prompt list | ground_truth string |
|---|---|
[
{
"content": "Suppose $\\{X_n\\}$ and $\\{Y_n\\}$ converge in probability to $X$ and $Y$, respectively. Does $X_n Y_n$ converge in probability to $X Y$?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $X$ be an exponential random variable with parameter $\\lambda=4$ and $Y$ be an exponential random variable with parameter $\\lambda=5$. $X$ and $Y$ are independent. Find the probability that $3X < 2Y$. Express your answer as a single probability value.",
"role": "user"
}
] | \dfrac{8}{23} |
[
{
"content": "Let \\( \\{x_k\\}_{k\\in\\mathbb{N}} \\) be a sequence of i.i.d. random variables with \\( x_k\\sim U_{[[ 1,n]]} \\) for all \\( k\\in\\mathbb{N} \\), where \\( U_{[[1,n]]} \\) is the uniform distribution on the discrete set \\( [[1,n]]:=\\{1,2,\\ldots,n\\} \\). Fix \\( a\\in[[1,n]] \\). What is t... | 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... | 142.52mH |
[
{
"content": "Five people are passing a ball amongst themselves, starting with Alonzo. Each person who has the ball passes it to someone else. After the eighth pass, the ball must return to Alonzo. How many possible sequences of passes are there?",
"role": "user"
}
] | 13108 |
[
{
"content": "Determine the number of pairs of vectors \\( \\vec{V}_1 = a_1 \\mathbf{i} + b_1 \\mathbf{j} + c_1 \\mathbf{k} \\) and \\( \\vec{V}_2 = a_2 \\mathbf{i} + b_2 \\mathbf{j} + c_2 \\mathbf{k} \\) such that \\( \\vec{V}_1 \\) and \\( \\vec{V}_2 \\) are perpendicular, where all components \\( a, b, c \\)... | 288 |
[
{
"content": "Two irrational numbers in the interval $(0,1)$ are chosen at random, and their decimal parts are considered up to 2014 places. The probability that the smaller number can be subtracted from the larger one without borrowing is expressed as $p^k$, where $p$ is a rational number and $k > 1000$ is a p... | 455 |
[
{
"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... | 1.20e29 m |
[
{
"content": "Define $\\#N$ by the formula $\\#N = .5(N) + 1$. Calculate $\\#(\\#(\\#58))$.",
"role": "user"
}
] | 9 |
[
{
"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... | 5.762 J/K |
[
{
"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... | 1.235*P_atm |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to \\infty} \\int_x^{x+1} \\frac{t^2+1}{t^2+20t+8}\\, dt \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Find the maximum value of the expression \\( \\frac{1}{\\sin^2 \\theta + 3\\sin\\theta \\cos\\theta + 5\\cos^2 \\theta} \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Given an algebraically closed field $F$ and a polynomial $p \\in F[X_1, ..., X_n]$ that is irreducible, determine if $p$ remains irreducible in $K[X_1, ..., X_n]$, where $K$ is a transcendental field extension of $F$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty}\\left(\\left(1-\\frac{1}{2^2}\\right)\\left(1-\\frac{1}{3^2}\\right)\\cdots\\left(1-\\frac{1}{n^2}\\right)\\right)$$",
"role": "user"
}
] | \dfrac{1}{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... | $311,424.02 |
[
{
"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... | $8,947.63 |
[
{
"content": "Suppose you have $n$ red balls, $n$ green balls, and an arbitrary number of white balls. Each ball is selected into a set with a probability of $p = \\frac{1}{n}$ and excluded with a probability of $1 - p = 1 - \\frac{1}{n}$, independently of one another. Determine a constant lower bound (independ... | \dfrac{3}{16} |
[
{
"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 g O2/ 1 g S |
[
{
"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... | 260 L |
[
{
"content": "Evaluate the limit: \\[ I = \\lim_{x\\to 0}\\dfrac{\\displaystyle\\int_{x}^{x^2}e^{x\\sqrt{1-y^2}}dy}{\\arctan{x}} \\]",
"role": "user"
}
] | -1 |
[
{
"content": "In a simple symmetric random walk (SSRW) on the 1D integer lattice, does the walk hit every integer infinitely many times with probability one?",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the limit of the sequence as \\( n \\to \\infty \\):\n\\[\n\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{\\left(n^{3}+1\\right)\\left(n^{2}+3\\right)}-\\sqrt{n\\left(n^{4}+2\\right)}}{2 \\sqrt{n}}\n\\]",
"role": "user"
}
] | \dfrac{3}{4} |
[
{
"content": "Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$.",
"role": "user"
}
] | \left(\frac{1}{2},\frac{2}{3}\right) |
[
{
"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... | MR = a + 2bQ |
[
{
"content": "Evaluate the limit of the improper integral: $$ \\lim_{n \\to \\infty}\\int_{0}^{n}x\\,\\sin\\left(\\frac{1}{nx}\\right)\\, \\frac{\\mathrm{d}x}{\\sqrt{x^{2} + 1}} $$",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether there exist two distinct number fields $E$ and $F$ such that for every prime $p$, the tensor products of their rings of integers with $\\mathbb{F}_p$ are isomorphic, i.e., $\\mathcal{O}_E \\otimes \\mathbb{F}_p \\cong \\mathcal{O}_F \\otimes \\mathbb{F}_p$. Provide a justificatio... | 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... | $63,986.43 |
[
{
"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... | 7.95 g |
[
{
"content": "Determine the smallest odd natural number \\( a > 1 \\) such that \\( \\int_{10}^{19} \\frac{\\sin x}{1+x^a} \\, dx < \\frac{1}{9} \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Suppose $A \\subset \\mathbb{R}^n$ has a positive $n$-dimensional Lebesgue measure, i.e., $L^n(A) > 0$. Does the closure of $A$, denoted by $\\overline{A}$, necessarily contain an open set? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Let $A$ be a commutative ring with unit and $M$ a module over $A$. Suppose the rank of $M_{P}$ over $A_{P}$ is constant for every $P \\in \\mathrm{Spec}(A)$. Does this property hold for $M_{Q}$ over $A_{Q}$ for any prime ideal $Q$ of $A$?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate \\( \\lim_{(x,y) \\to (0,0)} x \\ln \\sqrt{x^2+y^2} \\).",
"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... | 9.77 |
[
{
"content": "Calculate the surface integral of the vector field \\( \\mathbf{G} = \\begin{pmatrix} y \\\\ z \\\\ x \\end{pmatrix} \\) over a sphere \\( S \\) using the divergence theorem.",
"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... | 2702.73 hp |
[
{
"content": "Find the sum of the series \\(\\sum\\limits_{n=1}^\\infty \\frac{n^3}{n!}\\) by demonstrating the convergence of the series using the definition that a series converges if its sequence of partial sums converges.",
"role": "user"
}
] | 5e |
[
{
"content": "Find the value of \\( k \\) such that \\( \\int^{\\pi}_0 x f (\\sin x) \\, dx = k \\int^{\\pi/2}_0 f (\\sin x) \\, dx \\).",
"role": "user"
}
] | \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... | 905971.26 |
[
{
"content": "Given the function \\( H(s) = \\frac{\\zeta(s)}{\\phi(q)} \\sum_{\\chi \\mod{q}} L(s,\\chi) = \\sum_{n=1}^{\\infty} \\frac{h(n)}{n^s} \\), determine the smallest integer \\( n \\) as a function of \\( q \\) such that \\( h(n) \\neq 1 \\).",
"role": "user"
}
] | q + 1 |
[
{
"content": "Determine whether \\( \\lfloor \\sqrt{n!} \\rfloor \\) can be a prime number for any integer \\( n > 1 \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Solve for \\(x\\) in the equation: \\[ \\sqrt{x + \\sqrt{4x + \\sqrt{16x + \\sqrt{64x + 5}}}} - \\sqrt{x} = 1 \\]",
"role": "user"
}
] | \dfrac{1}{16} |
[
{
"content": "A unit square is rotated $45^\\circ$ about its center. What is the area of the region swept out by the interior of the square?",
"role": "user"
}
] | \frac \pi4 + 2 - \sqrt2 |
[
{
"content": "Find the Fourier transform of the function \\( \\delta(x)f(x) \\).",
"role": "user"
}
] | f(0) |
[
{
"content": "Find the inverse of the polynomial \\(x^2 + x + 1\\) in the field \\(\\mathbb{Z}_2[x]/\\langle x^3 + x + 1 \\rangle\\).",
"role": "user"
}
] | x^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... | 0.344 moles |
[
{
"content": "Consider the set \\( S = \\{T:\\mathbb{R}^3\\to\\mathbb{R}^3:T\\text{ is a linear transformation with }T(1,0,1)=(1,2,3),T(1,2,3)=(1,0,1)\\} \\). Determine the nature of the set \\( S \\):\n\n(a) a singleton set\n(b) a finite set containing more than one element\n(c) a countably finite set\n(d) an ... | 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... | 1.52e-3 M |
[
{
"content": "Find the value of the limit: \\[ \\lim_{n\\to \\infty}\\frac{(2n)!}{4^n(n!)^2} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Compute the value of the infinite series \\( \\sum\\limits_{n=0}^\\infty \\frac{n^2}{n!} \\).",
"role": "user"
}
] | 2e |
[
{
"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... | 9.41% |
[
{
"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... | 9.69% |
[
{
"content": "Find the area of the region $R$ defined by the inequalities $x^2 + y^2 \\leq 100$ and $\\sin(x + y) \\geq 0$. Express your answer in terms of $\\pi$ if necessary.",
"role": "user"
}
] | 50\pi |
[
{
"content": "Consider the equation \\(x^{2}+y^{2}-3z^{2}-3t^{2}=0\\). Determine the total number of integral solutions for this equation where \\(1 \\leq x, y, z, t \\leq 10000\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Compute $\\cos 150^\\circ$.",
"role": "user"
}
] | -\frac{\sqrt{3}}{2} |
[
{
"content": "Compute the value of \\(1234^{1234} \\pmod{5379}\\) given that \\(5379 = 3 \\times 11 \\times 163\\). Do not use Euler's Theorem.",
"role": "user"
}
] | 4603 |
[
{
"content": "Let \\(\\left( x_{n} \\right)_{n \\geq 1}\\) be a sequence defined by:\n\\[ x_{1} = \\frac{1}{2014} \\quad \\text{and} \\quad x_{n+1} = x_{n} + x_{n}^{2}, \\quad \\forall n \\geq 1 \\]\nCompute the integer part of the sum:\n\\[ S = \\frac{x_1}{x_2} + \\frac{x_2}{x_3} + \\cdots + \\frac{x_{2014}}{x... | 2013 |
[
{
"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... | 61000 |
[
{
"content": "Find the last two digits of \\(9^{9^9}\\).",
"role": "user"
}
] | 89 |
[
{
"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... | 2.14 N |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi} f(e^{it}) \\cos t \\, dt \\) where \\( f(z) \\) is an analytic function.",
"role": "user"
}
] | \pi f'(0) |
[
{
"content": "If $R$ is a finite subring of a division ring $D$, is $R$ necessarily a division ring?",
"role": "user"
}
] | 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... | Trough (contractionary gap) |
[
{
"content": "Let \\( A \\) and \\( B \\) be \\( n \\times n \\) matrices. Which of the following is equal to \\( \\text{trace}(A^2 B^2) \\)?\n\n(i) \\( (\\text{trace}(AB))^2 \\)\n(ii) \\( \\text{trace}(AB^2 A) \\)\n(iii) \\( \\text{trace}((AB)^2) \\)\n(iv) \\( \\text{trace}(BABA) \\)",
"role": "user"
}
] | ii |
[
{
"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... | The force exerted by the ground on each rear wheel is: 2982.56 N
The force exerted by the ground on each front wheel is: 4129.670 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... | 33920 nm |
[
{
"content": "Find the value of $\\log_{p^5}\\left(3^5\\times 5^3\\right)$ given the equations:\n\n$$\\frac{3+\\log_{p}3}{5^2}=\\frac{-5+\\log_{p}5}{3^2}=\\frac{10}{3^3+5^3}$$",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the limit of the sequence defined by successive compositions of the natural logarithm function, starting with a positive number greater than 1. Does this sequence converge to zero as the number of compositions approaches infinity?",
"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... | $k^2$ |
[
{
"content": "Given the sequences \\(y_t\\) and \\(z_t\\) defined by the equations:\n\\[ y_t = 6y_{t-1} - 20z_{t-1} \\]\n\\[ z_t = -17y_{t-1} - 12z_{t-1} \\]\nDerive a second-order difference equation of the form:\n\\[ y_{t+1} + ny_t + ry_{t-1} = 0 \\]\nwhere \\(n\\) and \\(r\\) are constants. The auxiliary equ... | 17.52 |
[
{
"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... | 877.9 m at 68.7 degrees South of West |
[
{
"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... | 85 N/m |
[
{
"content": "Find the exact sum of the series \\( \\sum_{n=1}^{\\infty}\\frac{1}{2^n n} \\).",
"role": "user"
}
] | \ln 2 |
[
{
"content": "Determine the number of integer values of $a$ for which there exists at least one complex number $z$ that satisfies both $|z-ai|=a+4$ and $|z-2|<1$. Provide your answer as a single integer.",
"role": "user"
}
] | 2 |
[
{
"content": "Given real numbers $x$ and $y$ such that \\(5x^2 + 5y^2 - 6xy = 8,\\) find the maximum value of \\(|x - y|\\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Determine whether a $6 \\times 6$ matrix $\\mathbf{A}$ can satisfy the equation $\\mathbf{A}^4 = -I$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the limit: \\[ \\lim_{x\\to0}\\left(\\frac{1}{x^5}\\int_0^x e^{-t^2}\\,dt-\\frac{1}{x^4}+\\frac{1}{3x^2}\\right) \\]",
"role": "user"
}
] | \dfrac{1}{10} |
[
{
"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... | 3.5 s |
[
{
"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... | 280.9 kJ/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... | $125,000 |
[
{
"content": "What is the value of $y$ such that $\\lceil y\\rceil+y=15.5?$",
"role": "user"
}
] | 7.5 |
[
{
"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... | 693333.41 |
[
{
"content": "Let $f(x)$ be a polynomial on $\\mathbb{R}$ such that $f(2x) = f'(x)f''(x)$. Find the value of $f(3)$. \\(\\text{Note: } f'(x) \\text{ and } f''(x) \\text{ are the first and second derivatives of } f(x).\\)",
"role": "user"
}
] | 12 |
[
{
"content": "Determine for which values of \\( p > 0 \\) the integral \\( \\int_0^1 \\sin(x^{-p})\\, dx \\) 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... | 188000 |
[
{
"content": "Find the limit of the function \\( f(x,y) = \\frac{x \\cdot \\sin\\left(\\frac{y}{\\sqrt{x}}\\right)}{\\sqrt{x^2 + y^2}} \\) as \\((x,y) \\rightarrow (0,0)\\), given that \\(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... | 10.78% |
[
{
"content": "In an isosceles trapezoid $\\square ABCD$ with $AD$ as the longer base, point $P$ lies inside the trapezoid and bisects angles $A$, $B$, $C$, and $D$. Given that $PA = 3$ and $\\angle APD = 120^\\circ$, find the area of the trapezoid. Note: Connect the midpoints of $AB$ and $CD$ to form an equilat... | 6\sqrt{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... | 14.8 s |
[
{
"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... | -10.72 degrees Celsius |
[
{
"content": "Let $k$ be a non-algebraically closed field and $I \\subset k[x_1, \\dots, x_n]$ be a maximal ideal. Is the set $V_{\\bar{k}}(I) = \\{x \\in \\bar{k}^n \\mid \\forall f \\in I, f(x) = 0\\}$ necessarily finite?",
"role": "user"
}
] | 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... | $272,726 |
[
{
"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... | 19.06 s |
[
{
"content": "Find all real values of $x$ that satisfy $\\frac{x(x+1)}{(x-4)^2} \\ge 12.$ (Give your answer in interval notation.)",
"role": "user"
}
] | [3, 4) \cup \left(4, \frac{64}{11}\right] |
[
{
"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... | 1.75 mol/L |
[
{
"content": "In 2003, the average monthly rainfall in Mathborough was $41.5\\text{ mm.}$ In 2004, the average monthly rainfall in Mathborough was $2\\text{ mm}$ more than in 2003. What was the total amount of rain that fell in Mathborough in 2004?",
"role": "user"
}
] | 522 |
[
{
"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... | 1.81e-17 J |
[
{
"content": "Let $f(x,y)$ be a polynomial in $x$ and $y$ satisfying the following conditions: \\( (i) \\) $f(1,2) = 2$ and \\( (ii) \\) $yf(x,f(x,y)) = xf(f(x,y), y) = (f(x,y))^2$. Determine all such polynomials $f(x,y)$. ",
"role": "user"
}
] | xy |
[
{
"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... | -1.9 cm |
[
{
"content": "Find the minimum positive integer \\( n \\geq 4 \\) such that among \\( n \\) people, each playing a table tennis game with every other person (where each game has a winner), there always exists an ordered group of four people \\((a_1, a_2, a_3, a_4)\\) such that \\( a_i \\) wins against \\( a_j \... | 8 |
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