prompt list | ground_truth string |
|---|---|
[
{
"content": "Given that \\(a < b\\) and \\(\\frac{f(b)/b}{f(a)/a}<2\\), find a tighter upper bound for the expression \\(\\frac{a}{b}+\\frac{b}{a}\\).",
"role": "user"
}
] | \dfrac{5}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.27% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.93e-3 |
[
{
"content": "Can the Gaussian integral be evaluated using complex analysis techniques, such as contour integrals?",
"role": "user"
}
] | \sqrt{\pi} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9.43e-2 N |
[
{
"content": "Does the limit \\( \\lim_{x\\rightarrow \\infty} f(x) \\) exist for a function \\( f:[0,\\infty) \\rightarrow \\mathbb{R} \\), given that for any positive \\( a \\), the sequence \\( f(an) \\) converges to zero?",
"role": "user"
}
] | No |
[
{
"content": "Let $a$ be the greatest natural number such that $4^{2016} + 4^a + 4^{37}$ is the square of a natural number. Determine $a \\pmod{37}$. ",
"role": "user"
}
] | 35 |
[
{
"content": "Determine the upper bound for the length of the shortest word that can distinguish two non-equivalent NFAs, one with $n$ states and the other with $k$ states.",
"role": "user"
}
] | 2^{n + k} - 1 |
[
{
"content": "Determine the value of \\( t \\) for which the series \\( \\sum _ { n = 1 } ^ { \\infty } \\frac { n ^ { 3 n } } { ( n + 2 ) ^ { 2 n + t } ( n + t ) ^ { n + 2 t } } \\) is convergent.",
"role": "user"
}
] | t > \dfrac{1}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Net income $1,310 |
[
{
"content": "In triangle \\(ABC\\), where \\(AB = 13\\), \\(BC = 14\\), and \\(CA = 15\\), let \\(D\\) be the foot of the altitude from \\(A\\) to \\(BC\\). The inscribed circles of triangles \\(ABD\\) and \\(ACD\\) are tangent to \\(AD\\) at points \\(P\\) and \\(Q\\), respectively, and to \\(BC\\) at points ... | \dfrac{25}{4} |
[
{
"content": "Consider two real-valued sequences \\(\\{x_n\\}\\) and \\(\\{y_n\\}\\) satisfying the condition \\(x_n^3 - y_n^3 \\rightarrow 0\\) as \\(n \\rightarrow \\infty\\). Determine which of the following statements is true:\n\n(A) \\(x_n - y_n \\rightarrow 0\\)\n\n(B) \\(x_n - y_n \\rightarrow 0\\) only ... | A |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to 0} [x - x^2 \\ln(1+\\frac{1}{x})] \\) using the Taylor series expansion of the function \\( f(x) = x^2 \\ln(1+\\frac{1}{x}) \\) at \\( x = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Method C and Method D |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $13,698,630.14 |
[
{
"content": "Let $R=\\mathbb{R}[x]$ be the ring of polynomials and $I=(x^2+1)$, $J=(x+1)$, and $K=(x^3+x^2+x+1)$ be ideals of $R$. There is a ring isomorphism between $R/K$ and $R/I \\times R/J$. Find the element in $R/K$ whose image in $R/I \\times R/J$ is $(2x+1+I,3+J)$.",
"role": "user"
}
] | 2x^2 + 2x + 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $40 |
[
{
"content": "Find the remainder when $50!$ is divided by $10^{12}$, expressed as $N \\equiv a \\pmod{10}$, where $a$ is the remainder.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the following statement is true or false: Every infinite subset of the subgroup of permutations of an infinite set that move only a finite number of elements must contain an even permutation.",
"role": "user"
}
] | False |
[
{
"content": "Using Rouche's theorem, determine the number of roots of the polynomial \\( p(z) = 2z^4 - 2z^3 + 2z^2 + 7 \\) that lie inside the region \\( \\{z \\in \\mathbb{C} : |z| < 2\\} \\).",
"role": "user"
}
] | 4 |
[
{
"content": "A regular triangular prism \\( A B C A_{1} B_{1} C_{1} \\) is inscribed in a sphere. The base of the prism is \\( A B C \\), and the lateral edges are \\( A A_{1}, B B_{1}, C C_{1} \\). The segment \\( C D \\) is a diameter of this sphere, and the point \\( K \\) is the midpoint of the edge \\( A ... | 36 |
[
{
"content": "Determine whether the statement \"If \\(A = PBP^{-1}\\), then the spectral norm is the same for similar matrices\" is true or false.",
"role": "user"
}
] | B |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -9.65e13 N/C |
[
{
"content": "Evaluate the ratio of the following two definite integrals:\n\\[ \\frac{\\int _0 ^\\pi x^3 \\ln (\\sin x) \\, \\mathrm{d} x}{\\int _0 ^\\pi x^2 \\ln(\\sqrt{2} \\sin x) \\, \\mathrm{d} x} \\]",
"role": "user"
}
] | \dfrac{3\pi}{2} |
[
{
"content": "For any positive integer $n$, let $S_n$ denote the set of positive integers that cannot be expressed in the form $an + 2017b$ for nonnegative integers $a$ and $b$. Define $A_n$ as the average of the elements of $S_n$ if $S_n$ is non-empty and finite, and $0$ otherwise. Compute \\[ \\left\\lfloor \... | 1007 |
[
{
"content": "Determine whether the following statement is true for a bounded differentiable function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$: For any $\\varepsilon > 0$, there exists an $x \\in \\mathbb{R}$ such that $|f'(x)| < \\varepsilon$. Provide a justification for your answer.",
"role": "user"
}
... | True |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Hydrogen Burning, a fusion reaction where hydrogen nuclei fuse to form helium, releasing energy. |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.119 |
[
{
"content": "Let $E$ be an arbitrary closed subset of $\\mathbb{R}$. Is it possible to construct a function $f: \\mathbb{R} \\to \\mathbb{R}$ such that the set of all points where $f$ is not bounded is precisely $E$?",
"role": "user"
}
] | Yes |
[
{
"content": "Is there a function $f(x)$ defined on the interval $(a, b)$ such that $f'(c_i) = 0$ for an infinite ordered set of points $\\{c_i\\}$ where $c_i \\in (a, b)$, and for each pair of consecutive points $c_i$ and $c_{i+1}$, there exists at least one point $d \\in (c_i, c_{i+1})$ where $f'(d) \\neq 0$?... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 98 g/mol |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 20 units |
[
{
"content": "Find the minimum value of the expression \\( x^2 + 4xy + 4y^2 + 2z^2 \\) given that \\( x, y, \\) and \\( z \\) are positive real numbers satisfying \\( xyz = 32 \\). Use the AM-GM inequality and algebraic manipulation to solve the problem.",
"role": "user"
}
] | 96 |
[
{
"content": "Determine the total number of linear functions from \\( \\{0, 1\\}^n \\) to \\( \\{0, 1\\} \\).",
"role": "user"
}
] | 2^n |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 439247 |
[
{
"content": "Let $G = C_6 = \\langle g \\mid g^6 = 1 \\rangle$ and let $V_1, V_2$ be two $\\mathbb{C}G$-modules with bases $\\mathcal{B}_1 = \\{v_1, v_2, v_3\\}$ and $\\mathcal{B}_2 = \\{w_1, w_2\\}$ respectively. The action of $G$ on $V_1$ and $V_2$ is given by $[g]_{V_1} = \\begin{pmatrix}0 & 1 & 0\\\\ 0 & 0... | \{\} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 556 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 23.23 mL |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $599.441 |
[
{
"content": "Find the value of \\( \\lambda > 0 \\) such that the function \\( f(x) = \\begin{cases} 0 & x = 0 \\\\ |x|^{\\lambda} \\sin\\left(\\frac{1}{x}\\right) & x \\neq 0 \\end{cases} \\) is differentiable at zero and has a continuous derivative at zero.",
"role": "user"
}
] | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.31e-3 |
[
{
"content": "Determine if any general symmetric \\((n \\times n)\\) matrix can be expressed as a sum of products of the form \\(M=\\sum_{i=1}^k j_iv_iv_i^T\\), where \\(j_i\\) are scalars and \\(v_i\\) are \\((n \\times 1)\\) vectors. If possible, find the minimum number of vectors \\(k\\) required.",
"rol... | n |
[
{
"content": "Solve the equation $2^x = -2x + 11$ for $x$.",
"role": "user"
}
] | 2.56 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $33,000 |
[
{
"content": "Find the five-digit number that, when divided by each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13, results in different remainders.",
"role": "user"
}
] | 83159 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $312,298.52 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 8.94427 ft/s |
[
{
"content": "Let $X$ be a connected subset of a metric space $M$ where for any $y, z \\in X$, the distance $d(y,z) = 2$. Must there exist an $x \\in X$ such that $d(x,z) = 1$ for some $z \\in X$?",
"role": "user"
}
] | No |
[
{
"content": "Find the 20th derivative of the function $\\arcsin(x)$ at $x = 0$. Provide your answer as a single number.",
"role": "user"
}
] | 0 |
[
{
"content": "Find all values of $k$ so that the graphs of $x^2 + y^2 = 4 + 12x + 6y$ and $x^2 + y^2 = k + 4x + 12y$ intersect. Enter your answer using interval notation.",
"role": "user"
}
] | [-36,104] |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 93.29% |
[
{
"content": "Determine the values of $p$ for which the improper integral \\( \\int_1^{\\infty} x^p \\cos x \\, \\mathrm{d}x \\) converges.",
"role": "user"
}
] | p < 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 73*10^9J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 236 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | (7, -8) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Counterclockwise |
[
{
"content": "Given the sequence \\(a_{n}\\) defined as the solution to the equation \\(x^{n} = \\cos x\\) for \\(x > 0\\), find the value of \\(\\lim_{n\\rightarrow\\infty}n\\left ( 1- a_{n} \\right )\\).",
"role": "user"
}
] | -\ln(\cos 1) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 369 mL |
[
{
"content": "Find the covariance of the random variables $XZ$ and $Z$, where $X \\sim R(0,1)$ and $Z \\sim b(1,1/2)$. Express your answer as a single numerical value.",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 134.4g |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.905 m |
[
{
"content": "Do commutators, defined as \\([X, Y] = XY - YX\\), satisfy the Jacobi identity?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $K$ be an algebraically closed field of characteristic zero, equipped with an involution $x \\mapsto \\overline{x}$, an order-2 field automorphism. The fixed field $F := \\{x \\in K: \\overline{x} = x\\}$ is a proper subfield of $K$. Is $F$ always real-closed for any such involution?",
"ro... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.22 s |
[
{
"content": "Simply the expression\n\\[\\frac{(\\sqrt{2} - 1)^{1 - \\sqrt{3}}}{(\\sqrt{2} + 1)^{1 + \\sqrt{3}}},\\]writing your answer as $a - b \\sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of a prime.",
"role": "user"
}
] | 3 - 2 \sqrt{2} |
[
{
"content": "Determine whether the sequence \\( \\left\\{ \\left(\\frac{\\ln(n+1)}{\\ln n}\\right)^n \\right\\} \\) is convergent. If it is, find its limit.",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 499 Hz |
[
{
"content": "Let \\( A = 2012^{2012} \\). Define \\( B \\) as the sum of the digits of \\( A \\), \\( C \\) as the sum of the digits of \\( B \\), and \\( D \\) as the sum of the digits of \\( C \\). Determine the value of \\( D \\).",
"role": "user"
}
] | 7 |
[
{
"content": "Calculate the relative homology group $H_2(D^2,S^1)$ using the long exact sequence.",
"role": "user"
}
] | \mathbb{Z} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2644.9kg |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9.85206 |
[
{
"content": "Determine whether the equation $n+1 = x^{\\varphi(n)}$ has a solution for every positive integer $n$, where $\\varphi(n)$ is Euler's totient function.",
"role": "user"
}
] | No |
[
{
"content": "Let $q$ be a prime power and $n$ be a positive integer. Consider $U$, an $\\mathbb{F}_q$-vector subspace of the finite field $\\mathbb{F}_{q^n}$ with dimension $n-1$. Let $V$ be the kernel of the trace map $Tr: \\mathbb{F}_{q^n} \\rightarrow \\mathbb{F}_q$, which is also a vector subspace of dimen... | Yes |
[
{
"content": "Suppose the series \\( \\sum_{n=1}^{\\infty} \\frac{a_n}{n} \\) converges to a finite number \\( a \\), where \\( a_n \\geq 0 \\) for all positive integers \\( n \\). Determine if the limit \\( \\lim_{n \\to \\infty} \\sum_{i=1}^{n} \\frac{a_i}{i+n} \\) converges to 0.",
"role": "user"
}
] | 0 |
[
{
"content": "Given the current ELO score, the opponent's ELO score, the result of the match (win or loss), and the constant K, determine the previous ELO score. Use the following equations based on the match result:\n\nFor a win:\n\\[ ELO_{Current} = ELO_{Previous} + K \\cdot \\left(1 - \\frac{1}{1 + 10^{\\fra... | ELO_{Previous} |
[
{
"content": "Find the limit of the sequence \\(x_n=\\frac{1}{n}(n+\\frac{n-1}{2}+\\frac{n-2}{3}+...+\\frac{2}{n-1}+\\frac{1}{n}-\\log(n!))\\) as \\(n\\) approaches infinity, using asymptotic equivalences.",
"role": "user"
}
] | \gamma |
[
{
"content": "Calculate the value of \\( q(q(q(2000^{2000}))) \\), where \\( q(n) \\) is the sum of the digits of the natural number \\( n \\).",
"role": "user"
}
] | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -3.0 \mu C |
[
{
"content": "Mary and her brothers play a game called \"all or one\" to decide who will do the dishes. The game is played by at least three people, and in each round, each player chooses a natural number simultaneously. A player wins if they choose a certain number and all others choose the same different numb... | 2 |
[
{
"content": "Calculate the force exerted by water against a dam with a cross-section in the shape of an isosceles trapezoid. The density of water is \\( \\rho = 1000 \\, \\text{kg/m}^3 \\), and the acceleration due to gravity is \\( g = 10 \\, \\text{m/s}^2 \\). The pressure at depth \\( x \\) is given by \\( ... | 333000 |
[
{
"content": "Consider a linear transformation $T : V \\longrightarrow V$ with distinct eigenvalues $r_1, r_2, ..., r_m$ and corresponding eigenvectors $v_1, v_2, ..., v_m$. Determine if any of the eigenvalues $r_1, r_2, ..., r_m$ can be equal to 0.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the limit of the sequence \\(b_n = \\int_{n+\\sqrt{n}}^{2n}\\frac{\\cos x}{x}\\, dx\\).",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.78% |
[
{
"content": "Find the last three digits of \\(23^{320}\\).",
"role": "user"
}
] | 201 |
[
{
"content": "Let $n$ be the smallest positive integer such that $mn$ is a perfect $k$th power of an integer for some $k \\ge 2$, where $m=2^{1980} \\cdot 3^{384} \\cdot 5^{1694} \\cdot 7^{343}$. What is $n+k$?",
"role": "user"
}
] | 13 |
[
{
"content": "Find the inverse Laplace transform of \\( \\frac{3}{(s-1)^2} \\).",
"role": "user"
}
] | 3te^t |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9.10 |
[
{
"content": "Evaluate the infinite series \\( \\sum_{x=0}^\\infty P(X>x) \\), where \\( X \\) is a discrete random variable defined on the non-negative integers (0, 1, 2, 3, ...).",
"role": "user"
}
] | E[X] |
[
{
"content": "Prove that 26 is the only number that satisfies the equation \\(x^2 + 1 = z = y^3 - 1\\).",
"role": "user"
}
] | 26 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $93,600 |
[
{
"content": "Determine the cardinality of the set \\( N=\\{(x,y)\\in\\mathbb{R}^+\\times\\mathbb{R}^+\\mid (x\\cdot y\\in\\mathbb{N})\\text{ and }(y^2\\in\\mathbb{N})\\} \\).",
"role": "user"
}
] | \aleph_0 |
[
{
"content": "Let\n\\[\\sum_{n = 0}^{123456789} \\frac{3n^2 + 9n + 7}{(n^2 + 3n + 2)^3} = \\frac{a}{b},\\]where $a$ and $b$ are relatively prime positive integers. Find $b - a.$",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.135g/m |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 246 N |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 322.07 N |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5 years |
[
{
"content": "In a full binary tree, determine the difference between the sum of the depths of all leaves and the sum of the depths of all internal nodes, expressed in terms of the number of internal nodes.",
"role": "user"
}
] | 2n |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.48 ng |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.85 m/s |
[
{
"content": "Determine the number of non-isomorphic quotient groups of the symmetric group $S_4$.",
"role": "user"
}
] | 4 |
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