prompt list | ground_truth string |
|---|---|
[
{
"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... | $82.03 |
[
{
"content": "Let \\(X\\) be a continuous random variable with the probability density function \\(f(x)= \\frac{1}{(2+x^2)^{3/2}}\\) for \\(-\\infty<x<\\infty\\). Determine \\(E(X^2)\\). Choose the correct option:\n(a) \\(0\\) \n(b) \\(1\\) \n(c) \\(2\\) \n(d) does not exist.",
"role": "user"
}
] | D |
[
{
"content": "Calculate the degree of the map \\( S: \\mathbb{R}^{m+n} \\rightarrow \\mathbb{R}^{m+n} \\) defined by \\( S(x, y) = (y, x) \\) using the oriented bases of the tangent spaces.",
"role": "user"
}
] | (-1)^{mn} |
[
{
"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... | 67108.8 J |
[
{
"content": "In a game played inside the unit disc D, Player A chooses a point p0 ∈ D. At each step n, Player B draws a line ℓn through pn−1, and Player A chooses a point pn ∈ D ∩ ℓn. Player B wins if they can force the sequence (pn) to converge. If Players A and B play randomly, what is the probability that P... | 0 |
[
{
"content": "Calculate the limit: $$\\lim_{x \\to \\frac{\\pi}{6}}{(1-2\\sin(x))}^{\\tan(\\frac{\\pi}{6}-x)}$$",
"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... | 716kJ |
[
{
"content": "Determine the degree \\([K_p : \\mathbb{Q}_p]\\) for the number field \\(K = \\mathbb{Q}(\\theta)\\) with \\([K : \\mathbb{Q}] = n\\), where \\(\\mathbb{Q}_p\\) denotes the field of \\(p\\)-adic numbers and \\(K_p = \\mathbb{Q}_p(\\theta)\\).",
"role": "user"
}
] | 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... | 5.31g |
[
{
"content": "Determine whether the following statement is true or false: The affine hull of the intersection of two convex sets is equal to the intersection of the affine hulls of these sets, provided that the intersection of the two convex sets is non-empty.",
"role": "user"
}
] | False |
[
{
"content": "Compute the determinant \\(\\begin{vmatrix} bc - a^2 & ac - b^2 & ab - c^2 \\\\ ac - b^2 & ab - c^2 & bc - a^2 \\\\ ab - c^2 & bc - a^2 & ac - b^2\\end{vmatrix}\\) and express it in the form \\(k(a^3 + b^3 + c^3 - 3abc)^2\\). Given that \\(\\begin{vmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b\\... | 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... | 0.011 mol 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... | 10.34kJ |
[
{
"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... | Household and Business Sectors |
[
{
"content": "Calculate the determinant of the matrix \\(A_{2n}:=\\begin{pmatrix} a &0 &\\cdots & \\cdots &0 &b \\\\ 0& a & & & b&0 \\\\ \\vdots &\\vdots &\\ddots &\\ddots &\\vdots &\\vdots \\\\ \\vdots&\\vdots & \\ddots & \\ddots &\\vdots &\\vdots \\\\ 0& b & & & a&0 \\\\ b&0 &\\cdots &\\cdots &0 &a \\end{pmat... | (a^2 - b^2)^n |
[
{
"content": "On an island, there are liars, who always lie, and knights, who always tell the truth. Each of them made two statements:\n1) \"Among my friends, there is an odd number of knights.\"\n2) \"Among my friends, there is an even number of liars.\"\n\nDetermine whether the total number of inhabitants on ... | even |
[
{
"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... | trans-2-Butene |
[
{
"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... | 13.7% |
[
{
"content": "Determine whether the direct sum of quasi-coherent sheaves on a scheme of finite type over a field commutes with global sections. Answer true or false.",
"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... | 33,555.78 |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\sqrt[n^2]{1! + 2! + 3! + \\dots + n!}$$",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to-7} \\frac{\\lfloor x\\rfloor^2+15\\lfloor x\\rfloor+56}{\\sin(x+7)\\sin(x+8)}$$ where \\(\\lfloor x\\rfloor\\) denotes the greatest integer less than or equal to \\(x\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim _{x\\to 0}\\left(\\dfrac{\\sin\\alpha x-\\sin\\beta x}{e^{\\alpha x}-e^{\\beta x}}\\right) \\]",
"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... | 4.5 x 10^3 V |
[
{
"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... | 20.23% |
[
{
"content": "Determine the value of the constant \\( c \\) such that the random variable \\( Y = \\frac{c(X_1 + X_2)}{\\sqrt{(X_3)^2 + (X_4)^2}} \\) follows a t-distribution.",
"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... | 19.62 m |
[
{
"content": "We have 2019 empty boxes. In one operation, you can add exactly 100 stones to 100 boxes and exactly one stone to each of several other boxes. What is the smallest number of operations needed so that all boxes contain the same positive number of stones?",
"role": "user"
}
] | 2019 |
[
{
"content": "Determine whether it is possible to construct a set of subsets of natural numbers such that the set has an uncountable number of elements, but the intersection of any two distinct elements is 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... | 5.135% |
[
{
"content": "In the board game Tsuro, each tile has 8 entry points, with each point connecting to exactly one other point. Given that there are 35 unique tiles, determine the number of distinct configurations of these tiles, considering rotations as equivalent.",
"role": "user"
}
] | 35 |
[
{
"content": "If $AB = I$, where $A$ and $B$ are square matrices, is it necessarily true that $BA = I$? Provide a justification for your answer.",
"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... | 19.4 m/s |
[
{
"content": "Find the natural number $n$ for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$.",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the maximum value of the real number \\( m \\) such that for any set of positive numbers \\( a, b, c \\) satisfying \\( a b c \\leq \\frac{1}{4} \\) and \\( \\frac{1}{a^{2}} + \\frac{1}{b^{2}} + \\frac{1}{c^{2}} < m \\), there exists a triangle with side lengths \\( a, b, c \\).",
"r... | 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... | 16.67 m |
[
{
"content": "Given a monic cubic polynomial \\( P(x) \\) with real roots, where \\( P(0) = 0 \\) and \\( P(1) = 1 \\), let \\( S \\) denote the sum of its roots. How many integer values cannot be taken by \\( S \\)? The polynomial can be expressed as \\( P(x) = x(x - r)(x - s) \\), and applying \\( P(1) = 1 \\... | 3 |
[
{
"content": "Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ as shown with $\\overline{ED}\\perp\\overline{BC}.$ What is the ratio of the area of $\\triangle DEF$ to the area of $\\triangle\nABC$? [asy]\nimport olympiad; import geometry; size(100); defaultpen(linewidth(0.8));\ndraw(dir(90... | \frac 13 |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty} \\left[\\prod_{k=1}^{n}\\left(1+\\frac{(k+1)^{n}}{n^{n+1}}\\right)\\right]$$",
"role": "user"
}
] | 1 |
[
{
"content": "Let $\\{\\Omega_m\\}_m$ be a sequence of simply connected domains with $1\\in\\Omega_m\\subset\\mathbb{D}$. If $\\text{dist}(1,\\partial\\Omega_m)\\to0$ as $m\\to\\infty$ and $g_m\\in\\mathcal{H}(\\mathbb{D})$ is bijective over $\\Omega_m$ with $g_m(1)=1$ and $g'_m(1)>0$, determine if $g_m \\overs... | g(z) = z |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{140\\pi} | \\sin^2(x) \\cos(x)|dx \\).",
"role": "user"
}
] | \dfrac{280}{3} |
[
{
"content": "Determine whether the sequence \\( \\left(1 + \\frac{1}{n+2}\\right)^{3n\\log(n)} \\) converges or diverges. If it converges, find its limit.",
"role": "user"
}
] | \infty |
[
{
"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... | 3.2% |
[
{
"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... | -25 m/s^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... | +1.8 m/s^2 downwards |
[
{
"content": "Let $R$ be a commutative ring with unity. An ideal $I$ of $R$ is called a regular ideal if $I$ contains an element which is not a zero-divisor. If $I$ and $J$ are ideals of $R$ such that their product $IJ$ is a regular ideal, must $I$ and $J$ also be regular ideals?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether there exists a continuous map \\( f: S^1 \\to \\mathbb{R} \\) such that for every odd number \\( o \\in \\mathbb{N} \\), there are finitely many points in \\( f(S^1) \\) with the property that the preimage of each one has exactly \\( o \\) points in \\( S^1 \\).",
"role": "us... | Yes |
[
{
"content": "Calculate the value of the improper integral: \\[ \\int_0^{\\infty} \\frac{ \\cos{x}}{x^2+1}dx. \\]",
"role": "user"
}
] | \dfrac{\pi}{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... | 95 |
[
{
"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... | 52.1 m/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... | 2CrO_4^{2-} + 5S^{2-} + 6H_2O o 2CrO^{-} + 5S + 12OH^- |
[
{
"content": "Find the remainder when \\(1^{2016} + 2^{2016} + \\cdots + 2016^{2016}\\) is divided by 2016.",
"role": "user"
}
] | 48 |
[
{
"content": "Evaluate the integral \\( \\int_{-\\pi}^{\\pi} \\sin(nx) \\cos(mx) \\, 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... | 3.79% |
[
{
"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... | 64 s |
[
{
"content": "Calculate the sum of the series \\(\\sum_{n=1}^{\\infty} \\frac{(-1)^{n}}{2n+3}\\) to within an error of \\(\\pm 0.02\\).",
"role": "user"
}
] | -0.12 |
[
{
"content": "If \\( f(x)=\\frac{1}{x^2+x+1} \\), find \\( f^{(36)} (0) \\).",
"role": "user"
}
] | 36! |
[
{
"content": "Given a prime integer \\(p\\), consider \\(q = p^r\\) and \\(q' = p^k\\). Determine the values of \\(r\\) and \\(k\\) such that \\(x^{q'} - x\\) divides \\(x^q - x\\) in the ring \\(\\mathbb{Z}[x]\\). Justify your answer by demonstrating that \\(\\mathbb{F}_{q'} \\subset \\mathbb{F}_q\\).",
"r... | k \mid r |
[
{
"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... | 5523 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... | $7,604,850 |
[
{
"content": "Determine the number of distinct continuous functions that can be defined from a separable Hausdorff space \\((X, \\tau)\\) to the interval \\([0, 1]\\).",
"role": "user"
}
] | \mathfrak{c} |
[
{
"content": "Calculate the probability that the median of the sample $X_1, X_2, \\dots, X_n$ is less than the median of the sample $Y_1, Y_2, \\dots, Y_n$, given that both samples are independent and identically distributed (i.i.d.).",
"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... | 1000 V/m |
[
{
"content": "Let $E$ be an open set in $[0,1]^n$. Is it possible for the Lebesgue measure of $E$ to differ from the Lebesgue measure of its closure $\\bar{E}$?",
"role": "user"
}
] | Yes |
[
{
"content": "Sue can either borrow $10,\\!000$ dollars for $5$ years with a simple interest of $7\\%$ annually or an interest which compounds annually for $6\\%$. How much more money, rounded to the nearest dollar, would she have to pay back for the more expensive interest than the less expensive interest?",
... | 118 \text{ dollars} |
[
{
"content": "Determine the number of terms in the expansion of \\((a + b + c)^{15}\\) that have coefficients not divisible by 3.",
"role": "user"
}
] | 18 |
[
{
"content": "Evaluate the integral \\( \\displaystyle \\int_{\\partial \\mathcal{D}(0,2)} \\frac{8z^2-8z+1}{4z^3-8z^2+z-2}dz \\) using the residue theorem.",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Let $P(x)$ be a polynomial of degree $d$ with integer coefficients. Define the set\n\n$$ A=\\{(m,n)\\in \\mathbb{Z}^2: m^2-n^2=P(m)-P(n) \\} $$\n\nand let\n\n$$ L=\\max_{(m,n)\\in A}\\left|\\frac{m}{n}\\right|. $$\n\nIs $L$ bounded by a constant that depends only on $d$, the degree of $P$?",
"... | No |
[
{
"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... | 0.046 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... | 20*10^3 N m |
[
{
"content": "Evaluate the infinite series \\( \\sum \\limits_{k=1}^{\\infty} \\frac{\\zeta(2k)-1}{k} \\), where \\( \\zeta(s) \\) is the Riemann zeta function defined as \\( \\sum \\limits_{n=1}^{\\infty} n^{-s} \\).",
"role": "user"
}
] | \ln 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... | 44.9 m/s, 31.3 degrees west of north |
[
{
"content": "In parallelogram $ABCD$, $AB = 38$ cm, $BC = 3y^3$ cm, $CD = 2x +4$ cm, and $AD = 24$ cm. What is the product of $x$ and $y$?",
"role": "user"
}
] | 34 |
[
{
"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... | 9.46e7 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... | 5.6e18 electron/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... | 1680 mL |
[
{
"content": "Let \\( n \\) and \\( m \\) be fixed natural numbers, and let \\( N, X_1, X_2, \\ldots \\) be independent random variables such that \\( N \\sim \\text{Poisson}(n) \\) and \\( X_k \\sim \\text{Poisson}(m) \\) for all \\( k \\). Define \\( S_N = \\sum_{k=1}^N X_k \\) with the assumption that \\( S_... | mn |
[
{
"content": "Determine the smallest integer $n$ such that a square $ABCD$ can be decomposed into $n$ non-overlapping triangles, each having only acute angles.",
"role": "user"
}
] | 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... | $\sqrt{\dfrac{2h}{g \sin^2{20^{\circ}}}}$ |
[
{
"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... | 1/144 |
[
{
"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... | R = v * sqrt(2h/g) |
[
{
"content": "True or False: If $A$ is an invertible square matrix and $\\det(ABA) = 0$ for some square matrix $B$, then $\\det(A) = \\det(A^{-1})$ implies that $\\det(B) = 0$ for all square matrices $B$.",
"role": "user"
}
] | B |
[
{
"content": "Determine the total charge on the spherical surface defined by \\(x^2+y^2+z^2=R^2\\) with a charge density given by \\(\\rho(x,y,z)=|z|\\) C/m\\(^2\\).",
"role": "user"
}
] | 2\pi R^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... | 1.779e-22 g |
[
{
"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... | premium |
[
{
"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... | 6.61 g |
[
{
"content": "Given that \\( x\\sin a = y\\cos a = \\frac{2z\\tan a}{1-\\tan^2 a} \\), find the value of \\( 4z^2(x^2+y^2) \\).\n\n(a) \\((x^2+y^2)^{3}\\)\n(b) \\((x^2-y^2)^3\\)\n(c) \\((x^2-y^2)^2\\)\n(d) \\((x^2+y^2)^2\\)\n\nSelect the correct option.",
"role": "user"
}
] | C |
[
{
"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... | 3.9 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... | V2O5 |
[
{
"content": "Determine the number of ways to parenthesize the tensor product of $n$ elements $a_1, a_2, \\ldots, a_n$ in numerical order, where $n \\in \\mathbb{N}$. Assume that the tensor product is associative but not strictly so. For example, for $n=3$, there are two ways: $(a_1 \\otimes a_2) \\otimes a_3$ ... | C_{n-1} |
[
{
"content": "Find the infimum of the functional \\( \\phi(f) = \\int_{-1}^{1}((f(x)-x^2)^2 -x^4)dx \\) over the set \\( \\mathcal{M} \\).",
"role": "user"
}
] | -\dfrac{2}{5} |
[
{
"content": "Given the planar vectors $\\overrightarrow{\\mathrm{a}}$ and $\\overrightarrow{\\mathrm{b}}$ with $|\\vec{a}|=3$, $|\\vec{b}|=4$, and $|\\vec{a}-\\vec{b}|=2 \\sqrt{3}$, find the minimum value of the function $\\mathrm{f}(\\mathrm{x})=\\left|\\mathrm{x} \\overrightarrow{\\mathrm{a}}+\\frac{1}{\\mat... | \sqrt{37} |
[
{
"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... | 11.27 m |
[
{
"content": "Given a linear transformation $L: V_n \\to V_{n-2}$, where $V_n$ is the space of homogeneous polynomials of degree $n$ in two variables $x$ and $y$, and $L$ is defined as the Laplacian operator: $L(f) = \\frac{\\partial^2\\!f}{\\partial x^2} + \\frac{\\partial^2\\!f}{\\partial y^2}$. The basis for... | n - 1 |
[
{
"content": "Determine whether the set \\(X = \\{f \\in C^\\infty[0,1] : f^{(n)}(0)=f^{(n)}(1)=0 \\ \\forall n\\geq0\\}\\) is dense in \\(L^2[0,1]\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Given two positive real sequences $a_n$ and $b_n$ that both diverge to infinity, determine if it is possible to choose two subsequences $a_{s_n}$ and $b_{t_n}$ such that $\\lim_{n \\to \\infty} \\frac{a_{s_n}}{b_{t_n}} = 1$.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the degree of the splitting field of the polynomial \\( g(x) = x^4 + 2 \\) over \\( \\mathbb{Q} \\).",
"role": "user"
}
] | 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... | 28 degrees |
[
{
"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... | 267.59 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... | 0.973 |
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