of n-subsets makes a valid split exist with probability →1. | 0.1871 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-53896_7 | Find all prime numbers \( p \) such that the last \( p \) digits of \( p^p \) form a prime number. It is known that \( p = 433 \) satisfies this condition. Are there any other such prime numbers? | 433 | 433 | true | hard | mid | 6,674 | [
"Let me think about this problem carefully. We need to find all prime numbers $p$ such that the last $p$ digits of $p^p$ form a prime number.",
"The \"last $p$ digits of $p^p$\" means $p^p \\mod 10^p$. Let's call this number $N = p^p \\mod 10^p$.",
"We need $N$ to be a prime number.\n\nFirst, let me think about... | # Finding Primes $p$ Where the Last $p$ Digits of $p^p$ Form a Prime
## Setting Up the Problem
I need to find all primes $p$ such that $N = p^p \bmod 10^p$ is itself prime.
## Eliminating Small Primes
For $p^p$ to have **more than** $p$ digits, we need $p\log_{10}p > p-1$, which requires $p \geq 11$.
For $p \leq 7... | [
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"bran... | Eliminate small primes via digit-count argument, check p=11, then rely on known computational result that 433 is the unique solution. | 0.363 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-106924_0 | Let \( f(x) \) be a polynomial with integer coefficients such that \( f(15) f(21) f(35) - 10 \) is divisible by 105. Given \( f(-34) = 2014 \) and \( f(0) \ge 0 \), find the smallest possible value of \( f(0) \). | 620 | 620 | true | hard | mid | 9,265 | [
"Let me work through this problem step by step.\n\nWe need to find a polynomial $f(x)$ with integer coefficients such that:\n1. $f(15) \\cdot f(21) \\cdot f(35) - 10 \\equiv 0 \\pmod{105}$\n2. $f(-34) = 2014$\n3. $f(0) \\geq 0$",
"We want to minimize $f(0)$.\n\nSince 105 = 3 × 5 × 7, we need:\n- $f(15) \\cdot f(2... | Looking at this problem, I need to find a polynomial $f(x) \in \mathbb{Z}[x]$ satisfying all constraints and minimize $f(0)$.
## Setting Up Modular Constraints
Since $105 = 3 \times 5 \times 7$, I need $f(15) \cdot f(21) \cdot f(35) \equiv 10 \pmod{3, 5, 7}$.
**Using the property** $f(a) \equiv f(b) \pmod{m}$ when $... | [
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"bran... | Decompose mod 105 into mod 3, 5, 7 constraints on f(0); use f(-34)=2014 for divisibility-by-34 constraint; combine via CRT to find smallest valid f(0). | 0.2844 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-60202_3 | Find the value of $A + B + C$ such that \[ S = \sum_{n=1}^{\infty} \frac{1}{n^2 \binom{2n}{n}} = \frac{A}{B} \zeta(C). \] | 6 | 6 | true | hard | mid | 5,807 | [
"We need solve problem: sum_{n=1}^\\infty 1/(n^2 binom(2n,n)) = (A/B) zeta(C). Find A+B+C. Need step by step. Need think carefully, likely known series. Need identify value. Let's analyze.",
"S = sum_{n>=1} 1/[n^2 C(2n,n)]. We need express as rational multiple of zeta(C) maybe zeta(3)? Or zeta(2)? Let's see.",
... | We want to evaluate
\[
S=\sum_{n=1}^{\infty}\frac{1}{n^2\binom{2n}{n}}.
\]
Use the Beta integral
\[
\int_0^1 x^{n-1}(1-x)^n\,dx
= B(n,n+1)
= \frac{(n-1)!\,n!}{(2n)!}.
\]
Since
\[
\binom{2n}{n}=\frac{(2n)!}{n!\,n!},
\]
we have
\[
\frac{(n-1)!\,n!}{(2n)!}
= \frac{1}{n\binom{2n}{n}}.
\]
Thus
\[
\frac{1}{n^2\binom... | [
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"bran... | Beta function integral representation converts the series to a single integral, then factorization 1−t+t²=(1+t³)/(1+t) reduces it to known logarithmic integrals yielding ζ(2)/3. | 0.0334 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101240_0 | Consider the Erdős–Rényi random graph model $G(n, p)$ with $n$ vertices and edge probability $p \in [0,1]$. Denote $\mathbb{E}_{n,p}[k]$ as the average number of clusters, including isolated vertices as $1$. Analyze the asymptotic behavior of
$$ p \frac{1}{n}\frac{d}{dp} \mathbb{E}_{n,p}[k] \quad \text{as} \quad n \to... | 0 | 0 | true | hard | mid | 8,449 | [
"Let me carefully analyze this problem about the Erdős–Rényi random graph model $G(n,p)$.",
"First, let me understand what's being asked. We have $n$ vertices, each pair connected with probability $p$. We're looking at the expected number of connected components (clusters), including isolated vertices.",
"Let m... | # Analyzing the Derivative of Expected Cluster Count in $G(n,p)$
## Setting Up the Expected Number of Components
The expected number of connected components (clusters) in $G(n,p)$ is:
$$\mathbb{E}_{n,p}[k] = \sum_{j=1}^{n} \binom{n}{j}\, q_j(p)\,(1-p)^{j(n-j)}$$
where $q_j(p) = \Pr(G(j,p) \text{ is connected})$.
#... | [
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"bran... | Direct computation of dE[c]/dp via component-size sum, approximating q_k(p) by tree probability, showing both derivative terms scale as n². | 0.1672 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-53896_1 | Find all prime numbers \( p \) such that the last \( p \) digits of \( p^p \) form a prime number. It is known that \( p = 433 \) satisfies this condition. Are there any other such prime numbers? | 433 | 433 | true | hard | mid | 7,668 | [
"Let me think about this problem carefully. We need to find all prime numbers $p$ such that the last $p$ digits of $p^p$ form a prime number.",
"The last $p$ digits of $p^p$ is $p^p \\bmod 10^p$.",
"Let me denote $q = p^p \\bmod 10^p$. We need $q$ to be prime.",
"First, let me think about what we can say abou... | Looking at this problem, I need to find all primes $p$ where $q = p^p \bmod 10^p$ is itself prime.
## Setting Up the Problem
The last $p$ digits of $p^p$ is the number $q = p^p \bmod 10^p$, where $0 \leq q < 10^p$.
**Eliminating $p = 2$ and $p = 5$:**
- $p = 2$: $2^2 = 4$, last 2 digits: $04 = 4$ → not prime.
- $p =... | [
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"bran... | Eliminates small primes by direct computation, establishes basic properties of q, but ultimately relies on the given fact and problem phrasing to assert 433 is unique. | 0.3971 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-102541_3 | Given the sets \( A=\{(x, y) \mid |x| + |y| = a, a > 0\} \) and \( B=\{(x, y) \mid |xy| + 1 = |x| + |y|\} \), if \( A \cap B \) forms the vertices of a regular octagon in the plane, find the value of \( a \). | 2 + \sqrt{2} | 2+\sqrt{2} | true | hard | mid | 4,663 | [
"Let me work through this problem carefully.\n\nSet A: The set of points (x, y) such that |x| + |y| = a, where a > 0. This is a diamond (square rotated 45°) with vertices at (a, 0), (-a, 0), (0, a), (0, -a).",
"Set B: The set of points (x, y) such that |xy| + 1 = |x| + |y|.",
"Let me analyze set B. Let u = |x| ... | # Finding the Value of *a* for a Regular Octagon
## Analyzing Set A
The set $A = \{(x,y) : |x| + |y| = a\}$ is a **diamond** (rotated square) with vertices at $(\pm a, 0)$ and $(0, \pm a)$.
## Analyzing Set B
For $B = \{(x,y) : |xy| + 1 = |x| + |y|\}$, substituting $u = |x|$, $v = |y|$:
$$uv + 1 = u + v \implies u... | [
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"bran... | Factor B's equation, find 8 intersection points with the diamond, order them, compute alternating side lengths, set them equal to solve for a. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-60314_3 | Find the non-zero value of $k$ such that the average area between the curves $x^2$ and $ax$ for $x \ge 0$, as $a$ varies from $0$ to $k$, equals $x^2$. What is this value of $k$? | 24 | k = 24 | true | hard | mid | 3,528 | [
"Let me carefully parse this problem. We need to find the non-zero value of $k$ such that the average area between the curves $x^2$ and $ax$ for $x \\ge 0$, as $a$ varies from $0$ to $k$, equals $x^2$.",
"Wait, I need to think about what \"the average area between the curves $x^2$ and $ax$ for $x \\ge 0$\" means.... | # Finding the Value of $k$
## Setting Up the Area Between the Curves
For a fixed value of $a > 0$, the curves $y = x^2$ and $y = ax$ intersect at:
$$x^2 = ax \implies x(x - a) = 0 \implies x = 0 \text{ or } x = a$$
The bounded area between them for $x \geq 0$ is:
$$A(a) = \int_0^a (ax - x^2)\,dx = \left[\frac{ax^2}{... | [
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"bran... | Compute area between x² and ax as a³/6, average over [0,k] to get k³/24, set equal to k² (interpreting x² as k²), solve for k=24. | 0.7562 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-54629_1 | Find all continuous functions \( f \) such that \( f(x) - 1999f\left(\frac{2x}{1-x^2}\right) = 18 \) for \(|x| \neq 1\). | -\dfrac{1}{111} | f(x)=-\frac{1}{111} | true | hard | mid | 9,185 | [
"We need solve functional equation for continuous functions f such that for |x| != 1:\nf(x) - 1999 f(2x/(1-x^2)) = 18.\nWe need find all continuous functions (presumably real-valued on R? Domain where |x| != 1; equation for all real x except ±1. Need continuity? likely continuous on R? Need infer. f continuous mayb... | Let
\[
T(x)=\frac{2x}{1-x^2}.
\]
We are given that \(f\) is continuous and
\[
f(x)-1999f(T(x))=18 \qquad (|x|\ne 1).
\]
First plug in \(x=0\):
\[
f(0)-1999f(0)=18
\]
so
\[
-1998f(0)=18
\]
and therefore
\[
f(0)=-\frac{1}{111}.
\]
Now define
\[
g(x)=f(x)+\frac{1}{111}.
\]
Then \(g\) is continuous, \(g(0)=0\),... | [
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"bra... | Evaluate at x=0 for particular solution; subtract to get homogeneous equation; use inverse branch H(y)=tan(½arctan y) to iterate backward, showing g≡0 by continuity at 0. | 0.2592 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-58743_6 | Determine the number of countable models of ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). | 2^{\aleph_0} | 2^{\aleph_0} | true | hard | mid | 4,433 | [
"The question asks for the number of countable models of ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice).",
"Let me think about this carefully.\n\nFirst, I need to clarify what \"countable models of ZFC\" means. A model of ZFC is a structure $(M, \\in^M)$ that satisfies all axioms of ZFC. A \"countable... | # Number of Countable Models of ZFC
## Setting Up the Problem
A **countable model** of ZFC is a structure $(M, \in^M)$ where $M$ is a countable set satisfying all axioms of ZFC. I'll assume ZFC is consistent (the standard working assumption).
## Upper Bound: At Most $2^{\aleph_0}$
The language of set theory contain... | [
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"bran... | Upper bound by counting structures on a countable domain; lower bound by showing ZFC has 2^{aleph_0} complete extensions, each yielding a non-isomorphic countable model. | 0.1369 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55650_6 | Evaluate the integral \( \int_0^{2 \pi} \ln(1+e^{i \theta}) \, d\theta \). | 0 | 0 | true | hard | mid | 6,701 | [
"We need to evaluate I = ∫_0^{2π} ln(1 + e^{iθ}) dθ. This is a complex integral. We need to be careful about the branch of the logarithm. Let me think about this carefully.",
"First, let me consider the integrand. We have 1 + e^{iθ}. When does this equal zero? e^{iθ} = -1, which happens at θ = π. So the integrand... | Looking at this integral, I need to carefully handle the branch of the logarithm, since $1 + e^{i\theta} = 0$ when $\theta = \pi$ (a point on the integration path).
## Factorization
I factor:
$$1 + e^{i\theta} = e^{i\theta/2}\left(e^{-i\theta/2} + e^{i\theta/2}\right) = 2\cos\!\left(\frac{\theta}{2}\right) e^{i\theta... | [
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"bran... | Factor 1+e^{iθ}=e^{iθ/2}·2cos(θ/2), split at θ=π for the log branch, compute four sub-integrals separately. | 0.2308 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-63216_4 | Compute the limit \( \lim_{t \to + \infty} \int_0^{+ \infty} \frac{ \mathrm{d} x}{e^x+ \sin (tx)} \). | \dfrac{\pi}{2} | \dfrac{\pi}{2} | true | hard | mid | 4,802 | [
"We need to compute limit as t -> +infinity of integral from 0 to infinity of dx / (e^x + sin(tx)).",
"Let me denote I(t) = ∫_0^∞ dx / (e^x + sin(tx)).",
"First, let me check that the integrand is well-defined. The denominator is e^x + sin(tx). Since e^x ≥ 1 for x ≥ 0 and sin(tx) ≥ -1, we have e^x + sin(tx) ≥ 1... | Looking at this problem, I need to evaluate:
$$I(t) = \int_0^{\infty} \frac{dx}{e^x + \sin(tx)} \quad \text{as } t \to +\infty$$
## Setting Up the Oscillation Argument
**Well-definedness:** Since $e^x \geq 1$ for $x \geq 0$ and $\sin(tx) \geq -1$, the denominator satisfies $e^x + \sin(tx) \geq 0$, with equality only... | [
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"bran... | Average the rapidly oscillating sin(tx) over its period to reduce the limit to ∫₀^∞ dx/√(e^{2x}−1), then evaluate by u=e^x and u=sec θ substitutions. | 0.2475 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-104942_6 | Find the maximum value of the expression \[ A = \frac{\left(x^{2} - y\right) \sqrt{y + x^{3} - x y} + \left(y^{2} - x\right) \sqrt{x + y^{3} - x y} + 1}{(x - y)^{2} + 1} \] for \( x, y \in (0,1] \). | 1 | 1 | true | hard | mid | 8,427 | [
"Let me analyze this expression carefully.\n\n$A = \\frac{(x^2 - y)\\sqrt{y + x^3 - xy} + (y^2 - x)\\sqrt{x + y^3 - xy} + 1}{(x-y)^2 + 1}$",
"for $x, y \\in (0, 1]$.",
"Let me first check what happens at specific points.\n\n**Case: $x = y$**",
"If $x = y$, the denominator is $0 + 1 = 1$.",
"The numerator be... | Looking at this problem, I need to find the maximum of $A$ over $x,y \in (0,1]$.
## Setting Up the Key Inequality
Since $A \leq 1$ is equivalent to:
$$(x^2 - y)\sqrt{y + x^3 - xy} + (y^2 - x)\sqrt{x + y^3 - xy} \leq (x-y)^2$$
I'll prove this by defining $f(x,y)$ as the left side and analyzing by cases. Note the expr... | [
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"bran... | Case analysis on signs of (x²-y) and (y²-x); in the critical case, bound each radical term using y≤x² and x≥y², then reduce to x³+y³≤x²+y². | 0.5432 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-63054_7 | Find a continuous function \( f: [0,1] \rightarrow \mathbb{R} \) that can be expressed as two distinct infinite polynomial series:
\[ f(x) = \sum_{n=0}^\infty c_n x^n \]
where the sequence \((c_n)_n\) is not unique. | 0 | f(x) = 0 | true | hard | mid | 6,059 | [
"The user is asking for a continuous function $f: [0,1] \\to \\mathbb{R}$ that can be expressed as two distinct infinite polynomial series (power series), meaning there exist two different sequences $(c_n)$ and $(d_n)$ such that:",
"$$f(x) = \\sum_{n=0}^\\infty c_n x^n = \\sum_{n=0}^\\infty d_n x^n$$",
"for all... | Looking at this problem, I need to find a continuous function on $[0,1]$ that admits **two different** power series representations $\sum c_n x^n$.
## Setting Up the Core Question
If $f(x) = \sum_{n=0}^{\infty} c_n\, x^n = \sum_{n=0}^{\infty} d_n\, x^n$ for all $x \in [0,1]$ with $(c_n) \neq (d_n)$, then the differen... | [
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"bran... | Applies the power series identity/uniqueness theorem to show no non-zero continuous function on [0,1] admits two distinct power series representations, concluding f(x)=0. | 0.5268 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-53295_5 | Find the quadratic variation of the process $X(s) = W_{s+\epsilon} - W_s$, where $W_s$ is a standard Wiener process. | 2t | 2t | true | hard | mid | 9,201 | [
"We need to find the quadratic variation of the process $X(s) = W_{s+\\epsilon} - W_s$, where $W_s$ is a standard Wiener process and $\\epsilon > 0$ is a fixed constant.",
"The quadratic variation of a process $X(s)$ on $[0, T]$ is defined as:\n$$[X]_T = \\lim_{\\|\\Pi\\| \\to 0} \\sum_{i=0}^{n-1} (X(t_{i+1}) - X... | # Quadratic Variation of $X(s) = W_{s+\epsilon} - W_s$
## Setting Up the Problem
I need to compute the quadratic variation:
$$[X]_T = \lim_{\|\Pi\|\to 0} \sum_{i=0}^{n-1} \big(X(t_{i+1}) - X(t_i)\big)^2$$
where $\|\Pi\| = \max_i |t_{i+1}-t_i| \to 0$.
## Computing the Increment
Each increment factors cleanly:
$$\Del... | [
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"bran... | Compute X's covariance (triangular kernel), show increment variance is 2h, then prove L² convergence of the QV sum to 2T via explicit variance bound. | 0.2796 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-102889_3 | Evaluate the triple integral \( I = \iiint_{G} \frac{d x d y}{1-x-y} \), where the region \( G \) is bounded by the planes: \( x + y + z = 1 \), \( x = 0 \), \( y = 0 \), \( z = 0 \), \( x = 1 \), \( y = 2 \), \( y = 5 \), \( z = 2 \), and \( z = 4 \). | \dfrac{1}{2} | \dfrac{1}{2} | true | hard | mid | 5,652 | [
"Let me carefully analyze this problem. We need to evaluate the triple integral:",
"$$I = \\iiint_G \\frac{dx\\,dy\\,dz}{1-x-y}$$",
"Wait, the integral is written as $\\iiint_G \\frac{dx\\,dy}{1-x-y}$. This is a triple integral, so the integrand should be $\\frac{1}{1-x-y}$ with respect to $dx\\,dy\\,dz$.",
"... | Looking at this problem, I need to identify the region $G$ and evaluate the triple integral.
## Identifying the Region
The region $G$ is bounded by the planes $x=0$, $y=0$, $z=0$, and $x+y+z=1$ (with $x \leq 1$ automatically satisfied at the vertex). The additional planes ($y=2$, $y=5$, $z=2$, $z=4$) lie outside the ... | [
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"bra... | Identify region as tetrahedron x+y+z≤1 in first octant; integrate 1/(1-x-y) over z first, reducing to a simple 2D integral yielding 1/2. | 0.525 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55316_1 | Let $a,b \in \mathbb{R}_+$ be real positive numbers with $\frac{1}{2}<a<1$, and let $I=[0,b]$ be a closed real interval. Define the sequence of functions $f_n(x) : I \to \mathbb{R}$ by \[ f_n(x) = \frac{n}{(2n-1)^{a+x}} - \frac{n}{(2n)^{a+x}} \] and the sequence of functions $g_n(x) : I \to \mathbb{R}$ by \[ g_n(x) = \... | 0 | 0 | true | hard | mid | 7,023 | [
"Let me work through this problem carefully.\n\nWe have $f_n(x) = \\frac{n}{(2n-1)^{a+x}} - \\frac{n}{(2n)^{a+x}}$ where $a \\in (1/2, 1)$ and $x \\in [0, b]$.",
"The function $g_n(x) = f_n'(x) = \\frac{n \\ln(2n)}{(2n)^{a+x}} - \\frac{n \\ln(2n-1)}{(2n-1)^{a+x}}$.",
"Wait, let me verify this derivative. \n\n$f... | Looking at this problem, I need to analyze whether $h(x) = \sum_{n=1}^{\infty} \left|\frac{f_n(x)-f_n(0)}{x} - g_n(0)\right|^2 \to 0$ as $x \to 0^+$.
## Setting Up the Key Representation
By Taylor's theorem with integral remainder applied to $f_n$:
$$f_n(x) = f_n(0) + x \cdot f_n'(0) + x^2 \int_0^1 (1-t)\, f_n''(tx)... | [
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"bran... | Taylor integral remainder reduces to bounding Σ|f_n''|²; MVT exploits cancellation in f_n'' to get summable n^{-2a}(ln n)⁴, giving h(x) ≤ Cx². | 0.1577 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-58752_7 | Find the smallest integer \( n \) such that:
\[ 2 \cdot \sum_{k = 1}^{n} \frac{\left(\frac{n}{100}\right)^k}{k!} \geq e^{\frac{n}{100}} \] | 70 | 70 | true | hard | mid | 4,773 | [
"Let me work through this problem carefully.\n\nWe need to find the smallest integer $n$ such that:\n$$2 \\cdot \\sum_{k=1}^{n} \\frac{\\left(\\frac{n}{100}\\right)^k}{k!} \\geq e^{\\frac{n}{100}}$$",
"Let me denote $x = \\frac{n}{100}$. Then the inequality becomes:\n$$2 \\cdot \\sum_{k=1}^{n} \\frac{x^k}{k!} \\g... | # Finding the Smallest Integer $n$
## Setting Up the Problem
Let $x = \frac{n}{100}$. The inequality becomes:
$$2\sum_{k=1}^{n}\frac{x^k}{k!} \geq e^x$$
## Key Insight: Relating to the Exponential Series
Since $e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!} = 1 + \sum_{k=1}^{\infty}\frac{x^k}{k!}$, the infinite sum gives:... | [
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"bran... | Approximate the finite sum by e^x−1 (remainder negligible), reduce to e^x≥2, solve x≥ln2 to get n≥69.31, verify boundary. | 0.4038 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101338_6 | A broken clock moves the minute hand each minute randomly one minute ahead or one minute behind with equal probabilities. Each minute it does so independently of all other times it moved the minute hand. The hour hand is synchronized with the minute hand. The clock starts running at noon. Let T denote the first time th... | 3600 | 3600 | true | hard | mid | 5,740 | [
"Let me carefully analyze this problem.\n\nA broken clock starts at noon. Each minute, the minute hand moves either 1 minute ahead or 1 minute behind with equal probability (1/2 each), independently each minute. The hour hand is synchronized with the minute hand.",
"First, let me understand what \"synchronized wi... | # Finding the Expected Time to 1:00 or 11:00
## Setting Up the Model
I'll model the clock's minute hand position as a **simple random walk on the integers**. Starting at noon, I assign position $M_0 = 0$. Each minute, $M_{t+1} = M_t + 1$ or $M_t - 1$ with equal probability $\frac{1}{2}$.
The clock display is determi... | [
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"bran... | Reduce to simple symmetric random walk on integers hitting ±60, apply standard hitting time formula E[τ]=a². | 0.543 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-59251_2 | Determine the Hausdorff dimension of the set $A$ of non-recurrent walks on $\mathbb{Z}$, where walks are identified with infinite binary sequences in the Cantor space $2^\mathbb{N}$. A non-recurrent walk is defined as a sequence $x$ for which $\frac{\sum_{k=1}^n x(k)}{n} = \frac{1}{2}$ for only finitely many $n$. What ... | 1 | 1 | true | hard | mid | 4,687 | [
"Let me carefully work through this problem about the Hausdorff dimension of the set of non-recurrent walks on $\\mathbb{Z}$.",
"First, let me understand the setup. We're working in the Cantor space $2^\\mathbb{N}$ with the product topology (or equivalently, the metric $d(x,y) = 2^{-\\min\\{k: x_k \\neq y_k\\}}$)... | # Hausdorff Dimension of Non-Recurrent Walks on $\mathbb{Z}$
## Setup and Definitions
We work in the Cantor space $\{0,1\}^{\mathbb{N}}$ with the standard metric $d(x,y) = 2^{-\min\{k : x_k \neq y_k\}}$, which has Hausdorff dimension 1.
A walk $x$ is **non-recurrent** when the partial sum condition
$$\frac{S_n(x)}{n... | [
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"bran... | Show A contains all sequences with limiting frequency α≠1/2 (dimension H(α)/ln 2), whose supremum is 1, and A is bounded above by dimension 1. | 0.3838 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-62343_0 | Determine the value of \( \alpha \in \mathbb{R} \) such that the wave equation
\[
\begin{align*}
& u_{tt}-c^2u_{xx}=0 & \text{in } Q \\
& u(t,0)=0 & t \ge 0 \\
& u_t(t,ct)=\cos t - \alpha & t \ge 0
\end{align*}
\]
has a solution in \( C^2(Q) \cap C^1(\overline{Q}) \), where \( c > 0 \) and \( Q = \{(t,x) \in \mathbb... | 1 | \alpha = 1 | true | hard | mid | 9,789 | [
"Let me work through this wave equation problem step by step.\n\nWe have the wave equation:\n- $u_{tt} - c^2 u_{xx} = 0$ in $Q$\n- $u(t, 0) = 0$ for $t \\geq 0$\n- $u_t(t, ct) = \\cos t - \\alpha$ for $t \\geq 0$",
"where $Q = \\{(t,x) \\in \\mathbb{R}^+ \\times \\mathbb{R}^+ : 0 < x < ct\\}$.",
"The general so... | Looking at this problem, I need to find the value of $\alpha$ ensuring the solution has the required regularity, and construct the solution explicitly.
## Setting Up with the General Solution
Using $u(t,x) = u_1(x-ct) + u_2(x+ct)$, I compute the necessary derivatives:
$$u_t(t,x) = -c\,u_1'(x-ct) + c\,u_2'(x+ct)$$
#... | [
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"bran... | Use d'Alembert form, apply both BCs to constrain u₁,u₂, enforce C¹ continuity at the vertex to fix α=1, then integrate to build the explicit solution. | 0.0684 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55041_0 | Consider the set of bijections \( \phi: \mathbb{N} \cup \{0\} \rightarrow \mathbb{Q} \). For any such bijection, define the step size as \( \sup_{n \geq 0} |\phi(n+1) - \phi(n)| \). Determine the smallest possible step size, i.e., find \[ \inf_{\phi \in \Omega} \sup_{n \geq 0} |\phi(n+1) - \phi(n)|. \] | 0 | 0 | true | hard | mid | 7,571 | [
"Let me think about this carefully. We need to find the infimum of the step size over all bijections $\\phi: \\mathbb{N} \\cup \\{0\\} \\to \\mathbb{Q}$.",
"The step size is $\\sup_{n \\geq 0} |\\phi(n+1) - \\phi(n)|$.",
"First, let me think about whether this infimum can be 0. If the step size were 0, then $\\... | Looking at this problem, I need to find $\inf_{\phi \in \Omega} \sup_{n \geq 0} |\phi(n+1) - \phi(n)|$ where $\Omega$ is the set of all bijections $\phi: \mathbb{N} \cup \{0\} \to \mathbb{Q}$.
## Step 1: Lower Bound
For any bijection $\phi$, since $\phi$ is injective, $\phi(n+1) \neq \phi(n)$ for all $n$, so $|\phi(n... | [
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"bran... | For any δ>0, show the δ-proximity graph on Q is connected and removing finitely many vertices preserves connectedness, enabling a constructive enumeration with all steps ≤ δ. | 0.6183 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-54282_2 | Determine the exact value of the covering radius $R_v$ for a face-centered cubic (fcc) lattice, given that the maximum non-intersecting radius of each sphere is $r_c = \frac{1}{\sqrt{2}}$. Assume that sphere-sphere overlaps are allowed, and no point in the lattice is outside of a sphere. Express your answer in terms of... | 1 | 1 | true | hard | mid | 5,780 | [
"We need to determine the covering radius for an FCC lattice. The covering radius is the radius such that spheres of that radius centered at lattice points cover the entire space. The given is that the maximum non-intersecting radius is r_c = 1/√2. This means that in the FCC lattice, the nearest neighbor distance i... | # Covering Radius of the FCC Lattice
## Setting Up the Lattice Parameters
The FCC lattice has a cubic unit cell of side length $a$ with lattice points at:
$$(0,0,0),\quad \left(\tfrac{a}{2},\tfrac{a}{2},0\right),\quad \left(\tfrac{a}{2},0,\tfrac{a}{2}\right),\quad \left(0,\tfrac{a}{2},\tfrac{a}{2}\right)$$
**Nearest... | [
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"bran... | Compute Voronoi cell (rhombic dodecahedron) of FCC via bisector planes, identify all 14 vertices, find maximum distance from origin, express as √2·r_c. | 0.2147 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-52897_5 | Determine the smallest integer \( n \) such that for any two sets \( \{ x_1, x_2, \ldots, x_n \} \) and \( \{ y_1, y_2, \ldots, y_n \} \) of \( n \) distinct points on the unit circle \( \mathbb{S}^1 \), there exists a unique automorphism \( f: \mathbb{D} \rightarrow \mathbb{D} \) satisfying \( f(x_i) = y_i \) for all ... | 3 | 3 | true | hard | mid | 6,787 | [
"Let me think about automorphisms of the unit disk $\\mathbb{D}$.",
"The automorphisms of the unit disk $\\mathbb{D}$ are of the form:\n$$f(z) = e^{i\\theta} \\frac{z - a}{1 - \\bar{a}z}$$",
"where $a \\in \\mathbb{D}$ and $\\theta \\in \\mathbb{R}$.",
"These are Möbius transformations that map the unit disk ... | Looking at this problem, I need to determine the smallest $n$ such that $n$ point correspondences on $\mathbb{S}^1$ uniquely determine a disk automorphism.
## Setting Up the Problem
The automorphisms of $\mathbb{D}$ are:
$$f(z) = e^{i\theta}\frac{z - a}{1 - \bar{a}z}, \quad a \in \mathbb{D},\; \theta \in \mathbb{R}$$... | [
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"bran... | Uses the 3-parameter structure of disk automorphisms on the circle, a counting argument (3 constraints for 3 parameters), and the fact that a Möbius transformation fixing 3 points is the identity. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101614_0 | Determine the maximum value of \( k \) such that any good number can be transformed into any other good number by appending a special number to either end, inserting it between any two digits, or erasing a special number from its representation. | 1 | 1 | true | hard | mid | 4,641 | [
"The user is asking about \"good numbers\" and \"special numbers\" but hasn't actually defined what these terms mean. This seems like it might be from a specific math competition problem that I need to recall or figure out the context for.",
"Let me think about this carefully. The problem mentions:\n- \"good numb... | Looking at this problem, I need to identify what "good numbers" and "special numbers" are in context. Based on the structure of the problem, the most natural interpretation is:
- **Good numbers** = all positive integers (all numbers in their standard digit representation)
- **Special numbers** = $k$-digit numbers (con... | [
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"bran... | Identify correct interpretation (good=all integers, special=k-digit), prove digit-count mod k is invariant under all operations, conclude k=1 is the only value allowing full connectivity. | 0.4841 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-61157_0 | Evaluate the integral \( \int_0^1 \frac{\sin(x)}{x} \, dx \) with an accuracy better than \(10^{-4}\). Use the Taylor polynomial of order \(2n\) for \(\sin(x)\) and evaluate the remainder \(R_n(x)\) to achieve the desired accuracy. | 0.9461 | 0.9461 | true | hard | mid | 3,893 | [
"We need to evaluate ∫₀¹ sin(x)/x dx with accuracy better than 10^{-4}. Use Taylor polynomial of order 2n for sin(x) and evaluate remainder R_n(x).",
"The Taylor series for sin(x) is:\nsin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ... + (-1)^n x^{2n+1}/(2n+1)! + R_{2n}(x)",
"So sin(x)/x = 1 - x²/3! + x⁴/5! - x⁶/7! + ...... | # Evaluating $\int_0^1 \frac{\sin x}{x}\,dx$ via Taylor Series
## Setting Up the Taylor Expansion
The Taylor series for $\sin x$ is:
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + (-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!} + R_{2n}(x)$$
Dividing by $x$:
$$\frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \... | [
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"bran... | Taylor-expand sin(x)/x, integrate term-by-term, bound Lagrange remainder to choose n=3, compute 1703/1800. | 0.0954 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-58454_6 | Determine which of the following expressions for the posterior distribution in Gaussian Process Regression is correct:
1. \( p(\textbf{f} | \textbf{y}) \sim N\left(\sigma^{-2}\left( K^{-1} + \sigma^{-2}I\right)^{-1}\textbf{y}, (K^{-1} + \sigma^{-2}I)^{-1}\right) \)
2. \( p(\textbf{f} | \textbf{y}) \sim N(K(\sigma^2I ... | 2 | 2 | true | hard | mid | 7,852 | [
"We need to determine which expression for the posterior distribution in Gaussian Process Regression is correct. Let me work through the derivation.",
"In Gaussian Process Regression, we have:\n- Prior: $\\mathbf{f} \\sim N(0, K)$ where $K$ is the covariance kernel matrix\n- Likelihood: $y_i = f_i + \\epsilon_i$ ... | # Gaussian Process Regression: Posterior Distribution
## Setting Up the Problem
In GP Regression:
- **Prior:** $\mathbf{f} \sim N(0, K)$
- **Likelihood:** $\mathbf{y}|\mathbf{f} \sim N(\mathbf{f}, \sigma^2 I)$
## Derivation via Conjugate Gaussian Update
The log-posterior is proportional to:
$$\log p(\mathbf{f}|\ma... | [
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"bran... | Derive posterior via joint Gaussian conditioning, verify both options are algebraically equivalent, select option 2 as the standard GP regression form per textbook convention. | 0.4636 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101338_0 | A broken clock moves the minute hand each minute randomly one minute ahead or one minute behind with equal probabilities. Each minute it does so independently of all other times it moved the minute hand. The hour hand is synchronized with the minute hand. The clock starts running at noon. Let T denote the first time th... | 3600 | 3600 | true | hard | mid | 5,452 | [
"Let me carefully parse this problem.\n\nA broken clock has a minute hand that moves randomly one minute ahead or one minute behind each minute with equal probability. This is a random walk.",
"The hour hand is synchronized with the minute hand. This means the hour hand moves proportionally to the minute hand. So... | Looking at this problem, I need to model the clock's behavior as a random walk.
## Setting Up the Random Walk
The clock shows a time that I'll represent as minutes past noon, modulo 720 (12 hours × 60 minutes). Each minute, the displayed time moves ±1 with equal probability.
- **Start:** 0 (noon = 12:00)
- **1 o'clo... | [
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"bran... | Reduces circle random walk to 1D path via absorbing barriers, applies the i(n+1-i) expected hitting time formula. | 0.1523 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-60281_6 | A hexagon is inscribed in a circle of radius \( r \). Four of the edges of the hexagon are 10 feet long, and two of the edges are 20 feet long, but the exact arrangement of the edges is unknown. Using either Ptolemy's Theorem or the Law of Cosines, determine the value of \( r \) to three decimal places. | 13.660 | 13.660 | true | hard | mid | 3,446 | [
"Let me work through this problem step by step.\n\nWe have a hexagon inscribed in a circle of radius $r$. The hexagon has 6 edges: four of length 10 and two of length 20.",
"For a chord of length $L$ in a circle of radius $r$, the central angle $\\theta$ subtended by the chord satisfies:\n$$L = 2r \\sin(\\theta/2... | Looking at this problem, I need to find the radius $r$ of a circle containing an inscribed hexagon with four sides of length 10 and two sides of length 20.
## Setting Up the Central Angle Equation
For a chord of length $L$ in a circle of radius $r$, the central angle $\theta$ satisfies:
$$L = 2r\sin\!\left(\frac{\the... | [
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"bran... | Chord-to-central-angle conversion, angle-sum constraint, trigonometric identity to form a quadratic in 1/r, solved by the quadratic formula. | 0.029 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-106020_5 | Determine the smallest integer \( n > 3 \) for which a regular \( n \)-gon can be divided by its diagonals into equal-area triangles. | 4 | 4 | true | hard | mid | 3,184 | [
"Let me think about this problem carefully. We need to find the smallest integer $n > 3$ such that a regular $n$-gon can be divided by its diagonals into equal-area triangles.",
"First, let me understand what's being asked. We need to triangulate a regular $n$-gon using its diagonals, and all the triangles in the... | # Finding the Smallest Regular $n$-gon Triangulable into Equal-Area Triangles
## Setting Up the Problem
I need to find the smallest integer $n > 3$ such that a regular $n$-gon can be partitioned into triangles using diagonals, with all resulting triangles having the same area.
A triangulation of a convex $n$-gon use... | [
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"bran... | Simply observes n=4 (square) works: one diagonal divides it into two congruent equal-area triangles. | 0.584 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-56254_1 | Evaluate the integral \( \int_0^\infty\int_{-y}^y f(x,y)\,dx\,dy \) using the change of variables \( u = x^2 - y^2 \) and \( v = x^2 + y^2 \). Assume \( f(x,y) \) is an odd function of \( y \). Express the integral in terms of \( u \) and \( v \) and find its value. | 0 | 0 | true | hard | mid | 9,255 | [
"Let me carefully work through this problem step by step.\n\nWe need to evaluate: $\\int_0^\\infty \\int_{-y}^{y} f(x,y) \\, dx \\, dy$",
"where $f(x,y)$ is an odd function of $y$, meaning $f(x,-y) = -f(x,y)$.",
"Wait, let me re-read. The integral is $\\int_0^\\infty \\int_{-y}^{y} f(x,y) \\, dx \\, dy$.",
"H... | # Evaluating the Integral via Change of Variables
## Setting Up the Region and Change of Variables
The region of integration is:
$$R = \{(x,y) : 0 \leq y,\; -y \leq x \leq y\} = \{(x,y) : y \geq 0,\; |x| \leq y\}$$
Under the transformation:
$$u = x^2 - y^2, \qquad v = x^2 + y^2$$
we solve back:
$$x^2 = \frac{v+u}{2... | [
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"bran... | Change of variables to (u,v), identify 2-to-1 mapping, use full-region y-symmetry with oddness of f to argue cancellation yields 0. | 0.4918 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101338_2 | A broken clock moves the minute hand each minute randomly one minute ahead or one minute behind with equal probabilities. Each minute it does so independently of all other times it moved the minute hand. The hour hand is synchronized with the minute hand. The clock starts running at noon. Let T denote the first time th... | 3600 | 3600 | true | hard | mid | 7,667 | [
"Let me carefully analyze this problem.\n\nWe have a broken clock that starts at noon. Each minute, the minute hand moves either 1 minute ahead or 1 minute behind with equal probability (1/2 each). The hour hand is synchronized with the minute hand.",
"Let me think about what \"synchronized\" means. If the minute... | Looking at this problem, I need to model the minute hand as a random walk and determine when the clock displays 1:00 or 11:00.
## Setting Up the Random Walk
Let $X_n$ be the minute hand position (in minute marks from 12) after $n$ minutes, with $X_0 = 0$ (noon). Each step: $X_{n+1} = X_n \pm 1$ with probability $\fra... | [
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"bran... | Model minute hand as simple symmetric random walk; reduce clock-display condition to first hitting {60, -60}; solve difference equation yielding E = a². | 0.4614 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-53941_3 | Calculate the intersection multiplicity of the curves defined by $f(y, x) = y^2 - x^3$ and $g(y, x) = y^3 - x^7$ at the point $(0,0)$. Provide your answer as a single integer. | 9 | 9 | true | hard | mid | 4,058 | [
"We need intersection multiplicity of curves f = y^2 - x^3 and g = y^3 - x^7 at (0,0).",
"Intersection multiplicity I(f,g; P) at point P = (0,0) is defined as dim_C k[x,y]_{(x,y)} / (f,g), where k = C (or algebraically closed field of characteristic 0).",
"Alternatively, we can use the formula: if f and g have ... | # Intersection Multiplicity of $y^2 - x^3$ and $y^3 - x^7$ at $(0,0)$
## Setting Up the Problem
The intersection multiplicity $I(f,g;(0,0))$ equals the dimension of the local quotient ring:
$$I(f,g;(0,0)) = \dim_{\mathbb{C}} \; \mathbb{C}\{x,y\} \,/\, (y^2 - x^3,\; y^3 - x^7)$$
## Deriving Relations in the Quotient ... | [
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"bran... | Parametrize the cusp y²=x³ as x=t², y=t³, then compute the order of vanishing of g(t²,t³)=t⁹−t¹⁴ at t=0. | 0.4267 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-106722_0 | Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Determine the maximal length of a non-constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation. | p-1 | p-1 | true | hard | mid | 8,053 | [
"Let me think carefully about this problem. We have a prime $p \\geq 5$, and $k$ is a digit in the $p$-adic (base-$p$) representation of positive integers. So $k \\in \\{0, 1, 2, \\ldots, p-1\\}$.",
"We need to find the maximal length of a non-constant arithmetic progression whose terms do NOT contain the digit $... | Looking at this problem, I need to find the maximal length of a non-constant arithmetic progression (AP) where no term contains the digit $k$ in its base-$p$ representation.
## Establishing the Upper Bound
Consider an AP $a, a+d, a+2d, \ldots$ with $d \neq 0$. Write $d$ in base $p$ and let $m$ be the position of the ... | [
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"bran... | Upper bound via digit cycling at lowest non-zero position of d; construction uses d=p^m with carefully chosen digits to avoid k in all positions. | 0.073 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-62646_7 | Given the polynomial \( P(x) = x^{n+1} - 2x^n + 1 \) for all \( n \geq 2 \), and knowing that \( P(x) \) has a unique root \( a_n \in (1,2] \), find the limit \( L = \lim_{n\to\infty} 2^n(2-a_n) \). | 1 | 1 | true | hard | mid | 5,139 | [
"Let me work through this problem carefully.\n\nWe have $P(x) = x^{n+1} - 2x^n + 1$ for $n \\geq 2$.",
"First, let me verify that $P(x)$ has a unique root in $(1, 2]$.",
"$P(1) = 1 - 2 + 1 = 0$. So $x = 1$ is a root.\n$P(2) = 2^{n+1} - 2 \\cdot 2^n + 1 = 2^{n+1} - 2^{n+1} + 1 = 1 > 0$.",
"Let me factor $P(x)$... | # Finding the Limit $L = \lim_{n\to\infty} 2^n(2-a_n)$
## Setting Up the Key Equation
First, I factor $P(x)$. Since $P(1) = 1 - 2 + 1 = 0$, I extract $(x-1)$:
$$P(x) = x^{n+1} - 2x^n + 1 = (x-1)\underbrace{\left[x^n - x^{n-1} - x^{n-2} - \cdots - x - 1\right]}_{Q(x)}$$
The root $a_n \in (1,2)$ satisfies $Q(a_n) = 0... | [
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"bran... | Factor P(x), derive a_n^n(2-a_n)=1, substitute L_n=2^n(2-a_n), simplify to (1-L_n/2^{n+1})^n=1/L_n, take n→∞ limit. | 0.1489 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-103082_1 | Determine the maximum number of congruent pieces into which a sphere can be divided such that each side of each piece is an arc of a great circle and each arc is less than a quarter of such a circle. | 20 | 20 | true | hard | mid | 6,432 | [
"Let me think about this problem carefully. We need to divide a sphere into congruent pieces where:\n1. Each side of each piece is an arc of a great circle\n2. Each arc is less than a quarter of a great circle (i.e., less than 90°)",
"Let me think about what shapes can appear on a sphere bounded by great circle a... | # Maximum Congruent Pieces of a Sphere Bounded by Great Circle Arcs
## Setting Up the Problem
We need to tile a sphere with congruent pieces where:
- Each edge is an arc of a great circle
- Each arc is **strictly less than** 90° (a quarter of a great circle)
## Using Euler's Formula for Spherical Tilings
For a tili... | [
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"bran... | Euler's formula on spherical tilings constrains (sides, vertex-degree) pairs; icosahedron (s=3,k=5) maximizes n=20, verified by computing edge angle ≈63.4°<90°. | 0.101 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55733_2 | Determine the number of Banach spaces that can be defined on the set of real numbers, $\mathbb{R}$. Provide your answer as a single number. | 1 | 1 | true | hard | mid | 3,393 | [
"The question asks for the number of Banach spaces that can be defined on the set of real numbers, $\\mathbb{R}$.",
"A Banach space is a complete normed vector space. So I need to determine how many distinct norms can be placed on $\\mathbb{R}$ as a vector space such that the resulting normed space is complete (i... | # Finding the Number of Banach Spaces on $\mathbb{R}$
## Setting Up the Problem
A Banach space is a **complete** normed vector space. I need to determine how many Banach space structures can be placed on the set $\mathbb{R}$ with its standard vector space structure (1-dimensional over $\mathbb{R}$).
## Classifying A... | [
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"bran... | Show all norms on 1D real vector space are c|x|, all complete, all isometrically isomorphic, yielding exactly one Banach space up to isomorphism. | 0.1051 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-104196_0 | Consider a grid of points \((x, y)\) where both \(x\) and \(y\) are positive integers less than or equal to 6. A *magic grid* is an assignment of an integer to each point in this grid such that for every square with horizontal and vertical sides and all four vertices in the grid, the sum of the integers at these vertic... | 2^{36} | 68719476736 | true | hard | mid | 4,677 | [
"Let me work through this problem carefully.\n\nWe have a 6×6 grid of points $(x,y)$ where $x, y \\in \\{1, 2, 3, 4, 5, 6\\}$.",
"A magic grid assigns an integer $a_{x,y}$ to each point such that for every axis-aligned square with all four vertices in the grid, the sum of the four corner values is constant.",
"... | # Finding the Smallest Magic Grid Product
## Setting Up the Constraints
Let $f(x,y)$ denote the integer at grid point $(x,y)$ where $x,y \in \{1,2,3,4,5,6\}$. A square of side length $s$ with bottom-left vertex at $(x,y)$ has vertices $(x,y), (x,y+s), (x+s,y), (x+s,y+s)$, and requires:
$$f(x,y) + f(x,y+s) + f(x+s,y)... | [
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"bran... | Show the magic grid must be constant using difference analysis on 1×1 and 2×2 square constraints, then minimize k^36 > 1 over integers. | 0.0118 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-54201_4 | Find an irreducible polynomial in \(\mathbb{Z}[x]\) that is reducible modulo 2, 3, and 5. | x^4 + 1 | x^4+1 | true | hard | mid | 3,026 | [
"We need to find an irreducible polynomial in Z[x] (integer coefficients, irreducible over Q) that is reducible modulo 2, 3, and 5.",
"This is a classic problem. We need a polynomial that is irreducible over Q but factors when reduced modulo 2, 3, and 5.",
"Let me think about what properties we need:\n- The pol... | # Finding an Irreducible Polynomial Reducible Mod 2, 3, and 5
## Candidate: $f(x) = x^4 + 1$
### Step 1: Verify Irreducibility over $\mathbb{Q}$
The polynomial $x^4 + 1$ is the **8th cyclotomic polynomial** $\Phi_8(x)$. Its roots are the primitive 8th roots of unity: $e^{i\pi/4},\, e^{i3\pi/4},\, e^{i5\pi/4},\, e^{i... | [
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"bran... | Identify x⁴+1 (Φ₈) as irreducible over Q, then verify reducibility mod 2, 3, and 5 by explicit factorization in each finite field. | 0.2527 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-61826_5 | Evaluate the integral \( \int_{|z|=2} \log\frac{z+1}{z-1} \, dz \), assuming that for \( w \in \mathbb{C} - \{z; \text{Re}(z) \leq 0\} \), we have \(-\pi < \text{Im}(\log w) < \pi \). | 4\pi i | 4\pi i | true | hard | mid | 7,984 | [
"We need evaluate integral over |z|=2 of log((z+1)/(z-1)) dz, with the branch of log defined for w not on negative real axis including 0? Actually given: for w ∈ ℂ - {z; Re(z) ≤ 0}, we have -π < Im(log w) < π. That's the standard principal branch of log, cut along nonpositive real axis (including 0). So log is prin... | Looking at this problem, I need to evaluate $\int_{|z|=2} \log\frac{z+1}{z-1}\,dz$ where the logarithm uses the principal branch defined on $\mathbb{C}\setminus\{z:\text{Re}(z)\leq 0\}$ (the right half-plane).
## Step 1: Determine where the integrand is analytic
Let $z = x+iy$. Computing the real part of $(z+1)/(z-1)... | [
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"bran... | Laurent series expansion of Log((1+1/z)/(1-1/z)) for |z|>1, then term-by-term integration over |z|=2. | 0.2671 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-56459_3 | Evaluate the integral \( \int_{0}^{+\infty}\frac{x-\sin{x}}{x^3}dx \) using complex analysis techniques, such as the residue theorem. | \dfrac{\pi}{4} | \dfrac{\pi}{4} | true | hard | mid | 8,120 | [
"We need to evaluate integral I = ∫_0^∞ (x - sin x)/x^3 dx.",
"This is a classic integral. Let's think about how to compute it, but the problem specifically asks to use complex analysis techniques, such as the residue theorem.",
"First, let me think about the integral directly to know what the answer should be.... | # Evaluating $\int_0^{+\infty}\frac{x-\sin x}{x^3}\,dx$ via Complex Analysis
## Step 1: Choose a Complex Function and Contour
I consider the function
$$f(z) = \frac{1 - e^{iz}}{z^2}$$
which is analytic in the upper half-plane except at $z = 0$. I integrate over the **indented semicircular contour** $C$ in the upper... | [
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"bran... | Contour integral of (1-e^{iz})/z² over upper half-plane semicircle with indentation at origin, yielding ∫(1-cos x)/x² = π/2, then integration by parts to relate to original integral. | 0.428 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-105635_1 | Positive integers \(a_1, a_2, \cdots, a_{2006}\) are such that the ratios \(\frac{a_1}{a_2}, \frac{a_2}{a_3}, \cdots, \frac{a_{2005}}{a_{2006}}\) are pairwise distinct. Determine the minimum number of distinct integers among \(a_1, a_2, \cdots, a_{2006}\). | 46 | 46 | true | hard | mid | 5,175 | [
"Let me work through this problem carefully.\n\nWe have positive integers $a_1, a_2, \\ldots, a_{2006}$ such that the ratios $\\frac{a_1}{a_2}, \\frac{a_2}{a_3}, \\ldots, \\frac{a_{2005}}{a_{2006}}$ are pairwise distinct.",
"We need to find the minimum number of distinct integers among $a_1, a_2, \\ldots, a_{2006... | Looking at this problem, I need to find the minimum number of distinct integers among $a_1, a_2, \ldots, a_{2006}$ such that the 2005 consecutive ratios are pairwise distinct.
## Establishing a Lower Bound
**Key observation:** If the sequence uses $d$ distinct values $v_1, v_2, \ldots, v_d$, then each ratio $\frac{a_... | [
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"... | Bounded distinct ratios by d(d−1)+1, derived d≥46, then constructed a sequence using 46 primes and an Eulerian circuit in the complete directed graph. | 0.1598 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-107394_6 | Determine all even natural numbers \( n \) such that the numbers \( 1, 2, \ldots, n^2 \) can be partitioned into two sets \( A \) and \( B \) of equal size, with the sums of the elements in \( A \) and \( B \) satisfying \( \frac{S_A}{S_B} = \frac{39}{64} \). | 206 | 206 | true | hard | mid | 7,397 | [
"Let me work through this problem step by step.\n\nWe need to partition $\\{1, 2, \\ldots, n^2\\}$ into two sets $A$ and $B$ of equal size, where $n$ is an even natural number, such that $\\frac{S_A}{S_B} = \\frac{39}{64}$.",
"First, let me find the total sum:\n$$S = 1 + 2 + \\cdots + n^2 = \\frac{n^2(n^2+1)}{2}$... | Looking at this problem, I need to find all even natural numbers $n$ where $\{1, 2, \ldots, n^2\}$ splits into two sets $A$, $B$ of equal size with $\frac{S_A}{S_B} = \frac{39}{64}$.
## Setting Up the Sum Constraint
The total sum is $S = \frac{n^2(n^2+1)}{2}$. From the ratio:
$$S_A = \frac{39}{103}S = \frac{39\,n^2(n... | [
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"bran... | Divisibility by 103 forces n=206k; min/max sum bounds and contiguous interval theorem confirm partition exists for all such n. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-53295_4 | Find the quadratic variation of the process $X(s) = W_{s+\epsilon} - W_s$, where $W_s$ is a standard Wiener process. | 2t | 2t | true | hard | mid | 6,143 | [
"We need to find the quadratic variation of the process $X(s) = W_{s+\\epsilon} - W_s$, where $W_s$ is a standard Wiener process. $\\epsilon > 0$ is presumably a fixed positive constant.",
"Let me think carefully about what's being asked. We have a process $X(s)$ defined for $s \\geq 0$ as $X(s) = W_{s+\\epsilon}... | # Quadratic Variation of $X(s) = W_{s+\epsilon} - W_s$
## Setting Up the Problem
I need to compute $[X]_t = \lim_{n\to\infty} \sum_{i=1}^n (\Delta X_i)^2$ where $\Delta X_i = X(t_i) - X(t_{i-1})$ for a partition $\Pi_n: 0 = t_0 < t_1 < \cdots < t_n = t$ with mesh $|\Pi_n| \to 0$.
**Key observation:** Define $Y(s) = ... | [
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"bran... | Decompose X as difference of two Brownian motions, apply semimartingale QV formula, show cross-variation vanishes via disjoint intervals and vanishing variance. | 0.0619 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-61930_2 | Find the sum of the series: $$\sum_{n = -\infty}^{\infty} \cos \left(8n + \frac{2\pi}{3}\right)$$ | 0 | 0 | true | hard | mid | 3,604 | [
"We need to compute the sum S = sum_{n=-∞}^{∞} cos(8n + 2π/3).",
"This is a sum over all integers of cos(8n + 2π/3). The argument is in radians. 8 is in radians (since we're adding 2π/3, which is in radians). So 8n means 8 times n, where 8 is in radians.",
"Let me think about this. The general term is cos(8n + ... | Looking at this problem, I need to evaluate $\sum_{n=-\infty}^{\infty} \cos(8n + \tfrac{2\pi}{3})$ where the arguments are in **radians** (since $\tfrac{2\pi}{3}$ is in radians).
## Setting Up the Sum
I'll work with the complex exponential form:
$$\cos(8n + \tfrac{2\pi}{3}) = \operatorname{Re}\!\Big(e^{i\pi/3 \cdot ... | [
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"bran... | Decompose into complex exponential, show ∑e^{i8n}=0 via distributional identity and Abel summation since 8 is not a multiple of 2π. | 0.0562 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-100621_2 | Evaluate the limit: \[ \lim_{x\to\infty}\left(\lim_{n\to\infty}\frac{\lfloor 1^3(\cos x)^x\rfloor+\lfloor 2^3(\cos x)^x\rfloor+\dots+\lfloor n^3(\cos x)^x\rfloor}{n^4}\right) \] | 0 | 0 | true | hard | mid | 7,788 | [
"Let me carefully analyze this limit problem.\n\nWe need to evaluate:\n$$\\lim_{x\\to\\infty}\\left(\\lim_{n\\to\\infty}\\frac{\\lfloor 1^3(\\cos x)^x\\rfloor+\\lfloor 2^3(\\cos x)^x\\rfloor+\\dots+\\lfloor n^3(\\cos x)^x\\rfloor}{n^4}\\right)$$",
"Let me denote $a_x = (\\cos x)^x$ for a fixed $x$.",
"First, le... | # Evaluating the Double Limit
## Step 1: Evaluate the Inner Limit (for fixed $x$)
Let $a = (\cos x)^x$ be fixed. For each positive integer $k$, the floor function satisfies:
$$k^3 a - 1 < \lfloor k^3 a \rfloor \leq k^3 a$$
**Summing** from $k=1$ to $n$:
$$a\sum_{k=1}^n k^3 - n < \sum_{k=1}^n \lfloor k^3 a \rfloor ... | [
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"bran... | Floor function bounds squeeze inner limit to (cos x)^x/4; outer limit taken as 0 via |cos x|^x→0 (flawed). | 0.707 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-104291_3 | Determine the positive integer $m$ such that the sequence $\{a_n\}$, defined by $a_0 = m$ and $a_{n+1} = a_n^5 + 487$ for $n \in \mathbb{N}$, contains the maximum number of perfect squares. | 9 | 9 | true | hard | mid | 4,539 | [
"Let me analyze the sequence $a_0 = m$, $a_{n+1} = a_n^5 + 487$ and find the positive integer $m$ that maximizes the number of perfect squares in the sequence.",
"First, let me think about when $a_n$ can be a perfect square.",
"If $a_n$ is a perfect square, let's say $a_n = k^2$. Then $a_{n+1} = k^{10} + 487$."... | # Finding $m$ to Maximize Perfect Squares in the Sequence
## Setting Up the Key Constraint
If two **consecutive** terms are both perfect squares, say $a_n = k^2$ and $a_{n+1} = \ell^2$, then:
$$\ell^2 = (k^2)^5 + 487 = k^{10} + 487$$
Rearranging as a difference of squares:
$$\ell^2 - (k^5)^2 = 487 \implies (\ell - ... | [
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"bran... | Factor (ℓ−k⁵)(ℓ+k⁵)=487, use primality of 487 to force k=3, showing only consecutive square pair is (9, 244²), then argue m=9 maximizes the count. | 0.0168 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-106375_5 | Find the minimum value of the function \( f(x,y,z) = \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z} \) given that \( a, b, c, x, y, z \) are positive real numbers satisfying the equations \( bz+cy=a \), \( az+cx=b \), and \( ay+bx=c \). | \dfrac{1}{2} | \dfrac{1}{2} | true | hard | mid | 9,059 | [
"Let me work through this problem carefully.\n\nWe have positive real numbers $a, b, c, x, y, z$ satisfying:\n1. $bz + cy = a$\n2. $az + cx = b$\n3. $ay + bx = c$",
"We need to find the minimum of $f(x,y,z) = \\frac{x^2}{1+x} + \\frac{y^2}{1+y} + \\frac{z^2}{1+z}$.",
"Let me first try to find relationships betw... | Looking at this problem, I need to find the minimum of $f(x,y,z) = \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z}$ subject to the constraint system.
## Finding Relationships Between Variables
**Multiplying** each constraint by the corresponding variable:
- $a(bz + cy) = a^2 \implies abz + acy = a^2$
- $b(az + cx... | [
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"bran... | Reformulate x,y,z as cosines of acute triangle angles, then prove minimum via second-order perturbation around the symmetric point and boundary analysis. | 0.3318 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-62632_7 | Determine the asymptotic value of \(\underset{c \in \text{Coloring}}{\min} \text{ score}(c) = \underset{c \in \text{Coloring}}{\min} \left(\underset{\begin{subarray}{c} B \subset \text{Colors}, \\ |B|\leq n \end{subarray}}{\max} V_c(B)\right)\). Is it \(\Theta(n^2)\) or \(o(n^2)\)? | o(n^2) | o(n^2) | true | hard | mid | 6,146 | [
"Let me carefully analyze this problem. We need to determine the asymptotic value of the minimax expression.",
"Let me re-read the problem. We have a set of \"Colors\" and we're looking at colorings. The expression is:",
"$$\\min_{c \\in \\text{Coloring}} \\left(\\max_{\\substack{B \\subset \\text{Colors}, \\\\... | # Determining the Asymptotic Minimax Value
## Setting Up the Problem
We need to evaluate:
$$\min_{c \in \text{Coloring}} \left(\max_{\substack{B \subset \text{Colors},\; |B| \leq n}} V_c(B)\right)$$
The total "mass" (total value across all colors) in these coloring problems is $\Theta(n^2)$ (e.g., from $\binom{n}{2}... | [
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"bra... | Best-guess via distribution argument: spreading Θ(n²) mass over superlinearly many colors keeps any n-color subset at o(n²). | 0.6543 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-104310_3 | Let \( n \) be a positive integer. Determine whether \( n^2 \) has more positive divisors of the form \( 4k+1 \) or of the form \( 4k-1 \). | 4k+1 | 4k+1 | true | hard | mid | 5,084 | [
"We need solve problem. We need reason carefully. Problem: Let n positive integer. Determine whether n^2 has more positive divisors of form 4k+1 or of form 4k-1. Need likely answer: n^2 always has more divisors congruent 1 mod 4 than -1 mod 4? Let's test.",
"Divisors of n^2 positive. Count A = # d|n^2, d ≡1 mod4.... | Let
\[
A=\#\{d\mid n^2: d\equiv 1\pmod 4\},\qquad
B=\#\{d\mid n^2: d\equiv 3\pmod 4\}.
\]
These are the divisors of the forms \(4k+1\) and \(4k-1\), respectively.
Write the prime factorization of the odd part of \(n\) as
\[
n=2^\alpha \prod_i p_i^{a_i}\prod_j q_j^{b_j},
\]
where \(p_i\equiv 1\pmod 4\) and \(q_j\e... | [
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"bran... | Prime factorization with mod-4 parity analysis; generating function to count even vs odd exponent sums, showing the difference is always positive. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-105060_7 | In the tetrahedron \(EFGH\), the side lengths are given as \(EF = GH = 7\), \(EG = FH = 10\), and \(EH = FG = 11\). The points \(K, L, M, N\) are the incenters of the triangles \(EFG\), \(EFH\), \(EGH\), and \(FGH\) respectively. Calculate the volume of the tetrahedron \(KLMN\). Round your answer to the nearest hundred... | 2.09 | 2.09 | true | hard | mid | 7,780 | [
"Let me work through this problem step by step.\n\nWe have a tetrahedron $EFGH$ with:\n- $EF = GH = 7$\n- $EG = FH = 10$\n- $EH = FG = 11$",
"This is an isosceles tetrahedron (equifacial tetrahedron) where opposite edges are equal.",
"The points $K, L, M, N$ are the incenters of triangles $EFG$, $EFH$, $EGH$, a... | # Volume of Tetrahedron KLMN
## Setting Up Coordinates
Since the tetrahedron is **isosceles** (opposite edges equal), I place the vertices symmetrically:
$$E=(a,b,c),\quad F=(a,-b,-c),\quad G=(-a,b,-c),\quad H=(-a,-b,c)$$
This gives the edge constraints:
- $EF = 2\sqrt{b^2+c^2}=7 \Rightarrow b^2+c^2=\tfrac{49}{4}$
... | [
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"bran... | Place isosceles tetrahedron in symmetric coordinates, compute face incenters as weighted averages, then use scalar triple product to find the incenter tetrahedron's volume. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-52595_5 | Find the value of \(a \in \mathbb{R}\) such that the solution set of the linear system \[ \begin{cases} -x+2y+az=3 \\ y-z=a \\ x+y-z=-9 \end{cases} \] represents a line perpendicular to the plane \(K\) given by \[ \begin{cases} x=t+1 \\ y=-t+2s-1 \\ z=s-1 \end{cases} \]. | -2 | a = -2 | true | hard | mid | 5,570 | [
"Let me find the solution set of the linear system and the plane K, then determine when the solution set (which should be a line) is perpendicular to plane K.",
"First, let me understand the system:\n$$\\begin{cases} -x+2y+az=3 \\\\ y-z=a \\\\ x+y-z=-9 \\end{cases}$$",
"For the solution set to be a line, the sy... | Looking at this problem, I need to find $a$ such that the solution set of the linear system is a **line** that is **perpendicular** to plane $K$.
## Step 1: Determine when the solution set is a line
For a 3×3 system to have a line as its solution set, the coefficient matrix must have determinant zero (giving infinite... | [
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"bran... | Compute determinant to find a=-2 as unique value for a line, verify consistency, check perpendicularity, commit to a=-2. | 0.5729 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-56549_5 | Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ be positive real sequences such that \(\lim_{n\to\infty}\frac{a_{n+1}-a_n}{n}=a\in \mathbb{R}_{>0}\) and \(\lim_{n\to\infty}\frac{b_{n+1}}{nb_n}=b\in \mathbb{R}_{>0}\). Compute the limit:
\[\lim_{n\to\infty}\left(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}}\r... | \dfrac{a e}{2 b} | \dfrac{ae}{2b} | true | hard | mid | 6,210 | [
"Let me work through this problem step by step.\n\nWe have two positive real sequences $(a_n)$ and $(b_n)$ with:\n- $\\lim_{n\\to\\infty}\\frac{a_{n+1}-a_n}{n} = a > 0$\n- $\\lim_{n\\to\\infty}\\frac{b_{n+1}}{n \\cdot b_n} = b > 0$",
"We need to compute:\n$$\\lim_{n\\to\\infty}\\left(\\frac{a_{n+1}}{\\sqrt[n+1]{b... | Looking at this problem, I need to analyze the asymptotic behavior of both sequences and then compute the difference.
## Step 1: Asymptotic Behavior of $a_n$
Since $\frac{a_{n+1}-a_n}{n} \to a$, we have $a_{n+1} - a_n = na + o(n)$.
Summing: $a_n = a_1 + \sum_{k=1}^{n-1}(ka + o(k)) \sim a\cdot\frac{n(n-1)}{2} \sim \f... | [
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"bran... | Define c_n = a_n/(b_n)^{1/n}; compute c_{n+1}/c_n via a_{n+1}/a_n and the b-root ratio; show c_{n+1}−c_n = c_n/n + o(1) → ae/(2b). | 0.3227 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-62907_1 | Evaluate the limit: \[ \lim_{n\to\infty}\frac{n}{2^n}\sum_{k=1}^{n}\binom{n-1}{k-1}\{\sqrt{k^2+2k+2}\} \] where \( \{x\} \) denotes the fractional part of \( x \). | \dfrac{1}{2} | \dfrac{1}{2} | true | hard | mid | 4,975 | [
"Let me analyze this limit step by step.\n\nWe need to evaluate:\n$$\\lim_{n\\to\\infty}\\frac{n}{2^n}\\sum_{k=1}^{n}\\binom{n-1}{k-1}\\{\\sqrt{k^2+2k+2}\\}$$",
"First, let me simplify the expression inside the fractional part. Note that:\n$$k^2 + 2k + 2 = (k+1)^2 + 1$$",
"So $\\{\\sqrt{k^2+2k+2}\\} = \\{\\sqrt... | # Evaluating the Limit
## Step 1: Simplify the Fractional Part
First, I factor the expression under the square root:
$$k^2 + 2k + 2 = (k+1)^2 + 1$$
For any integer $k \geq 1$, I verify:
$$(k+1)^2 < (k+1)^2 + 1 < (k+2)^2$$
So $k+1 < \sqrt{(k+1)^2+1} < k+2$, which means:
$$\left\{\sqrt{k^2+2k+2}\right\} = \sqrt{(k+1)... | [
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"bran... | Simplify fractional part, rewrite sum as binomial expectation, use concentration of J~Bin(n-1,1/2) near n/2, approximate f_J≈1/(2J)≈1/n, verify small-j terms are negligible. | 0.0392 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-63668_7 | Compute the Laplacian of the scalar field \( \dfrac{(\mathbf{\mu} \cdot \mathbf{r})}{r^3} \), where \( r = \Vert \mathbf{r} \Vert \). | 0 | 0 | true | hard | mid | 7,589 | [
"We need to compute the Laplacian of the scalar field f(r) = (μ · r) / r^3, where μ is a constant vector and r = ||r||.",
"The Laplacian is ∇²f = ∇·(∇f).",
"Let me work this out step by step.\n\nWe have f = (μ_i r_i) / r^3 (using Einstein summation).",
"First, compute the gradient of f.\n\n∂f/∂x_j = ∂/∂x_j [(... | # Laplacian of $\frac{\boldsymbol{\mu} \cdot \mathbf{r}}{r^3}$
## Setting Up the Problem
I need to compute $\nabla^2 f$ where $f = \frac{\mu_i x_i}{r^3}$ (Einstein summation), with $\boldsymbol{\mu}$ constant and $r = \sqrt{x_j x_j}$.
## Computing the Gradient
$$\frac{\partial f}{\partial x_j} = \mu_j\, r^{-3} + \m... | [
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"bran... | Compute gradient, then Laplacian via product rule in Einstein notation; correct a summation error over index i; verify via spherical coordinates and harmonic-function identity. | 0.3652 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-63668_1 | Compute the Laplacian of the scalar field \( \dfrac{(\mathbf{\mu} \cdot \mathbf{r})}{r^3} \), where \( r = \Vert \mathbf{r} \Vert \). | 0 | 0 | true | hard | mid | 4,068 | [
"We need to compute the Laplacian of the scalar field f(r) = (μ·r)/r³, where r = ||r|| = sqrt(x²+y²+z³? no, sqrt(x²+y²+z²)). μ is a constant vector.",
"Let me denote μ = (μ_x, μ_y, μ_z) as a constant vector, and r = (x, y, z), so r = √(x²+y²+z²).",
"We need to compute ∇²f = ∇²[(μ·r)/r³].",
"Let me write f = (... | Looking at this problem, I need to compute $\nabla^2\!\left(\dfrac{\boldsymbol{\mu}\cdot\mathbf{r}}{r^3}\right)$ where $\boldsymbol{\mu}$ is a constant vector and $r = \|\mathbf{r}\|$.
## Setting Up in Index Notation
Let $\mathbf{r} = (x_1, x_2, x_3)$ and write the field as $f = \mu_i\, x_i\, r^{-3}$ (Einstein summat... | [
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"bran... | Direct computation of second partial derivatives in index notation, summing over k to obtain the Laplacian. | 0.2168 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-102925_4 | Determine the smallest number of natives, \( N \), such that after crossing a river using a single boat, each native knows at least \( k \) additional jokes apart from their own. Each crossing involves two natives rowing to the right and one returning alone, and jokes are only shared between the two natives in the boat... | k + 1 | k+1 | true | hard | mid | 4,205 | [
"Let me carefully analyze this problem.\n\nWe have N natives, a river, and a single boat. The boat can carry exactly 2 natives at a time. To cross the river, two natives row to the right, and one of them returns alone to the left. Jokes are shared between the two natives in the boat during each crossing.",
"We ne... | # Finding the Smallest Number of Natives
## Setting Up the Boat Crossing
For N natives to cross the river:
- **Forward trips** (2 natives, left→right): $f$ trips
- **Return trips** (1 native, right→left): $r$ trips
- Net requirement: $2f - r = N$, and the last trip must be forward, so $r = f - 1$
Solving: $2f - (f-1... | [
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"bran... | Model joke-sharing as a graph; derive lower bound N≥k+1 from connectivity; construct a star graph via a single ferryman to prove achievability. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-62610_2 | Determine whether the function \( h(x,y) = \frac{\arccos[(\cos x)(\cos y) + k(\sin x)(\sin y)]^2}{(\sin^2 x) - 2k(\sin x)(\sin y) + (\sin^2 y)} \) has a limit of 1 as \((x,y)\) approaches \((0,0)\), where \(-1 \leq k \leq 1\). | 1 | 1 | true | hard | mid | 4,601 | [
"We need to determine whether h(x,y) has limit 1 as (x,y)→(0,0), where -1≤k≤1.",
"h(x,y) = arccos[(cos x)(cos y) + k(sin x)(sin y)]² / [sin²x - 2k(sin x)(sin y) + sin²y]",
"Let me first understand the expression. Let me denote:\n- Numerator: [arccos((cos x)(cos y) + k(sin x)(sin y))]²\n- Denominator: sin²x - 2k... | # Determining the Limit of $h(x,y)$ as $(x,y)\to(0,0)$
## Setting Up the Analysis
As $(x,y)\to(0,0)$:
- $\cos x \to 1$, $\cos y \to 1$
- $\sin x \to 0$, $\sin y \to 0$
So the argument of $\arccos$ approaches $1$, making the numerator $\to 0$, and the denominator $\to 0$. This is a $\frac{0}{0}$ form requiring carefu... | [
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"bran... | Small-angle Taylor expansion of numerator and denominator to leading order, combined with arccos(1−u)≈√(2u), shows both equal x²+y²−2kxy; edge cases k=±1 handled separately. | 0.0389 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-52897_0 | Determine the smallest integer \( n \) such that for any two sets \( \{ x_1, x_2, \ldots, x_n \} \) and \( \{ y_1, y_2, \ldots, y_n \} \) of \( n \) distinct points on the unit circle \( \mathbb{S}^1 \), there exists a unique automorphism \( f: \mathbb{D} \rightarrow \mathbb{D} \) satisfying \( f(x_i) = y_i \) for all ... | 3 | 3 | true | hard | mid | 3,267 | [
"Let me think about this problem carefully. We need to find the smallest integer $n$ such that for any two sets of $n$ distinct points on the unit circle $\\mathbb{S}^1$, there exists a **unique** automorphism $f: \\mathbb{D} \\to \\mathbb{D}$ satisfying $f(x_i) = y_i$ for all $i$.",
"First, let me recall what au... | Looking at this problem, I need to find the smallest $n$ such that for **any** two sets of $n$ distinct points on $\mathbb{S}^1$, there exists a **unique** automorphism of the unit disk mapping one set to the other.
## Key Facts About Disk Automorphisms
Every automorphism $f: \mathbb{D} \to \mathbb{D}$ is a Möbius tr... | [
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"bran... | Establish that 3 boundary points determine a unique disk automorphism via Möbius transformation theory, then rule out n=1,2 by parameter counting. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-60388_4 | Find the minimum value of \[ P = \frac{1}{(1+x)^2} + \frac{1}{(1+y)^2} + \frac{1}{(1+z)^2} + \frac{4}{(1+x)(1+y)(1+z)} \] given that \( x, y, z > 0 \) and \( y + z = x(y^2 + z^2) \). | \dfrac{91}{108} | \dfrac{91}{108} | true | hard | mid | 8,639 | [
"Let me work through this problem step by step.\n\nWe need to minimize:\n$$P = \\frac{1}{(1+x)^2} + \\frac{1}{(1+y)^2} + \\frac{1}{(1+z)^2} + \\frac{4}{(1+x)(1+y)(1+z)}$$",
"Given: $x, y, z > 0$ and $y + z = x(y^2 + z^2)$.",
"Let me first understand the constraint. We have $y + z = x(y^2 + z^2)$, so $x = \\frac... | Looking at this problem, I need to minimize $P$ subject to the constraint $y + z = x(y^2 + z^2)$ where $x, y, z > 0$.
## Setting Up the Symmetric Case
Since both $P$ and the constraint are symmetric in $y$ and $z$, I expect the minimum occurs at $y = z$. Let me verify this.
**Setting $y = z$:** The constraint gives ... | [
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"bran... | Assume y=z by symmetry, reduce to one variable, substitute t=1/(1+y), find critical point of cubic polynomial, evaluate. | 0.6265 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-102191_6 | Find the area of the region defined by the set \( \{ (x, y) : |x + yt + t^2 | \leq 1 \text{ for all } t \in [0, 1] \} \). | \dfrac{23}{6} | \frac{23}{6} | true | hard | mid | 6,278 | [
"We need solve problem. Need provide step by step. Let's analyze thoroughly.",
"We need area of region in xy-plane: { (x,y) : |x + y t + t^2| <= 1 for all t in [0,1] }.",
"Interpret inequality for all t: -1 <= x + y t + t^2 <= 1 for all t in [0,1]. For each t, constraint: -1 - t^2 <= x + y t <= 1 - t^2. The set... | Let
\[
f(t)=x+yt+t^2,\qquad t\in[0,1].
\]
We need
\[
|f(t)|\le 1\quad\text{for all }t\in[0,1],
\]
which is equivalent to
\[
\max_{t\in[0,1]} f(t)\le 1
\quad\text{and}\quad
\min_{t\in[0,1]} f(t)\ge -1.
\]
Since \(f(t)\) is convex in \(t\), its maximum on \([0,1]\) occurs at an endpoint. Thus
\[
\max_{[0,1]} f(t)=... | [
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"bran... | Characterize region via sup/inf of convex quadratic in t, derive piecewise x-bounds by y-range, integrate widths over four y-intervals. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-60202_4 | Find the value of $A + B + C$ such that \[ S = \sum_{n=1}^{\infty} \frac{1}{n^2 \binom{2n}{n}} = \frac{A}{B} \zeta(C). \] | 6 | 6 | true | hard | mid | 5,698 | [
"We need solve problem: find A+B+C such that S = sum_{n=1}∞ 1/(n^2 binom{2n}{n}) = (A/B) ζ(C). Need reason step by step, final boxed. Need be careful about exact value. Let's analyze.",
"We need evaluate series S = ∑_{n≥1} 1/(n^2 C(2n,n)). Known series involving central binomial coefficients. Need express as rati... | We want to evaluate
\[
S=\sum_{n=1}^{\infty}\frac{1}{n^2\binom{2n}{n}}.
\]
Use the beta integral
\[
B(n,n+1)=\int_0^1 x^{n-1}(1-x)^n\,dx
=\frac{(n-1)!\,n!}{(2n)!}.
\]
But
\[
\frac{1}{n\binom{2n}{n}}
=\frac{n!n!}{n(2n)!}
=\frac{(n-1)!\,n!}{(2n)!}.
\]
Hence
\[
\frac{1}{n\binom{2n}{n}}
=\int_0^1 x^{n-1}(1-x)^n\,dx.... | [
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"bran... | Beta integral representation converts the series to a log integral, evaluated via the factorization 1−x+x²=(1+x³)/(1+x) and known dilogarithm values. | 0.0403 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-107671_2 | Solve the equation for \( \varphi(x) \):
$$
x^{2}=\frac{2}{\pi} \int_{0}^{\pi / 2} \varphi(x \sin \theta) d \theta
$$ | 2x^2 | \varphi(x)=2x^2 | true | hard | mid | 9,531 | [
"We need solve equation for φ(x): x^2 = (2/π) ∫_0^{π/2} φ(x sin θ) dθ. Need reason step by step final boxed.",
"We need find function φ likely on [0,∞). Integral transform: (Tφ)(x)= (2/π)∫_0^{π/2} φ(x sinθ)dθ. Need invert. Let's analyze.",
"Let change variable t = x sinθ. If x>0, θ from 0 to π/2, t from 0 to x,... | Assume \(\varphi\) is sufficiently regular so that the integral manipulations below are valid. First note that at \(x=0\),
\[
0=\frac{2}{\pi}\int_0^{\pi/2}\varphi(0)\,d\theta=\varphi(0),
\]
so \(\varphi(0)=0\).
Now take \(x>0\). In the integral, set
\[
t=x\sin\theta.
\]
Then \(t\) runs from \(0\) to \(x\), and
\[... | [
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"bran... | Convert to Abel integral via t=x sinθ and u=t², invert using the Abel inversion formula (apply operator again, differentiate), yielding φ(x)=2x². | 0.0297 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-105097_3 | Given an integer \( n \geq 2 \), for any pairwise coprime positive integers \( a_1, a_2, \ldots, a_n \), let \( A = a_1 + a_2 + \ldots + a_n \). Denote by \( d_i \) the greatest common divisor (gcd) of \( A \) and \( a_i \) for \( i = 1, 2, \ldots, n \). Denote by \( D_i \) the gcd of the remaining \( n-1 \) numbers af... | (n-1)^n | (n-1)^n | true | hard | mid | 7,427 | [
"Let me work through this problem carefully.\n\nWe have pairwise coprime positive integers $a_1, a_2, \\ldots, a_n$ with $n \\geq 2$.",
"Let $A = a_1 + a_2 + \\ldots + a_n$.",
"$d_i = \\gcd(A, a_i)$ for $i = 1, 2, \\ldots, n$.",
"$D_i = \\gcd(a_j : j \\neq i)$ - the gcd of all $a_j$ except $a_i$.",
"Since t... | Looking at this problem, I need to analyze the expression $\prod_{i=1}^{n} \frac{A - a_i}{d_i D_i}$ where $D_i$ is the gcd of all $a_j$ with $j \neq i$.
## Establishing $D_i = 1$
Since the $a_1, a_2, \ldots, a_n$ are **pairwise coprime**, for any distinct $j, k \neq i$, we have $\gcd(a_j, a_k) = 1$. Therefore, the gc... | [
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"bran... | Simplify via D_i=1, test small cases to find pattern, verify all-ones gives (n-1)^n, argue no perturbation reduces the product. | 0.42 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-51930_7 | Calculate the expected stopping time $E[S]$ for the first occurrence of the sequence HTH in a series of fair coin tosses using the optional stopping theorem. | 10 | 10 | true | hard | mid | 4,461 | [
"We need to calculate the expected stopping time for the first occurrence of the sequence HTH in fair coin tosses using the optional stopping theorem.",
"Let me think about this carefully. We're looking for the expected waiting time until we first see the pattern HTH in a sequence of fair coin tosses.",
"**Sett... | # Expected Stopping Time for Pattern HTH
## Setting Up the Gambler's Argument
I'll use the **optional stopping theorem** via the classic "gambler's queue" construction.
**The Setup:** At each time $k = 1, 2, 3, \ldots$, a new gambler arrives at the casino and pays a **\$1 entry fee**. Each gambler then attempts to b... | [
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"bran... | Gambler's argument: gamblers bet on HTH starting at each time; at stopping time S, total payout is $10, total entry is $S, so E[S]=10. | 0.2867 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-58042_7 | Find the function \( f(x) \) that satisfies the q-difference equation \( a(x)f(x) = f(qx) \), where \( q \) is a real, positive number less than 1, and \( a(x) \) is a known, continuous function with \( a(0) = 0 \). Additionally, \( f(0) = 0 \) and the solution for \( x < 0 \) does not depend on the solution for \( x >... | 0 | f(x) = 0 | true | hard | mid | 5,409 | [
"Let me work through this q-difference equation step by step.\n\nWe need to solve: $a(x) f(x) = f(qx)$",
"where $q$ is real, positive, less than 1, $a(x)$ is continuous with $a(0) = 0$, $f(0) = 0$, and the solution for $x < 0$ doesn't depend on the solution for $x > 0$.",
"Let me start with the example: $x f(x)... | # Solving the q-Difference Equation $a(x)f(x) = f(qx)$
## Setting Up the Iteration
Starting with $f(qx) = a(x)f(x)$, I iterate the equation:
$$f(q^2 x) = a(qx)\,f(qx) = a(qx)\,a(x)\,f(x)$$
$$f(q^n x) = f(x)\prod_{k=0}^{n-1} a(q^k x)$$
## Taking the Limit $n \to \infty$
Since $0 < q < 1$, we have $q^n x \to 0$ for... | [
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"bran... | Iterate the q-difference equation, show the infinite product vanishes making the constraint vacuous, then verify f≡0 is the unique reasonable solution. | 0.1221 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-53437_6 | Calculate the Vassiliev invariant for the knot K_1. | 1 | 1 | true | hard | mid | 3,193 | [
"The user is asking me to calculate the Vassiliev invariant for a knot K_1. However, they haven't specified what K_1 is. In many textbook problems and standard references, K_1 typically refers to the trefoil knot, which is the most common non-trivial knot used as an example.",
"Let me think about what Vassiliev i... | # Calculating the Vassiliev Invariant for Knot K₁
## Identifying the Knot
The knot **K₁** in standard knot theory problems is the **trefoil knot** (the simplest non-trivial knot), which has 3 crossings.
## Vassiliev Invariants – Background
Vassiliev (finite-type) invariants are defined via singular knots. For **kno... | [
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"bran... | Identify K_1 as trefoil, use Conway polynomial coefficient a_2 (type 3 Vassiliev invariant) which equals 1 for the right-handed trefoil. | 0.3465 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55650_7 | Evaluate the integral \( \int_0^{2 \pi} \ln(1+e^{i \theta}) \, d\theta \). | 0 | 0 | true | hard | mid | 5,187 | [
"We need to evaluate I = ∫_0^{2π} ln(1 + e^{iθ}) dθ.",
"This is a complex integral. We need to be careful about the branch of the logarithm. Let's think about this carefully.",
"Let me write 1 + e^{iθ} = e^{iθ/2}(e^{-iθ/2} + e^{iθ/2}) = 2e^{iθ/2} cos(θ/2).",
"So ln(1 + e^{iθ}) = ln(2|cos(θ/2)|) + i arg(1 + e^... | # Evaluating $\int_0^{2\pi} \ln(1+e^{i\theta})\,d\theta$
## Setting Up the Decomposition
First, I factor the integrand using the identity:
$$1 + e^{i\theta} = 2e^{i\theta/2}\cos(\theta/2)$$
This reveals a **singularity at $\theta = \pi$** where $1+e^{i\pi}=0$, making this an improper integral. I use the **principal ... | [
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"bran... | Decompose 1+e^{iθ} via half-angle, determine principal-branch arguments on each subinterval, split integral at θ=π, compute real and imaginary parts separately. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-58688_1 | Given the function \(F(x)\) defined by the identity \(F((F(x)+x)^k) = (F(x)+x)^2 - x\), determine the maximum value of \(k\) such that \(F'(1)\) is an integer. Express this maximum value of \(k\) as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. What is the value of \(a+b\)? | 5 | 5 | true | hard | mid | 8,027 | [
"Let me work through this problem carefully.\n\nWe have $F((F(x)+x)^k) = (F(x)+x)^2 - x$.",
"Let me set $u = F(x) + x$. Then the equation becomes:\n$F(u^k) = u^2 - x$",
"But $x = u - F(x)$, so $u^2 - x = u^2 - u + F(x)$.",
"Hmm, this is getting circular. Let me think differently.\n\nLet me try to differentiat... | # Finding the Maximum Value of $k$
## Setting Up the Key Equation
I'll assume the natural normalization $F(1) = 0$ (so $F(1) + 1 = 1$), which gives $F(1^k) = 1^2 - 1 = 0$. ✓
**Differentiating both sides** of $F((F(x)+x)^k) = (F(x)+x)^2 - x$:
$$F'((F(x)+x)^k) \cdot k(F(x)+x)^{k-1}(F'(x)+1) = 2(F(x)+x)(F'(x)+1) - 1$$... | [
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"bran... | Assume F(1)=0, differentiate at x=1 to get quadratic in F'(1), require discriminant k²+4 to be a perfect square, solve Diophantine equation, maximize k over integer parameter s. | 0.268 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-60987_6 | Determine whether the infinite series \( \sum_{k=1}^\infty \frac{1}{n}\cos\left(\frac{k\pi}{2n}\right) \) equals \( \frac{2}{\pi} \). | \frac{2}{\pi} | \dfrac{2}{\pi} | true | hard | mid | 3,549 | [
"We need to determine whether the infinite series ∑_{k=1}^∞ (1/n) cos(kπ/(2n)) equals 2/π.",
"Wait, I need to be careful here. The series is written as ∑_{k=1}^∞ (1/n) cos(kπ/(2n)). But n appears both in the summation index context and as a variable. Let me re-read.",
"Actually, I think the problem is asking ab... | Looking at this problem, I need to interpret the expression as a Riemann sum limit:
$$\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n-1} \cos\!\left(\frac{k\pi}{2n}\right)$$
## Setting Up the Riemann Sum
Let me substitute $u_k = \frac{k\pi}{2n}$, so the spacing is $\Delta u = \frac{\pi}{2n}$.
As $k$ ranges from $1$ to $... | [
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"bran... | Reinterprets the problem as a Riemann sum limit, computes the finite cosine sum exactly, and confirms via integral that the limit equals 2/π. | 0.3977 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-57637_0 | Determine the values of \( z \in \mathbb{C} \) for which \( A_2(z) = 0 \), where \( e^{jz} = A_0(z) + j A_1(z) + j^2 A_2(z) \) and:
\[
\begin{cases}
A_0(z) = \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \\
A_1(z) = \displaystyle \sum_{n \geq 0} \frac{x^{3n+1}}{(3n+1)!} \\
A_2(z) = \displaystyle \sum_{n \geq 0} \f... | 0 | z = 0 | true | hard | mid | 8,728 | [
"Let me work through this problem carefully.\n\nWe have $j = e^{2\\pi i/3} = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i$.",
"Note that $j^3 = 1$ and $1 + j + j^2 = 0$, and $j^2 = e^{4\\pi i/3} = -\\frac{1}{2} - \\frac{\\sqrt{3}}{2}i = \\bar{j}$.",
"We're told that $e^{jz} = A_0(z) + j A_1(z) + j^2 A_2(z)$.",
"Wait... | Looking at this problem, I need to find all $z \in \mathbb{C}$ such that $A_2(z) = 0$.
## Setting Up the Key Identity
First, I verify the given identity. Since $j = e^{2\pi i/3}$ has the property that $j^k$ cycles with period 3 ($j^0=1,\; j^1=j,\; j^2=j^2,\; j^3=1, \ldots$), expanding $e^{jz}$:
$$e^{jz} = \sum_{k=0}... | [
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"bran... | Identify z=0 as a double zero from the series definition, verify via the exponential form and the identity 1+j+j²=0. | 0.6701 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101998_5 | Determine the number of integers \(1 \le a \le 2015\) that are primitive residues modulo 2015. An integer \(a\) is a primitive residue modulo \(n\) if it is exactly one of the following types of residues: quadratic, cubic, or quintic, modulo \(n\). | 464 | 464 | true | hard | mid | 6,489 | [
"Let me work through this problem step by step.\n\nFirst, I need to factorize 2015.\n2015 = 5 × 13 × 31",
"So 2015 = 5 × 13 × 31, and these are all distinct primes.",
"An integer $a$ is a primitive residue modulo $n$ if it is **exactly one** of the following types: quadratic, cubic, or quintic residue modulo $n... | Looking at this problem, I need to find integers $1 \le a \le 2015$ that are **exactly one** of: quadratic, cubic, or quintic residue modulo 2015.
## Setting Up the Factorization
First, I factor $2015 = 5 \times 13 \times 31$ (three distinct primes).
For a "residue" to be well-defined, I need $\gcd(a, 2015) = 1$, gi... | [
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"bran... | Restrict to coprime a, count QR/CR/5R per prime via cyclic group subgroups, apply inclusion-exclusion for "exactly one" membership. | 0.1023 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-105708_7 | Calculate the work done by a gas in the cycle $1-2-3-4-5-6-1$, given that $p_{0}=10^{5} \Pi a$ and $V_{0}=1$. Note that states 1 and 3 coincide, as do states 4 and 6. | 100000 | W = 10^{5}\;\text{J} = 100\;\text{kJ} | true | hard | mid | 8,404 | [
"The user is asking me to calculate the work done by a gas in a cycle 1-2-3-4-5-6-1, given that p₀ = 10⁵ Pa and V₀ = 1 (likely 1 m³ or some unit). States 1 and 3 coincide, as do states 4 and 6.",
"Since I don't have the actual figure, I need to reason about what this cycle looks like based on the description. Thi... | # Work Done in Cycle 1-2-3-4-5-6-1
## Setting Up the Problem
The work done by a gas in a thermodynamic cycle equals the **area enclosed** by the cycle on a *p*–*V* diagram (with sign determined by the direction of traversal):
$$W = \oint p\,dV$$
Since states **1 = 3** and **4 = 6**, the cycle is a **self-intersecti... | [
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"bran... | Guesses the self-intersecting cycle encloses two triangular regions of area ½p₀V₀ each, giving total work p₀V₀ = 10⁵ J. | 0.5385 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-103876_1 | In quadrilateral \(ABCD\), the diagonals intersect at point \(M\) with an angle \(\alpha\) between them. Let \(O_{1}, O_{2}, O_{3}, O_{4}\) be the centers of the circumcircles of triangles \(ABM, BCM, CDM,\) and \(DAM\), respectively. Find the ratio of the areas of quadrilateral \(ABCD\) to quadrilateral \(O_{1} O_{2} ... | 2 \sin^2 \alpha | 2\sin^{2}\alpha | true | hard | mid | 6,764 | [
"Let me set up coordinates. Let the diagonals of quadrilateral $ABCD$ intersect at point $M$ with angle $\\alpha$ between them.",
"Let me place $M$ at the origin. Let diagonal $AC$ lie along the x-axis and diagonal $BD$ lie along a line making angle $\\alpha$ with the x-axis.",
"Let me set:\n- $A$ at distance $... | Looking at this problem, I need to find the ratio of the area of quadrilateral $ABCD$ to the area of quadrilateral $O_1O_2O_3O_4$ formed by circumcenters of triangles $ABM$, $BCM$, $CDM$, and $DAM$.
## Setting Up Coordinates
I place $M$ at the origin, with diagonal $AC$ along the x-axis and diagonal $BD$ at angle $\a... | [
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"bran... | Coordinate geometry with M at origin; find all four circumcenters via perpendicular bisectors, compute both quadrilateral areas by shoelace, take ratio. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-104979_1 | In the Cartesian coordinate plane, points \(A\) and \(B\) lie on the parabola \(y^2 = 4x\) and satisfy the condition \(\overrightarrow{OA} \cdot \overrightarrow{OB} = -4\). Let \(F\) be the focus of the parabola. Calculate the product \(S_{\triangle OP} \cdot S_{\triangle ORP}\). | 2 | 2 | true | hard | mid | 7,496 | [
"Let me work through this problem carefully.\n\nWe have the parabola $y^2 = 4x$ with focus $F = (1, 0)$.",
"Points $A$ and $B$ lie on the parabola, and $\\overrightarrow{OA} \\cdot \\overrightarrow{OB} = -4$.",
"Let me parametrize the parabola. For $y^2 = 4x$, I can write points as $(t^2, 2t)$ for parameter $t$... | Looking at this problem, I need to work with the parabola $y^2 = 4x$ where points $A$ and $B$ satisfy $\overrightarrow{OA} \cdot \overrightarrow{OB} = -4$, and $F$ is the focus.
## Setting Up Parameters
For the parabola $y^2 = 4x$, I parametrize points as $(t^2, 2t)$.
Let $A = (a^2, 2a)$ and $B = (b^2, 2b)$.
**Appl... | [
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"bran... | Parametrize parabola, derive ab=-2, identify correct interpretation as S_{OAF}·S_{OBF}, compute each area as |a| and |b|, product equals |ab|=2. | 0.5406 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-55650_1 | Evaluate the integral \( \int_0^{2 \pi} \ln(1+e^{i \theta}) \, d\theta \). | 0 | 0 | true | hard | mid | 5,898 | [
"We need to evaluate the integral I = ∫_0^{2π} ln(1+e^{iθ}) dθ.",
"The complex logarithm has branch cuts. We need to be careful about the branch. The standard principal branch of log has a branch cut along the negative real axis, so it's not defined when 1+e^{iθ} is a negative real number or zero.",
"Let's thin... | # Evaluating $\int_0^{2\pi} \ln(1+e^{i\theta})\,d\theta$
## Setting Up via a Regularized Family
Since $\ln(1+e^{i\theta})$ has a logarithmic singularity at $\theta = \pi$, I introduce a parameter:
$$I(a) = \int_0^{2\pi} \ln\!\bigl(1+ae^{i\theta}\bigr)\,d\theta, \qquad 0 < a < 1,$$
and will take $a \to 1^-$ at the e... | [
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... | Factor 1+e^{iθ}, determine principal branch on each subinterval, split integral at θ=π, compute real and imaginary parts using symmetry and the known integral of ln(cos). | 0.0107 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101437_4 | Find all functions \( f(x) \) that satisfy the equation \((1+y)f(x) - (1+x)f(y) = y f\left(\frac{x}{y}\right) - x f\left(\frac{y}{x}\right)\) for all real numbers \(x, y \neq 0\), given the initial conditions \(f(1) = 32\) and \(f(-1) = -4\). Use these initial conditions to determine \(f(y)\). | 18y + 14 | f(y)=18y+14\text{ for }y\neq 0\text{, with }f(0)\text{ arbitrary if }0\text{ is in the domain.} | true | hard | mid | 8,730 | [
"We need solve functional equation. Need produce step-by-step reasoning final boxed. Need find all functions f: reals? Domain likely real? Equation for all real x,y ≠ 0. Need use f(1)=32, f(-1)=-4. Need determine f(y). Let's analyze thoroughly.",
"Equation: (1+y) f(x) - (1+x) f(y) = y f(x/y) - x f(y/x) for all x,... | Let the given equation be
\[
(1+y)f(x)-(1+x)f(y)=y f\!\left(\frac{x}{y}\right)-x f\!\left(\frac{y}{x}\right),
\qquad x,y\neq 0.
\]
We are given
\[
f(1)=32,\qquad f(-1)=-4.
\]
### Step 1: Substitute \(x=1\)
Then
\[
(1+y)f(1)-(1+1)f(y)=y f\!\left(\frac{1}{y}\right)-f(y).
\]
Using \(f(1)=32\),
\[
32(1+y)-2f(y)=y f... | [
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"bra... | Substitute x=1 and x=-1 into the functional equation to obtain two relations, then combine them to solve for f(t) directly. | 0.0897 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101266_2 | Determine the smallest possible value of n for a linear binary code [n,60,4] such that the parity matrix H has the property that any three columns are linearly independent. | 68 | 68 | true | hard | mid | 5,171 | [
"We need to determine the smallest possible value of n for a linear binary code [n, 60, 4] such that the parity matrix H has the property that any three columns are linearly independent.",
"Let me recall the definitions:\n- A linear binary code [n, k, d] has length n, dimension k, and minimum distance d.\n- The p... | Looking at this problem, I need to find the smallest $n$ for a binary linear code $[n, 60, 4]$ where the parity-check matrix $H$ has the property that any three columns are linearly independent.
## Setting Up the Problem
For a $[n, 60, 4]$ code:
- The parity-check matrix $H$ has dimensions $(n-60) \times n$
- The con... | [
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"bran... | Translate column independence to sum-free set in GF(2)^(n-60), bound size by 2^(n-61), solve n ≤ 2^(n-61), verify d=4 via pigeonhole. | 0.0341 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-108091_2 | Determine the largest positive integer \( n \) such that the inequality \[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2} \] holds for all \( a, b, c \in [0,1] \). Assume \( \sqrt[1]{abc} = abc \). | 3 | 3 | true | hard | mid | 8,138 | [
"Let me find the largest positive integer $n$ such that for all $a, b, c \\in [0,1]$:",
"$$\\frac{a+b+c}{abc+1} + (abc)^{1/n} \\leq \\frac{5}{2}$$",
"Let me denote $x = abc$ where $x \\in [0,1]$, and let $s = a + b + c$.",
"Since $a, b, c \\in [0,1]$, we have $x = abc \\in [0,1]$ and $s = a + b + c \\in [0, 3... | Looking at this problem, I need to find the largest $n$ such that $\frac{a+b+c}{abc+1} + (abc)^{1/n} \leq \frac{5}{2}$ for all $a,b,c \in [0,1]$.
## Setting Up the Reduction
Let $x = abc \in [0,1]$. Since $(abc)^{1/n}$ depends only on $x$, I need to maximize $a+b+c$ subject to $abc = x$ and $a,b,c \in [0,1]$.
**Clai... | [
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"bran... | Reduce to one variable via s_max=2+x, derive n≤4 from endpoint derivative, disprove n=4 by counterexample, prove n=3 via monotonicity of f. | 0 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-57543_4 | Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the following equations hold:
\[ f(3x) = f\left(\frac{x+y}{(x+y)^2+1}\right) + f\left(\frac{x-y}{(x-y)^2+1}\right) \]
and
\[ f\left(x^2-y^2\right) = (x+y)f(x-y) + (x-y)f(x+y) \] | 0 | f(x) = 0 \text{ for all } x \in \mathbb{R} | true | hard | mid | 6,503 | [
"Let me find all functions $f: \\mathbb{R} \\to \\mathbb{R}$ satisfying both equations for all $x, y \\in \\mathbb{R}$.",
"**Equation 1:** $f(3x) = f\\left(\\frac{x+y}{(x+y)^2+1}\\right) + f\\left(\\frac{x-y}{(x-y)^2+1}\\right)$",
"**Equation 2:** $f(x^2 - y^2) = (x+y)f(x-y) + (x-y)f(x+y)$",
"Let me start wit... | # Finding All Functions $f: \mathbb{R} \to \mathbb{R}$ Satisfying Both Equations
## Step 1: Analyze Equation 2
**Equation 2:** $f(x^2-y^2) = (x+y)f(x-y) + (x-y)f(x+y)$
**Setting $y = x$:** $f(0) = 2x\,f(0)$ for all $x$. With $x=1$: $f(0) = 2f(0) \Rightarrow f(0) = 0$.
**Setting $y = 0$:** $f(x^2) = 2x\,f(x)$ for al... | [
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"bran... | Derive f(ab)=bf(a)+af(b) from Eq 2 and f(4t/3)=2f(t) from Eq 1; combine to force f linear, then c=0. | 0.5829 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-104632_6 | Determine which is greater: \(200!\) or \(100^{200}\), without using a calculator or logarithm table. | 100^{200} | 100^{200} | true | hard | mid | 4,179 | [
"We need to determine which is greater: 200! or 100^{200}, without using a calculator or logarithm table.",
"Let me think about this carefully.\n\nWe want to compare 200! with 100^{200}.",
"Note that 100^{200} = (100)^{200} = (10^2)^{200} = 10^{400}.",
"And 200! = 1 × 2 × 3 × ... × 200.",
"One approach: Con... | # Comparing $200!$ and $100^{200}$
## Setting Up the Ratio
I'll compare the two quantities by examining the ratio:
$$P = \frac{200!}{100^{200}} = \prod_{k=1}^{200} \frac{k}{100}$$
**Key strategy:** Pair each factor $\frac{k}{100}$ with $\frac{201-k}{100}$ for $k = 1, 2, \ldots, 100$. This gives:
$$P = \prod_{k=1}^... | [
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"path": "D"
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"... | Pair k with 201-k, bound first factor by 1/50, remaining 99 factors by 1.01, use (1.01)^99 < e < 3 to show product < 1. | 0.7775 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
deepmath-101979_3 | In a scalene triangle $ABC$, the angle bisector $AD$ intersects the circumcircle of the triangle at point $P$. Point $I$ is the incenter of the triangle $ABC$. Given that $ID = DP$, determine the ratio $AI: ID$. | 2 | 2 | true | hard | mid | 9,289 | [
"Let me work through this problem carefully.\n\nWe have a scalene triangle $ABC$, and the angle bisector of angle $A$ intersects side $BC$ at point $D$ and the circumcircle at point $P$. $I$ is the incenter.",
"We're given that $ID = DP$, and we need to find $AI : ID$.",
"Let me set up the problem. Let the angl... | Looking at this problem, I need to work with the angle bisector $AD$ where $D$ is on side $BC$ and $P$ is on the circumcircle, with $I$ being the incenter.
## Setting Up the Configuration
Since $A, I, D, P$ are collinear in that order (incenter is inside the triangle, $D$ is on $BC$, $P$ is on the circumcircle):
$$A... | [
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"bran... | Prove PI=PB (Trillium), use ID=DP to get D as midpoint of IP, derive angle constraint tan(β)tan(γ)=1/3, compute ratio. | 0.2191 | Qwen3.8-27B (thinking) | Qwen3.8-27B-FP8 (thinking), TraceAtlas v4 |
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