data_source
stringclasses 1
value | prompt
listlengths 1
1
| ability
stringclasses 1
value | reward_model
dict | extra_info
dict |
|---|---|---|---|---|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\sqrt{x} \\left( \\sqrt[3]{x+1} - \\sqrt[3]{x-1} \\right) \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 0,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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)\\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "m^2 + 1 = 0"
}
|
{
"index": 1,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left(\\dfrac{1}{\\tan^2 x}-\\dfrac{1}{x^2} \\right) \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "-\\dfrac{2}{3}"
}
|
{
"index": 2,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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."
}
] |
math
|
{
"style": "rule",
"ground_truth": "34"
}
|
{
"index": 3,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\infty"
}
|
{
"index": 4,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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} \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{2}"
}
|
{
"index": 5,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 $A$?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "|A|"
}
|
{
"index": 6,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 terms of the golden ratio?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\phi^4"
}
|
{
"index": 7,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the longest geometric progression with a common ratio greater than 1 that can be formed from the set \\( \\{100, 101, 102, \\ldots, 1000\\} \\). What is the length of this progression?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "6"
}
|
{
"index": 8,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 9,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the values of \\( k \\) for which the equation \\( x^3 - 3x^2 + 6x + k = 0 \\) has three real roots."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\emptyset"
}
|
{
"index": 10,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 \\} \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "0.5"
}
|
{
"index": 11,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}} \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{1}{p+1}"
}
|
{
"index": 12,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Provide an example of a field that properly contains the field of complex numbers \\( \\mathbb{C} \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\mathbb{C}(t)"
}
|
{
"index": 13,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 $T(u_n) \\rightarrow T(u)$?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "No"
}
|
{
"index": 14,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8, 4$. Define $n = (10a + b)^{10c + d}$. Calculate the probability that $1984!$ is divisible by $n$. Use Fermat's Little Theorem to assist in your calculations."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{5}{6}"
}
|
{
"index": 15,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 16,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{t\\to 0}\\left(\\frac{1}{\\ln(1 + t)}+\\frac{1}{\\ln(1-t)}\\right). \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "1"
}
|
{
"index": 17,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the limit of \\( \\frac{\\pi(x)}{x} \\) as \\( x \\to \\infty \\), where \\( \\pi(x) \\) is the prime counting function."
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 18,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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."
}
] |
math
|
{
"style": "rule",
"ground_truth": "1"
}
|
{
"index": 19,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Consider a locally path connected space $X$. If every open subset $U \\subseteq X$ is semi-locally simply connected, does it follow that $X$ is locally simply connected?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 20,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\ln(2\\pi + 1)"
}
|
{
"index": 21,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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) \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "1"
}
|
{
"index": 22,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac{1}{\\alpha} \\right| \\), where \\( \\alpha \\) is a fifth root of unity."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\log 2"
}
|
{
"index": 23,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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. \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 24,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 \\) are pairwise distinct."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\{\\mathbf{0}\\}"
}
|
{
"index": 25,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the minimum value of the product \\(abc\\) if the quadratic equation \\(ax^2 - bx + c = 0\\) has two distinct roots in the interval \\((0, 1)\\), where \\(a\\), \\(b\\), and \\(c\\) are natural numbers."
}
] |
math
|
{
"style": "rule",
"ground_truth": "25"
}
|
{
"index": 26,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 \\]\nCalculus is not allowed."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{5}{2}"
}
|
{
"index": 27,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine whether the following statement is true or false: \"Any bounded sequence in $L^4[0,1]$ has a convergent subsequence in $L^2[0,1]$.\""
}
] |
math
|
{
"style": "rule",
"ground_truth": "B"
}
|
{
"index": 28,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the infinite series \\( \\sum_{n=1}^{\\infty} \\frac{2n+1}{(n^{2}+n)^{2}}. \\)"
}
] |
math
|
{
"style": "rule",
"ground_truth": "1"
}
|
{
"index": 29,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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. \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "True"
}
|
{
"index": 30,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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."
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 31,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 $\\varphi_{mK}: S \\rightarrow \\mathbb{P}^N$, when defined, will contract $f$ to a point."
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 32,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the number of ways to make change for a dollar using generating functions. Specifically, determine the coefficient of the \\(x^{100}\\) term in the expansion of the generating function \\(\\frac{1}{(x-1)(x^5-1)\\cdots(x^{50}-1)}\\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "292"
}
|
{
"index": 33,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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{} \\)"
}
] |
math
|
{
"style": "rule",
"ground_truth": "23"
}
|
{
"index": 34,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim _{n\\to \\infty }n \\int_{-1}^0(x + e^x)^{n}dx. \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
}
|
{
"index": 35,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 $P(Y > X+1)$."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{1}{2e}"
}
|
{
"index": 36,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0}\\left(\\log\\frac{1}{x}\\right)^x \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "1"
}
|
{
"index": 37,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate \\(1234^{1234} \\pmod{5379}\\). Note that \\(5379 = 3 \\times 11 \\times 163\\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "4603"
}
|
{
"index": 38,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\mathbf{c}^\\top \\otimes \\mathbf{A}"
}
|
{
"index": 39,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the maximum value of the function \\( f(x) = \\int^{x}_{0} \\sqrt{(x^2-x)^2+y^4}~dy \\) for \\( 0 \\leq x \\leq 1 \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
}
|
{
"index": 40,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Is the space \\(X = \\prod_{t \\in \\mathbb{Z}} \\mathbb{R}\\), with the product topology \\(T\\), completely metrizable by a metric \\(d\\) such that \\(d(\\tau x, \\tau y) = d(x, y)\\) for the shift map \\(\\tau\\) defined by \\(\\tau x = (\\ldots, x_0, x_1, x_2, \\ldots)\\)?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "No"
}
|
{
"index": 41,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "9"
}
|
{
"index": 42,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find a matrix $X \\in M_n(\\mathbb{C})$ such that the linear functional $f(A) = \\text{tr}(XA)$ on $M_n(\\mathbb{C})$ preserves matrix multiplication."
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 43,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the minimum natural number \\( n \\) such that the expression \\( f(n) = \\sqrt{100+\\sqrt{n}} + \\sqrt{100-\\sqrt{n}} \\) is an integer."
}
] |
math
|
{
"style": "rule",
"ground_truth": "6156"
}
|
{
"index": 44,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Compute the limit \\( \\lim_{n\\to\\infty}{(\\sqrt[n]{e}-\\frac{2}{n})^n} \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{1}{e}"
}
|
{
"index": 45,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Calculate the integral: \\[ \\int_{0}^{2} \\sqrt{1+x^3} \\, dx \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "3.241"
}
|
{
"index": 46,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "For the function \\( u(x, y) = \\sinh x \\cos y \\), find the conjugate harmonic function \\( v(x, y) \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\cosh x \\sin y"
}
|
{
"index": 47,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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 the cardinality of any $\\epsilon$-equidistant set $F$ is at most $N$? Provide a proof or counterexample."
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 48,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"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."
}
] |
math
|
{
"style": "rule",
"ground_truth": "e^{-2}"
}
|
{
"index": 49,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the number of orientations of a smooth manifold with \\(n\\) maximal connected components."
}
] |
math
|
{
"style": "rule",
"ground_truth": "2^n"
}
|
{
"index": 50,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Let $X$ and $Y$ be orthogonal skew symmetric matrices. Determine if $X$ and $Y$ are orthogonally conjugate, i.e., if there exists an orthogonal matrix $U \\in O(n)$ such that $UX = YU$. Provide a justification for your answer."
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 51,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the double integral \\( \\iint_\\Omega \\sqrt{x^2+y^2} \\,\\mathrm{d}x\\mathrm{d}y \\), where \\( \\Omega \\) is the region defined by \\((x-1)^2+y^2 \\leq 1\\) and \\(0 \\leq y\\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{16}{9}"
}
|
{
"index": 52,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find all real harmonic functions $u$ on the unit disk $D$ centered at $0$ in the complex plane such that $u(0) = 0$ and $u^2$ is also harmonic on $D$."
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 54,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Let \\( A \\) be an integral domain. For every maximal ideal \\( m \\) in \\( A \\), consider \\( A_m \\) as a subring of the quotient field \\( K \\) of \\( A \\). Prove that \\( \\bigcap_{m \\text{: maximal ideal}} A_m = A \\), where the intersection is taken over all maximal ideals \\( m \\) of \\( A \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "A"
}
|
{
"index": 55,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the value of \\( \\lim_{x \\to 0} \\frac{4\\cos^2(f(x)) - 1}{1 - x^2} \\), given that the function \\( f(x) \\) satisfies \\( f(x) = f(2x) \\) for all \\( x \\in \\mathbb{R} \\) and \\( f(2017) = \\frac{\\pi}{4} \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "1"
}
|
{
"index": 56,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int_{\\gamma} \\frac{e^{2 \\pi z}}{(z+i)^3}dz \\) using the Cauchy Integration Formula, where \\( \\gamma(t)=2e^{it}, t \\in [0,2 \\pi] \\). Determine if the calculation \\( \\int_{\\gamma} \\frac{f(z)}{z-0}dz = 2 \\pi i f(0) = 0 \\) is correct."
}
] |
math
|
{
"style": "rule",
"ground_truth": "4\\pi^3 i"
}
|
{
"index": 57,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "A stock market trader buys 100 shares of stock A and 200 shares of stock B. Let X and Y be the price changes of stock A and B, respectively, over a certain time period. Assume that the joint probability density function (PDF) of X and Y is uniform over the set of integers (x, y) satisfying β2 β€ x β€ 4 and β1 β€ y β x β€ 1. Find the expected value of the trader's profit."
}
] |
math
|
{
"style": "rule",
"ground_truth": "300"
}
|
{
"index": 58,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Using the Fundamental Theorem of Line Integrals, evaluate \\( \\int_{C} e^x \\, dy + e^{x}y \\, dx, \\) where \\( C \\) is the parabola parameterized by \\( r(t)=\\langle t+1,t^2 \\rangle \\) for \\( t\\in[-1,3]. \\)"
}
] |
math
|
{
"style": "rule",
"ground_truth": "9e^4 - 1"
}
|
{
"index": 59,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the fundamental group of the space \\((S^1 \\times S^1)/(S^1 \\times \\{x\\})\\), where \\(x\\) is a point in \\(S^1\\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\mathbb{Z}"
}
|
{
"index": 60,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the radius of convergence for the function \\( \\frac{1}{z^2 + 2z + 2} \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\sqrt{2}"
}
|
{
"index": 61,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the surface area of the solid of revolution obtained by rotating the function \\( x=\\frac{1}{15}(y^2+10)^{3/2} \\) from \\( y=2 \\) to \\( y=4 \\) about the \\( x \\)-axis."
}
] |
math
|
{
"style": "rule",
"ground_truth": "36\\pi"
}
|
{
"index": 62,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = \\max_{t \\in [0,1]} |t^2 - tx| \\) for \\( x \\in (0,1) \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
}
|
{
"index": 63,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Assume that \\( f \\) is a non-negative real function, and let \\( a > 0 \\) be a real number. Define \\( I_a(f) \\) as follows:\n\\[ I_a(f) = \\frac{1}{a}\\int_{0}^{a} f(x) \\, dx \\]\nSuppose \\( \\lim_{x \\rightarrow \\infty} f(x) = A \\) exists. Determine whether \\( \\lim_{a \\rightarrow \\infty} I_a(f) = A \\) is always true. Provide a proof or a counterexample to support your conclusion."
}
] |
math
|
{
"style": "rule",
"ground_truth": "A"
}
|
{
"index": 64,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Given a linear transformation \\( T: M^R_{2x2} \\rightarrow R_4[x] \\) defined by the following mappings: \\( T\\begin{pmatrix} 2 & 3 \\\\ 1 & 0 \\end{pmatrix} = x^2 \\), \\( T\\begin{pmatrix} 1 & 0 \\\\ 0 & 2 \\end{pmatrix} = 3x - 4 \\), and \\( T\\begin{pmatrix} 0 & 2 \\\\ 4 & 5 \\end{pmatrix} = 2x^2 - 7 \\), find \\( T\\begin{pmatrix} 5 & 0 \\\\ -10 & -13 \\end{pmatrix} \\). Assume the given matrices are linearly independent."
}
] |
math
|
{
"style": "rule",
"ground_truth": "-4x^2 + 3x + 17"
}
|
{
"index": 65,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "A cube is to be colored using 6 distinct colors such that no two adjacent faces share the same color. How many distinct ways can this be done?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "30"
}
|
{
"index": 66,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Let $k$ be a number field, and $F/k$ a finite extension. Is it possible to find a countable family of extensions $k_i/k$ of degree 2 and a place $v_i$ of $k_i$ such that if $v$ is the place of $k$ lying below $v_i$, then $[k_{v_i}:k_v] = 2$, where $k_{v_i}$ and $k_v$ are the completions of $k_i$ and $k$ at $v_i$ and $v$, respectively? Furthermore, for some place $w$ of $F$ lying above $v$, is it possible that $F_w = k_v$?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 67,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit \\( \\lim_{n\\to \\infty} \\int_{1}^{\\pi}\\frac{\\cos(\\frac{x}{n})}{1-e^{-xn}}dx \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\pi - 1"
}
|
{
"index": 68,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the sum of all integer values of \\( a \\) such that \\( a(x^2+x-1) \\leq (x^2+x+1)^2 \\) for all real numbers \\( x \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "36"
}
|
{
"index": 69,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "If a vector space over the complex numbers has dimension $n$, is it possible to redefine the operations of addition and scalar multiplication such that the dimension of the vector space changes?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 70,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Consider a connected undirected simple non-planar graph $G$ with 15 vertices. If removing any edge from $G$ results in a planar graph, how many edges does $G$ have?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "40"
}
|
{
"index": 71,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty} \\sum_{k=1}^{n} \\sin\\left(\\frac{(2k-1)a}{n^2}\\right) \\) by expressing it as a Riemann sum and finding the corresponding integral, where \\( a \\in \\mathbb{R} \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "a"
}
|
{
"index": 72,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Calculate the first partial derivative with respect to $x$ of the function \\( f(x,y) \\) at the point \\((0,0)\\), where\n\\[\nf(x,y) =\n\\begin{cases}\n0 & (x,y)=(0,0)\\\\\n\\frac{xy}{|x|+|y|} & (x,y) \\neq (0,0)\n\\end{cases}\n\\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 73,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the minimal possible order of a group $G$ that contains a subset $A \\subset G$ with $|A| = 2n$, such that for every $a \\in A$, there exists a unique $b \\in A$ with $[a, b] \\neq e$. Provide your answer as a single integer."
}
] |
math
|
{
"style": "rule",
"ground_truth": "6^n"
}
|
{
"index": 74,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find all strictly increasing functions \\( f : \\Bbb{Z} \\rightarrow \\Bbb{Z} \\) such that \\( f(f(x)) = x + 2 \\) for all integers \\( x \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "f(x) = x + 1"
}
|
{
"index": 75,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Is the statement \"If $\\gcd(y,z)=1$, then $yz$ is perfect if and only if $D(y)D(z)=2s(y)s(z)$\" always true for odd perfect numbers $q^k n^2$? Here, $D(x) = 2x - \\sigma(x)$ is the deficiency of $x$, $s(x) = \\sigma(x) - x$ is the sum of the aliquot divisors of $x$, and $\\sigma(x)$ is the sum of divisors of $x \\in \\mathbb{N}$, the set of positive integers."
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 76,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Given that \\( f(\\phi) = \\Delta \\phi \\), where \\( \\Delta \\phi = \\nabla \\cdot \\nabla \\phi \\), find the expression for \\( \\frac{df(\\phi)}{d\\phi} \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\Delta"
}
|
{
"index": 77,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Calculate the minimum distance from the curve \\( f(x) = \\begin{pmatrix} \\cos(\\pi x) \\\\ \\sin(\\pi x) \\\\ 1-x^2 \\end{pmatrix} \\) to the origin in \\( \\mathbb{R}^3 \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "1"
}
|
{
"index": 78,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine whether the sequence \\( \\frac{2^n - 3n^3}{1-4^n} \\) converges or diverges using the ratio test."
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 79,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Compute \\( \\lim\\limits_{n\\to \\infty} \\int\\limits_0^1 x^{2019} \\{nx\\} \\, dx \\), where \\( \\{a\\} \\) denotes the fractional part of the real number \\( a \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{1}{4040}"
}
|
{
"index": 80,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Use Logarithmic Differentiation to find \\(\\frac{d}{dx} (x^{{x}^{x}})\\) at \\(x=1\\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "1"
}
|
{
"index": 81,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the number of integers in the range from 1 to \\(10^9\\) that are not perfect squares, perfect cubes, or perfect fifth powers."
}
] |
math
|
{
"style": "rule",
"ground_truth": "999967355"
}
|
{
"index": 82,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine whether the hyperoctahedral group, defined as the wreath product of $S_2$ and $S_n$ or equivalently $G = S_2^n \\rtimes S_n$, has only 3 maximal normal subgroups. Consider $G$ as a subgroup of $S_{2n}$ with its natural action on $[2] \\times [n]$. The group $G$ has two known normal subgroups of index two: $N_1$, the preimage of $A_n$ under the quotient map $G \\to S_n$, and $N_2$, the intersection in $S_{2n}$ of $G$ and $A_{2n}$. A third normal subgroup of index two arises from the diagonal in the Klein group $G/(N_1 \\cap N_2)$. Are there any additional maximal normal subgroups in $G$?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "3"
}
|
{
"index": 83,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Let $F$ be a finite Galois extension of the rational function field $\\mathbb{Q}(x)$. Let $k$ be the field of constants of $F$, defined as the algebraic closure of $\\mathbb{Q}$ in $F$. Is $k$ necessarily a Galois extension of $\\mathbb{Q}$? Provide a justification for your answer."
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 84,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the 9th derivative of \\( \\frac{\\cos(5 x^2)-1}{x^3} \\) and evaluate it at \\( x=0 \\) using the Maclaurin Series."
}
] |
math
|
{
"style": "rule",
"ground_truth": "-7875000"
}
|
{
"index": 85,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Consider the upper half-plane \\( \\mathbb{H} = \\{ x \\in \\mathbb{R}^n : x_n > 0 \\} \\). Let \\( u \\in C^2(\\mathbb{H}) \\cap C(\\bar{\\mathbb{H}}) \\) be a bounded harmonic function such that \\( u \\leq 0 \\) on \\( \\partial\\mathbb{H} = \\{ x_n = 0 \\} \\). Determine if it is possible to conclude that \\( u \\leq 0 \\) in all of \\( \\mathbb{H} \\) for \\( n \\geq 3 \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "Yes"
}
|
{
"index": 86,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "A ladder is being moved through a corridor in the first quadrant of the $x$-$y$ plane. The ladder is represented by the line $y = mx + c$, where $m < 0$ and $c > 0$. The length of the ladder is $L$. The goal is to find the maximum length $L$ such that the ladder just touches the corner of the corridor as it clears it. Given the coordinates of the corner are $(a, b)$, find the maximum length of the ladder when $a = 8$ and $b = 6$. Express your answer to the nearest whole number."
}
] |
math
|
{
"style": "rule",
"ground_truth": "20"
}
|
{
"index": 87,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the sign of the map induced by the covering map \\( p: S^n \\to \\mathbb{R}P^n \\) on homology, specifically \\( p_{*}: H_n(S^n, R) \\to H_n(\\mathbb{R}P^n, R) \\), where \\( R \\) is an arbitrary ring. Given that \\( H_n(S^n, R) \\cong R \\) is generated by the class \\([\\sigma_n^{(1)} + \\sigma_n^{(2)}]\\) and \\( H_n(\\mathbb{R}P^n, R) \\cong R \\) is generated by the class \\([\\tau_n]\\), find the sign in the expression \\( p_{*} [\\sigma_n^{(1)} + \\sigma_n^{(2)}] = \\tau_n \\pm \\tau_n \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "+"
}
|
{
"index": 88,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Given the fixed point iteration \\( p_n = \\frac{p_{n - 1}^2 + 3}{5} \\), which converges for any initial \\( p_0 \\in [0, 1] \\), estimate an expression for the number of iterations \\( n \\) required to achieve an absolute error \\( \\left| p_n - p \\right| < 10^{-4} \\) when \\( p_0 = 1 \\). Use the error bound \\( \\left| p_n - p \\right| \\leq k^n\\max\\{ p_0 - a, b - p_0 \\} \\), where \\([a, b]\\) is the interval of convergence and \\( k \\) is the bound on the derivative of the function within this interval."
}
] |
math
|
{
"style": "rule",
"ground_truth": "11"
}
|
{
"index": 89,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the unique polynomial \\( r(x) \\) such that for all \\( p(x) \\in \\mathbb{P}^2 \\), the functional \\( f(p(x)) = p'(-15) + 8p(-1) \\) can be expressed as \\( \\langle p, r \\rangle = \\int_0^1 p(x)r(x)dx \\). Here, \\( \\mathbb{P}^2 \\) is the space of polynomials of degree less than two."
}
] |
math
|
{
"style": "rule",
"ground_truth": "-132x + 74"
}
|
{
"index": 90,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\frac{\\left(\\int_0^x e^{t^2} \\, dt\\right)^2}{\\int_0^x e^{2t^2} \\, dt} \\]"
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 91,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Let \\( A \\in M_{5\\times 6}(\\mathbb{R}) \\) be a matrix with rank 4. Consider the block matrix \\( D = \\begin{pmatrix} I_5 & A \\\\ A^T & 0 \\end{pmatrix} \\). Determine the rank of \\( D \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "9"
}
|
{
"index": 92,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "A consumer is searching for 1 unit of a product across stores with prices 1, 2, 3, ..., each occurring with probabilities 1/2, 1/4, 1/8, ..., 1/2^n respectively. The consumer incurs a search cost of 1 for each store visited. Determine the price at which the consumer should stop searching."
}
] |
math
|
{
"style": "rule",
"ground_truth": "2"
}
|
{
"index": 93,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Determine the intervals where the function \\(f(x) = (x+\\frac{1}{x})^{x}\\) is increasing and where it is decreasing."
}
] |
math
|
{
"style": "rule",
"ground_truth": "(0, \\infty)"
}
|
{
"index": 94,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate \\( \\lim_{n \\rightarrow \\infty} \\int_0^1 \\frac{nx^{n-1}}{2+x} \\, dx \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
}
|
{
"index": 95,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Calculate the unconditional variance of the random variable $N$, where $N|\\Lambda$ follows a binomial distribution with parameters $\\Lambda$ and $q = 0.4$, and $\\Lambda$ has a probability function defined by $p(1) = p(2) = p(3) = p(4) = 0.25$. Provide your answer as a numerical value."
}
] |
math
|
{
"style": "rule",
"ground_truth": "0.8"
}
|
{
"index": 96,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$, the inequality $|P|^2 < |G|$ holds?"
}
] |
math
|
{
"style": "rule",
"ground_truth": "True"
}
|
{
"index": 97,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Calculate the limit: $$ \\lim_{n\\rightarrow\\infty}\\frac{2^n + n\\sin{n}}{\\log_2{n} + e^n} $$"
}
] |
math
|
{
"style": "rule",
"ground_truth": "0"
}
|
{
"index": 98,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Find the minimum value of the perimeter of a triangle whose area is 3 cmΒ²."
}
] |
math
|
{
"style": "rule",
"ground_truth": "6\\sqrt[4]{3}"
}
|
{
"index": 99,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
deepmath-103k
|
[
{
"role": "user",
"content": "Evaluate the surface integral \\( \\int \\int_S z \\, dS \\) where \\( S \\) is the surface of the hemisphere defined by \\( x^2 + y^2 + z^2 = a^2 \\) with \\( z \\geq 0 \\)."
}
] |
math
|
{
"style": "rule",
"ground_truth": "\\pi a^3"
}
|
{
"index": 100,
"original_dataset": "deepmath-103k",
"split": "train"
}
|
DeepMath-103K VERL
π Dataset Summary
This dataset contains 95,496 mathematical reasoning problems in VERL format, processed from zwhe99/DeepMath-103K.
Key Features:
- 95,496 high-quality math problems
- Converted to VERL format for reward modeling
- Verified ground truth answers
- Ready for reinforcement learning training
π Source Dataset
Original Repository
Repository: zwhe99/DeepMath-103K
License: MIT
Authors: Zhenwen He et al.
Dataset Description
DeepMath-103K is a curated collection of 103,000 mathematical problems covering various difficulty levels and mathematical domains. The dataset emphasizes step-by-step reasoning and verification.
π Preprocessing Pipeline
This dataset has been preprocessed and converted to the VERL (Verification and Reinforcement Learning) format for use in mathematical reasoning tasks with reward modeling.
Cleaning Methodology
Standard Processing:
- URL filtering (samples containing URLs removed)
- Format normalization to VERL schema
- Basic text cleaning and validation
Deduplication
Intra-dataset Deduplication:
- Method: SHA-256 hash-based with text normalization
- Before deduplication: 103,000 samples
- After deduplication: 95,496 samples
- Reduction: 7.3%
Inter-dataset Deduplication (v3.0):
- Priority level: 4
- Cross-dataset duplicates removed: 5,000
π‘ Preprocessing Examples
Example 1: Standard Processing (No Cleaning Preset)
Before Cleaning:
Calculate the derivative of $f(x) = x^2 + 3x + 2$ with respect to $x$.
After Cleaning:
Calculate the derivative of $f(x) = x^2 + 3x + 2$ with respect to $x$.
Changes Applied:
- β Format normalization to VERL schema
- β URL filtering (samples with URLs removed)
- β Basic text validation
- β No artifact removal applied
π VERL Schema
This dataset follows the standardized VERL (Verification and Reinforcement Learning) format:
{
"data_source": "openai/gsm8k",
"prompt": [
{
"content": "Calculate the sum of all odd numbers from 1 to 99.",
"role": "user"
}
],
"ability": "math",
"reward_model": {
"style": "rule",
"ground_truth": "\boxed{2500}",
"hash": "sha256:abc123..."
},
"extra_info": {
"split": "train"
}
}
Field Descriptions
| Field | Type | Description |
|---|---|---|
data_source |
string |
Original dataset identifier (e.g., openai/gsm8k, numina_aime) |
prompt |
list[dict] |
User query in chat format with role and content |
ability |
string |
Task type (always "math" for this dataset) |
reward_model.style |
string |
Reward computation method ("rule" for rule-based verification) |
reward_model.ground_truth |
string |
Expected answer for verification (often in \boxed{} format) |
reward_model.hash |
string |
SHA-256 hash of prompt content for deduplication |
extra_info.split |
string |
Original split identifier ("train", "test", etc.) |
π Dataset Statistics
Sample Distribution
- Total Samples: 95,496
- Dataset Size: 10.5 MB
- Average Problem Length: N/A characters
Data Sources
Distribution of problems by original data source:
| Source | Count | Percentage |
|---|---|---|
| Mixed Sources | 95,496 | 100% |
Note: Detailed source distribution statistics will be added in future updates.
π Usage
Loading the Dataset
from datasets import load_dataset
# Load the full dataset
dataset = load_dataset("sungyub/deepmath-103k-verl")
# Load with streaming (recommended for large datasets)
dataset = load_dataset("sungyub/deepmath-103k-verl", streaming=True)
# Preview first few examples
for example in dataset['train'].take(5):
print(example['prompt'][0]['content']) # User question
print(example['reward_model']['ground_truth']) # Answer
print("---")
Using with VERL
from datatrove.utils.reward_score import compute_score
# Compute reward score for a generated solution
score = compute_score(
data_source=example['data_source'],
solution_str=generated_solution,
ground_truth=example['reward_model']['ground_truth'],
format_type="auto" # Auto-detect XML or GPT OSS format
)
print(f"Reward score: {score}")
Integration with DataTrove
from datatrove.pipeline.readers import ParquetReader
from datatrove.pipeline.filters import LambdaFilter
from datatrove.executor import LocalPipelineExecutor
pipeline = [
ParquetReader("sungyub/deepmath-103k-verl", text_key="prompt"),
LambdaFilter(lambda doc: len(doc.text) > 100), # Filter short problems
# Add more processing steps...
]
executor = LocalPipelineExecutor(pipeline=pipeline, tasks=4)
executor.run()
π Citation
Original Dataset
@article{he2025deepmath,
title = {DeepMath: Deep Learning for Mathematical Reasoning},
author = {He, Zhenwen and others},
journal = {arXiv preprint arXiv:2504.11456},
year = {2025},
url = {https://arxiv.org/abs/2504.11456}
}
This Processed Version
@dataset{sungyub_math_verl_deepmath-103k-verl,
author = {Sungyub Kim},
title = DeepMath-103K VERL,
year = {2025},
publisher = {Hugging Face},
howpublished = {\url{https://huggingface.co/datasets/sungyub/deepmath-103k-verl}}
}
βοΈ License
- This processed dataset: mit
- Original dataset: MIT
π Acknowledgments
This dataset was processed using the DataTrove library.
Credits:
- Original dataset authors: Zhenwen He et al.
- Processing and VERL conversion: Sungyub Kim
- MathDatasetCleaner implementation: DataTrove contributors
Special thanks to: Zhenwen He and collaborators for the DeepMath dataset
π Version History
v1.0.0 (Initial Release)
- Processed 95,496 samples from zwhe99/DeepMath-103K
- Converted to VERL format
- Standard processing applied
- Ready for reinforcement learning training
π Related Resources
- Unified Collection: sungyub/math-verl-unified - All 9 math datasets with inter-dataset deduplication
- DataTrove Documentation: https://github.com/huggingface/datatrove
- VERL Format Specification: See VERL Schema section above
Questions or issues? Open an issue on the DataTrove GitHub repository
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