Capacity Gap
Collection
Artifacts for the Teacher-Signal Usability in Extreme-Gap Distillation thesis • 3 items • Updated
Datasets:
id stringlengths 35 40 | final_answer stringlengths 0 305 | difficulty float64 -1 10 | topic stringclasses 82
values | messages listlengths 2 2 |
|---|---|---|---|---|
HuggingFaceTB/DeepMath-103K/00000/9 | 0 | 8 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | [
{
"content": "Find the expectation \\( \\mathbb{E}[X_t] \\) where \\( X_t = \\sin(B_t) \\) and \\( B_t \\) is a standard Brownian motion for \\( t \\geq 0 \\).",
"role": "user"
},
{
"content": "We are asked to compute the expectation:\n\n$$\n\\mathbb{E}[X_t] = \\mathbb{E}[\\sin(B_t)]\n$$\n\nwhere $ ... |
HuggingFaceTB/DeepMath-103K/00000/6 | |A| | 3 | Mathematics -> Algebra -> Intermediate Algebra -> Other | [
{
"content": "Let $A$ be a proper infinite subset of a set $X$. If $x$ and $y$ are two distinct elements of $X$ that are not in $A$, and we define $B = \\{x, y\\} \\cup A$, what is the cardinality of $B$ in terms of the cardinality of $A$?",
"role": "user"
},
{
"content": "We are given the following... |
HuggingFaceTB/DeepMath-103K/00000/13 | \mathbb{C}(t) | 4 | Mathematics -> Algebra -> Abstract Algebra -> Field Theory | [
{
"content": "Provide an example of a field that properly contains the field of complex numbers \\( \\mathbb{C} \\).",
"role": "user"
},
{
"content": "The field of complex numbers \\( \\mathbb{C} \\) is **algebraically closed**, meaning that every non-constant polynomial with coefficients in \\( \\m... |
HuggingFaceTB/DeepMath-103K/00000/4 | \infty | 5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to \\infty} (x!)^{1/x}\n$$\n\nThis is a limit involving the factorial function raised to the power $1/x$, as $x \\to \\infty$. Since $x!$ gr... |
HuggingFaceTB/DeepMath-103K/00000/10 | \emptyset | 5.5 | Mathematics -> Algebra -> Algebra -> Polynomial Operations | [
{
"content": "Determine the values of \\( k \\) for which the equation \\( x^3 - 3x^2 + 6x + k = 0 \\) has three real roots.",
"role": "user"
},
{
"content": "We are given the cubic equation:\n\n$$\nx^3 - 3x^2 + 6x + k = 0\n$$\n\nWe are to **determine the values of $ k $** for which this equation ha... |
HuggingFaceTB/DeepMath-103K/00000/2 | -\dfrac{2}{3} | 4 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left(\\dfrac{1}{\\tan^2 x}-\\dfrac{1}{x^2} \\right) \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to 0} \\left(\\dfrac{1}{\\tan^2 x}-\\dfrac{1}{x^2} \\right)\n$$\n\n---\n\n### Step 1: Use Taylor se... |
HuggingFaceTB/DeepMath-103K/00000/1 | m^2 + 1 = 0 | 5 | Mathematics -> Differential Equations -> Ordinary Differential Equations (ODEs) | [
{
"content": "Find the auxiliary equation for the ordinary differential equation with constant coefficients: \\((x^2D^2 + xD + 1)y = \\sin(2\\log x)\\sin(\\log x)\\).",
"role": "user"
},
{
"content": "We are given the ordinary differential equation (ODE):\n\n$$\n(x^2D^2 + xD + 1)y = \\sin(2\\log x)\... |
HuggingFaceTB/DeepMath-103K/00000/12 | \dfrac{1}{p+1} | 4.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}} \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}}\n$$\n\nwhere $ p $ is a fixed real number (typically assumed to b... |
HuggingFaceTB/DeepMath-103K/00000/18 | 0 | 8 | Mathematics -> Number Theory -> Prime Numbers | [
{
"content": "Determine the limit of \\( \\frac{\\pi(x)}{x} \\) as \\( x \\to \\infty \\), where \\( \\pi(x) \\) is the prime counting function.",
"role": "user"
},
{
"content": "We are asked to determine the limit:\n\n$$\n\\lim_{x \\to \\infty} \\frac{\\pi(x)}{x}\n$$\n\nwhere $ \\pi(x) $ is the **p... |
HuggingFaceTB/DeepMath-103K/00000/5 | \dfrac{\pi}{2} | 5 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Find the length of the polar curve given by \\( r = \\sqrt{1 + \\cos(2\\theta)} \\) for \\( 0 \\leq \\theta \\leq \\frac{\\pi\\sqrt{2}}{4} \\).",
"role": "user"
},
{
"content": "We are given a polar curve:\n\n$$\nr = \\sqrt{1 + \\cos(2\\theta)}, \\quad \\text{for } 0 \\leq \\theta \\le... |
HuggingFaceTB/DeepMath-103K/00000/0 | 0 | 4.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\sqrt{x} \\left( \\sqrt[3]{x+1} - \\sqrt[3]{x-1} \\right) \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to \\infty} \\sqrt{x} \\left( \\sqrt[3]{x+1} - \\sqrt[3]{x-1} \\right)\n$$\n\n---\n\n###... |
HuggingFaceTB/DeepMath-103K/00000/3 | 34 | 4.5 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations | [
{
"content": "Determine the minimum sample size required such that the probability of at least two people being a match at all six genetic markers exceeds 0.001, given that the probability of a match at each individual marker is 1/9.",
"role": "user"
},
{
"content": "We are given:\n\n- There are **6... |
HuggingFaceTB/DeepMath-103K/00000/14 | No | 8.5 | Mathematics -> Algebra -> Linear Algebra -> Linear Transformations | [
{
"content": "Let $T$ be a continuous linear operator. Suppose $(u_n)$ is a sequence that converges weakly to $u$, denoted as $(u_n) \\rightharpoonup u$. Additionally, assume $T(u_n) \\rightharpoonup T(u)$ and there exists a subsequence $(u_{n_k})$ such that $T(u_{n_k}) \\rightarrow T(u)$. Does it follow that $... |
HuggingFaceTB/DeepMath-103K/00000/22 | 1 | 6.5 | Mathematics -> Algebra -> Intermediate Algebra -> Other | [
{
"content": "Using the generating function \\( \\sum_{n=0}^{\\infty}P_n(x)r^n=(1-2rx+r^2)^{-\\frac{1}{2}} \\), find the value of \\( P_n(1) \\).",
"role": "user"
},
{
"content": "We are given the **generating function** for the **Legendre polynomials** $ P_n(x) $:\n\n$$\n\\sum_{n=0}^{\\infty} P_n(x... |
HuggingFaceTB/DeepMath-103K/00000/21 | \ln(2\pi + 1) | 5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "Find parametric equations for a unit circle with a speed of \\( e^t \\), starting from \\( x=1 \\), \\( y=0 \\). Determine when the circle is completed.",
"role": "user"
},
{
"content": "We are asked to find **parametric equations** for a **unit circle** traversed with **speed** $ e^t ... |
HuggingFaceTB/DeepMath-103K/00000/7 | \phi^4 | 5.5 | Mathematics -> Geometry -> Plane Geometry -> Polygons | [
{
"content": "Consider a regular pentagon circumscribed in a circle. Connect each vertex of this pentagon to every other non-adjacent vertex with a straight line segment to form a pentagram, which contains a smaller pentagon. What is the ratio of the area of the original (large) pentagon to the smaller one in t... |
HuggingFaceTB/DeepMath-103K/00000/25 | \{\mathbf{0}\} | 6 | Mathematics -> Algebra -> Linear Algebra -> Matrices | [
{
"content": "Find the kernel of the Vandermonde matrix \\( A \\) given by\n\\[\nA = \\begin{pmatrix} x_1^0 & x_1^1 & \\ldots & x_1^n \\\\ x_2^0 & x_2^1 & \\ldots & x_2^n \\\\ \\vdots & \\vdots & \\ldots & \\vdots \\\\ x_m^0 & x_m^1 & \\ldots & x_m^n \\end{pmatrix}\n\\]\nwhere \\( n < m - 1 \\) and the \\( x_i ... |
HuggingFaceTB/DeepMath-103K/00000/19 | 1 | 8 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | [
{
"content": "Consider a sequence $(r_n)$ resulting from infinite coin flips, where $R_n=1$ if $r_n$ is a head and $R_n=-1$ if $r_n$ is a tail. Determine if $P\\left(\\sum \\frac{R_n}{n} < \\infty\\right) = 1$. Provide a justification for your answer.",
"role": "user"
},
{
"content": "We are given a... |
HuggingFaceTB/DeepMath-103K/00000/29 | 1 | 6 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Evaluate the infinite series \\( \\sum_{n=1}^{\\infty} \\frac{2n+1}{(n^{2}+n)^{2}}. \\)",
"role": "user"
},
{
"content": "We are asked to evaluate the infinite series:\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{2n+1}{(n^2 + n)^2}\n$$\n\n---\n\n### Step 1: Simplify the denominator\n\nNote that... |
HuggingFaceTB/DeepMath-103K/00000/16 | Yes | 9 | Mathematics -> Differential Equations -> Partial Differential Equations (PDEs) | [
{
"content": "Let \\( A \\) be a C* algebra of operators on a Hilbert space \\( H \\). Determine if there exists an \\( x \\) in \\( H \\) such that the set \\( Ax \\) is dense in \\( H \\) but not equal to the whole \\( H \\).",
"role": "user"
},
{
"content": "We are given a C*-algebra $ A $ of bou... |
HuggingFaceTB/DeepMath-103K/00000/31 | Yes | 8 | Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers | [
{
"content": "Let $E \\subset \\mathbb{C}$ be a set such that for any sequence of distinct elements $(e_n)_{n \\in \\mathbb{N}}$ from $E$, $e_n \\to 0$ in norm. Is $E$ necessarily countable? Provide a justification for your answer.",
"role": "user"
},
{
"content": "We are given a set $ E \\subset \\... |
HuggingFaceTB/DeepMath-103K/00000/36 | \dfrac{1}{2e} | 5.5 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | [
{
"content": "Given the joint probability distribution of $X$ and $Y$:\n\\[ f(x,y) = \\begin{cases} \n e^{-(x+y)}, & x > 0, y > 0 \\\\\n 0, & \\text{otherwise} \n\\end{cases} \\]\ncompute the probability $P(Y > X+1)$.",
"role": "user"
},
{
"content": "We are given the joint probability density fun... |
HuggingFaceTB/DeepMath-103K/00000/24 | 0 | 7.5 | Mathematics -> Number Theory -> Prime Numbers | [
{
"content": "Verify that for \\(n = 2^kN\\), where \\(N\\) is odd, the following identity holds:\n\\[ \\sum_{d\\mid n}(-1)^{n/d}\\phi(d) = \\sum_{d\\mid 2^{k-1}N}\\phi(d) - \\sum_{d\\mid N}\\phi(2^kd) = 0. \\]",
"role": "user"
},
{
"content": "We are given that $ n = 2^k N $, where $ N $ is **odd**... |
HuggingFaceTB/DeepMath-103K/00000/28 | B | 8 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Determine whether the following statement is true or false: \"Any bounded sequence in $L^4[0,1]$ has a convergent subsequence in $L^2[0,1]$.\"",
"role": "user"
},
{
"content": "We are asked to determine whether the following statement is **true** or **false**:\n\n> \"Any bounded sequen... |
HuggingFaceTB/DeepMath-103K/00000/17 | 1 | 3.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{t\\to 0}\\left(\\frac{1}{\\ln(1 + t)}+\\frac{1}{\\ln(1-t)}\\right). \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{t \\to 0} \\left( \\frac{1}{\\ln(1 + t)} + \\frac{1}{\\ln(1 - t)} \\right)\n$$\n\n---\n\n### Step 1... |
HuggingFaceTB/DeepMath-103K/00000/30 | True | 6.5 | Mathematics -> Calculus -> Integral Calculus -> Integrals | [
{
"content": "Determine whether the following inequality is true for all real numbers $0 < r < 1$ and $t \\geq 0$:\n\\[ \\int_t^{t+r} \\sin(x)\\, dx \\leq \\int_{\\frac{\\pi}{2}-\\frac{r}{2}}^{\\frac{\\pi}{2}+\\frac{r}{2}}\\sin(x)\\, dx. \\]",
"role": "user"
},
{
"content": "We are asked to determin... |
HuggingFaceTB/DeepMath-103K/00000/11 | 0.5 | 6.5 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Verify that the distance from the function \\( f = x \\) to the set \\( Y = \\{ f \\in C[0,1] : \\int_0^1 f = 0 \\} \\) is 0.5, given that \\( Y \\) is a closed subset of \\( X = \\{ f \\in C[0,1] : f(0) = 0 \\} \\).",
"role": "user"
},
{
"content": "We are asked to verify that the **d... |
HuggingFaceTB/DeepMath-103K/00000/37 | 1 | 5.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0}\\left(\\log\\frac{1}{x}\\right)^x \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to 0^+} \\left( \\log \\frac{1}{x} \\right)^x\n$$\n\n> **Note:** The expression $\\log \\frac{1}{x}$ is only defined for... |
HuggingFaceTB/DeepMath-103K/00000/27 | \dfrac{5}{2} | 5.5 | Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions | [
{
"content": "Find the maximum value of $x^2 + y^2$ given that $(x, y)$ satisfy the following equations:\n\\[ 2x^2 + 5xy + 3y^2 = 2 \\]\n\\[ 6x^2 + 8xy + 4y^2 = 3 \\]\nNote: Calculus is not allowed.",
"role": "user"
},
{
"content": "We are given two equations:\n\n$$\n(1) \\quad 2x^2 + 5xy + 3y^2 = 2... |
HuggingFaceTB/DeepMath-103K/00000/23 | \log 2 | 5 | Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers | [
{
"content": "Evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac{1}{\\alpha} \\right| \\), where \\( \\alpha \\) is a fifth root of unity.",
"role": "user"
},
{
"content": "We are asked to evaluate:\n\n$$\n\\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac... |
HuggingFaceTB/DeepMath-103K/00000/34 | 23 | 6.5 | Mathematics -> Algebra -> Intermediate Algebra -> Other | [
{
"content": "Let $a$ and $b$ be positive integers such that the range of the function \\( y = \\frac{x^2 + ax + b}{x^2 + 2x + 3} \\) is the interval \\(-5 \\leq y \\leq 4\\) for all real numbers $x$. Find the value of $a + b$. \\( \\boxed{} \\)",
"role": "user"
},
{
"content": "We are given that th... |
HuggingFaceTB/DeepMath-103K/00000/47 | \cosh x \sin y | 5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "For the function \\( u(x, y) = \\sinh x \\cos y \\), find the conjugate harmonic function \\( v(x, y) \\).",
"role": "user"
},
{
"content": "We are given a function:\n\n$$\nu(x, y) = \\sinh x \\cos y\n$$\n\nWe are to find the **conjugate harmonic function** $ v(x, y) $, such that $ f(z... |
HuggingFaceTB/DeepMath-103K/00000/46 | 3.241 | 5.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Calculate the integral: \\[ \\int_{0}^{2} \\sqrt{1+x^3} \\, dx \\]",
"role": "user"
},
{
"content": "The integral\n\n$$\n\\int_{0}^{2} \\sqrt{1+x^3} \\, dx\n$$\n\ndoes **not** have an elementary antiderivative. That is, there is no closed-form expression in terms of elementary function... |
HuggingFaceTB/DeepMath-103K/00000/39 | \mathbf{c}^\top \otimes \mathbf{A} | 6 | Mathematics -> Algebra -> Linear Algebra -> Matrices | [
{
"content": "Find the derivative of the matrix product $\\mathbf{A}\\mathbf{B}\\mathbf{c}$ with respect to the matrix $\\mathbf{B}$, where $\\mathbf{A}$ is an $n\\times m$ matrix, $\\mathbf{B}$ is an $m\\times k$ matrix, and $\\mathbf{c}$ is a $k\\times 1$ vector.",
"role": "user"
},
{
"content": "... |
HuggingFaceTB/DeepMath-103K/00000/48 | Yes | 8 | Mathematics -> Geometry -> Differential Geometry -> Curvature | [
{
"content": "In a compact metric space $X$, consider a finite subset $F \\subset X$ that is $\\epsilon$-equidistant, meaning that the distance $d(x,y) = \\epsilon$ for all distinct $x,y \\in F$. For a given $\\epsilon > 0$, does there exist a maximum number $N$, depending on both $\\epsilon$ and $X$, such that... |
HuggingFaceTB/DeepMath-103K/00000/32 | Yes | 9 | Mathematics -> Algebra -> Abstract Algebra -> Field Theory | [
{
"content": "Let $S$ be an algebraic smooth surface over $\\\\mathbb{C}\\\\). Suppose there is a fibration $p: S \\rightarrow C$ onto a smooth curve, and let $f$ be a fiber of this fibration. Let $K$ be a canonical divisor on $S$ such that $K \\cdot f = 0$. Determine whether each $m$-canonical map $\\varphi_{m... |
HuggingFaceTB/DeepMath-103K/00000/38 | 4603 | 5.5 | Mathematics -> Number Theory -> Congruences | [
{
"content": "Evaluate \\(1234^{1234} \\pmod{5379}\\). Note that \\(5379 = 3 \\times 11 \\times 163\\).",
"role": "user"
},
{
"content": "We are asked to evaluate:\n\n$$\n1234^{1234} \\mod 5379\n$$\n\nGiven that:\n\n$$\n5379 = 3 \\times 11 \\times 163\n$$\n\nThese are all **prime** factors. So we ca... |
HuggingFaceTB/DeepMath-103K/00000/35 | \dfrac{1}{2} | 7.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Evaluate the limit: \\[ \\lim _{n\\to \\infty }n \\int_{-1}^0(x + e^x)^{n}dx. \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{n \\to \\infty} n \\int_{-1}^0 (x + e^x)^n \\, dx\n$$\n\n---\n\n### Step 1: Understand the integrand\n\nLet:\n$$\nf(x) =... |
HuggingFaceTB/DeepMath-103K/00000/45 | \dfrac{1}{e} | 6 | Mathematics -> Precalculus -> Limits | [
{
"content": "Compute the limit \\( \\lim_{n\\to\\infty}{(\\sqrt[n]{e}-\\frac{2}{n})^n} \\).",
"role": "user"
},
{
"content": "We are asked to compute the limit:\n\n$$\n\\lim_{n\\to\\infty} \\left( \\sqrt[n]{e} - \\frac{2}{n} \\right)^n\n$$\n\n---\n\n### Step 1: Understand $\\sqrt[n]{e}$\n\nWe know ... |
HuggingFaceTB/DeepMath-103K/00000/42 | 9 | 5 | Mathematics -> Geometry -> Plane Geometry -> Triangulations | [
{
"content": "A triangle is inscribed in a circle with radius 1. What is the maximum value of the sum of the squares of the sides of the triangle?",
"role": "user"
},
{
"content": "We are given a triangle inscribed in a **circle of radius 1**, and we are to find the **maximum value of the sum of the... |
HuggingFaceTB/DeepMath-103K/00000/49 | e^{-2} | 7 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: $$\\lim_{x\\to\\infty}\\left(\\frac1{x^2\\sin^2\\frac 1x}\\right)^\\frac 1{x\\sin\\frac 1x-1}$$ without using L'Hospital's Rule or Series expansion.",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x\\to\\infty}\\left(\\frac{1}{x^2... |
End of preview. Expand in Data Studio
Dataset Card for HuggingFaceTB/DeepMath-103K-Qwen3-235B-A22B-Instruct-2507
Dataset Summary
Synthetic data generated by DataForge:
- Model:
Qwen/Qwen3-235B-A22B-Instruct-2507(main) - Source dataset:
HuggingFaceTB/DeepMath-103K/all(trainsplit). - Generation config:
temperature=0.6,top_p=0.8,top_k=20,max_tokens=8192,model_max_context=32768 - Speculative decoding:
disabled - System prompt:
None - User prompt: Column
question
The run produced 103,022 samples and generated 173,029,532 (~173.0M) tokens.
You can load the dataset using
from datasets import load_dataset
ds = load_dataset("HuggingFaceTB/DeepMath-103K-Qwen3-235B-A22B-Instruct-2507")
Dataset Stats
| Metric | Value |
|---|---|
| Documents processed | 103,022 |
| Avg. source chars | 203.46 |
| Total prompt tokens | 5,092,007 (~5.1M) |
| Total completion tokens | 173,029,532 (~173.0M) |
| Mean prompt tokens | 79.45 |
| Mean completion tokens | 2699.75 |
Licensing Information
License: other
Contributions
Thanks to @cmpatino for adding this dataset.
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