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... | 55 Hz |
[
{
"content": "Determine the radius of convergence of the series \\( \\sum_{n = 1}^{\\infty} \\frac{x^n}{n^{\\sqrt{n}}} \\).",
"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... | 104.33 cm |
[
{
"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... | $1,397,985.65 |
[
{
"content": "Calculate the variance of the return on Investment $X$, denoted as $\\text{Var}(X)$, given its joint probability density function with Investment $Y$:\n\n$$ f(x,y) = \\begin{cases} \\frac{1}{2}, &\\text{if }0 < |x| + |y| < 1,\\\\ 0, &\\text{otherwise.} \\end{cases} $$\n\nThe region of the joint pr... | \dfrac{1}{6} |
[
{
"content": "In triangle $ABC,$ $\\angle A,$ $\\angle B,$ and $\\angle C$ form an arithmetic sequence. The length of the altitude from $B$ to $\\overline{AC}$ is equal to $AB - BC.$ Find all possible values of\n\\[\\sin \\frac{C - A}{2}.\\]Enter all possible values, separated by commas.",
"role": "user"
... | \frac{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... | {eq}\vec{v} = \bigg[ ( a(\cos{\theta} - \sin{\theta}) + b( \cos{2\theta} ) ) \hat{i} + (a(\cos{\theta} + \sin{\theta}) + b( \sin{2\theta})) \hat{j} - ( b \sin{\theta} ) \hat{k} \;\bigg] \frac{d \theta }{dt}{/eq} |
[
{
"content": "Find all functions \\( f: \\mathbb{Q} \\rightarrow \\mathbb{Q} \\) such that for all real numbers \\( x \\) and \\( y \\):\n\n\\[\nf\\left(\\frac{x+y}{2}\\right)=\\frac{f(x)}{2}+\\frac{f(y)}{2}\n\\]",
"role": "user"
}
] | f(x) = ax + b |
[
{
"content": "Choose the largest of the following sums, and express it as a fraction in simplest form:\n$$\\frac{1}{4} + \\frac{1}{5}, \\ \\ \\frac{1}{4} + \\frac{1}{6}, \\ \\ \\frac{1}{4} + \\frac{1}{3}, \\ \\ \\frac{1}{4} + \\frac{1}{8}, \\ \\ \\frac{1}{4} + \\frac{1}{7}$$",
"role": "user"
}
] | \frac{7}{12} |
[
{
"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... | 42.66% |
[
{
"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... | 2.88e-17 J |
[
{
"content": "Suppose $f(x)$ is a differentiable function on the entire real line $\\mathbb{R}$ and satisfies $\\lim_{x\\rightarrow \\infty} f(x) = \\lim_{x\\rightarrow -\\infty} f(x) = 17$. Can Rolle's Theorem be applied to demonstrate the existence of a point $\\xi \\in (-\\infty, \\infty)$ such that $f'(\\xi... | 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... | $1,453,577 |
[
{
"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... | 7.116 |
[
{
"content": "Given that $x$ is a root of the equation $x^{2017} - 2x + 1 = 0$ and $x \\neq 1$, find the value of the sum $$\\sum_{k=0}^{2016} x^{k}.$$",
"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... | 4.73 L |
[
{
"content": "Find the real polynomial \\( p(x) \\) such that for any real polynomial \\( q(x) \\), the equation \\( p(q(x)) = q(p(x)) \\) holds.",
"role": "user"
}
] | x |
[
{
"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... | 48.50 mi/hr |
[
{
"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... | 1.26 m/s^2 |
[
{
"content": "A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$ and $S(m, 1) = 1$, along with the recurrence relation $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m \\geq 2$ and $n \\geq 2$. Determine the largest integer $k$ such that $2^k$ divides $S(7, 7)$. \\( \\text{Provide your answer as an in... | 370 |
[
{
"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... | 3.2e-19 J |
[
{
"content": "Determine whether the matrix \\( A = \\begin{bmatrix} a & b \\\\ c & -a \\end{bmatrix} \\) with determinant 1 and \\( c \\neq 0 \\) is similar to the matrix \\( B = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix} \\) up to sign. State whether the statement is true or false.",
"role": "user"
... | True |
[
{
"content": "Suppose that $\\alpha$ is inversely proportional to $\\beta$. If $\\alpha = -3$ when $\\beta = -6$, find $\\alpha$ when $\\beta = 8$. Express your answer as a fraction.",
"role": "user"
}
] | \frac{9}{4} |
[
{
"content": "Evaluate the contour integral \\( \\dfrac{1}{2\\pi i}\\int_{\\Gamma} \\dfrac{1}{\\sin(z)z^{21}}\\,dz \\) to find the coefficient of the 20th term in the Laurent series expansion of the function \\( f(z) = \\dfrac{1}{\\sin(z)} \\). Can the residue theorem be applied in this case?",
"role": "use... | 0 |
[
{
"content": "Let $f \\in \\mathbb{Z}[x]$ be an irreducible polynomial. Suppose $f(x)$ has two roots in $\\mathbb{C}$ with a product of 1. Is it true that the degree of $f$ is even?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the definite integral \\( \\int_{-a}^a \\frac{1}{\\cos^2 x (e^x + 1)} \\, dx \\) in closed form.",
"role": "user"
}
] | \tan a |
[
{
"content": "A spherical scoop of vanilla ice cream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the ice cream spreads out uniformly forming a cylindrical region 8 inches in radius. Assuming the density of the ice cream remains constant, how many inches dee... | \frac{1}{6} |
[
{
"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... | somatic symptom |
[
{
"content": "Does the existence of the limit \\(\\lim_{h\\to0}\\dfrac{f(x+h)+f(x-h)-2f(x)}{h}\\) for all \\(x\\in\\mathbb{R}\\) imply that \\(f\\) is twice differentiable on \\(\\mathbb{R}\\)? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the line integral of the vector field \\( \\mathbf{F} = [xy, x^2y^2] \\) along the curve \\( C \\), which is a quarter-circle from \\((2,0)\\) to \\((0,2)\\) with center at \\((0,0)\\). Use the parametric representation \\( \\mathbf{r}(t) = [2\\cos(t), 2\\sin(t)] \\) for the curve. Comput... | \dfrac{8}{5} |
[
{
"content": "Determine the minimal polynomial of $\\sqrt{5} + \\sqrt{7}$ over the field $\\mathbb{Q}(\\sqrt{3})$. Provide your answer as a polynomial with rational coefficients.",
"role": "user"
}
] | x^4 - 24x^2 + 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... | 127 kJ |
[
{
"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... | 5.39 N/C |
[
{
"content": "Let $X$ be an inner product space and $M\\subset X$. Determine whether the following statement is true or false: $$(M^{\\perp})^{\\perp}=\\overline{\\operatorname{span} M}.$$",
"role": "user"
}
] | False |
[
{
"content": "Determine the dimension of the subset of pairs of 2x2 matrices \\((c_1, c_2)\\) that generate a 2-dimensional Lie algebra, where the commutator \\([c_1, c_2] = k_1 c_1 + k_2 c_2\\) for some scalars \\(k_1\\) and \\(k_2\\).",
"role": "user"
}
] | 6 |
[
{
"content": "Find the flux of the vector field \\( \\vec{G}=\\operatorname{curl} \\vec{F} \\), where\n\\[ \\vec{F}(x, y, z)=y^{3} \\vec{i}+x^{3} \\vec{j}+z^{3} \\vec{k} \\]\nthrough the upward oriented part of the paraboloid\n\\[ z=1-x^{2}-y^{2} \\]\nlying over the plane \\( z=0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given the polynomial \\(P(x) = x^3 - 10x + 11\\), whose zeros are \\(u, v, w\\), find the value of \\(\\arctan(u) + \\arctan(v) + \\arctan(w)\\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Find the value of $x$ such that the following equation holds: $$\\frac{2}{\\sqrt{\\pi}}\\int\\limits_{0}^{x}e^{-t^2}dt = 0.7$$",
"role": "user"
}
] | 0.733 |
[
{
"content": "Evaluate the limit \\( \\lim_{a\\to \\infty}\\int_0^1 a^x x^a \\,dx \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether there exists an injective, self-adjoint operator on $\\ell_2(\\mathbb{Z})$ with spectrum $[0,1]$ and empty point spectrum.",
"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... | 0.0413 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... | 6.28e+04 N/C |
[
{
"content": "Consider a set \\( E \\) of \\( 2n-1 \\) distinct points on a circle, where \\( n \\geq 3 \\). Suppose exactly \\( k \\) of these points are colored black. A coloring is defined as \"good\" if there exists at least one pair of black points such that the interior of one of the arcs between them con... | 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... | 0.974 rad/s |
[
{
"content": "Evaluate the integral \\( \\int^{1}_{0}\\left(\\frac{1}{1-x}+\\frac{1}{\\ln x}\\right)dx \\).",
"role": "user"
}
] | \gamma |
[
{
"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... | -$22,000 |
[
{
"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... | $3,391,212 |
[
{
"content": "Determine whether a 15-degree angle can be constructed using only a compass and straightedge.",
"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... | -1.92e-6 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... | C_{2v} |
[
{
"content": "Calculate the limit as \\( n \\) approaches infinity: \\[ \\lim_{n\\to\\infty}{\\dfrac{1}{n}\\sum_{k=1}^{n}{\\left(\\dfrac{n}{k}-\\left\\lfloor\\dfrac{n}{k}\\right\\rfloor\\right)}} \\]",
"role": "user"
}
] | 1 - \gamma |
[
{
"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... | 54618 |
[
{
"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... | 1.68*10^-3 W/m^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... | 29.4e5 kJ/h |
[
{
"content": "Determine whether the following statement is true or false: If the closure of a subset $A$ in a topological space $X$ is connected, then $A$ must also be connected.",
"role": "user"
}
] | B |
[
{
"content": "Find all functions \\( f: \\mathbb{N}^* \\to \\mathbb{N}^* \\) that satisfy the functional equation \\( f(n) + 2f(f(n)) = 3n + 5 \\) for all positive integers \\( n \\). Given that \\( f(1) = 2 \\), \\( f(2) = 3 \\), and \\( f(3) = 4 \\), determine if \\( f(n) = n + 1 \\) is the only solution.",
... | f(n) = n + 1 |
[
{
"content": "Let \\( p(x) \\) be the polynomial given by:\n\\[\np(x) = [(x-2)^{2016}(x+2016)]+[(x-2)^{2015}(x+2015)]+\\cdots+(x-2)(x+1).\n\\]\nWhat is the sum of the coefficients of \\( p(x) \\)?",
"role": "user"
}
] | 1008 |
[
{
"content": "Determine all group homomorphisms from the group \\((\\mathbb{R}, +)\\) to the group \\((\\mathbb{Z}, +)\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the set of positive integers that are divisible by $n^2$ for every integer $n \\geq 2$. Express your answer as a set.",
"role": "user"
}
] | \emptyset |
[
{
"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 workers |
[
{
"content": "Find the maximum value of \\( a \\) such that for any \\( k \\in [-1,1] \\), the inequality \\( 6 \\ln x + x^2 - 8x + a \\leq kx \\) holds for all \\( x \\in (0,6] \\).",
"role": "user"
}
] | 6(1 - \ln 6) |
[
{
"content": "Find all functions \\( f: \\mathbf{N} \\rightarrow \\mathbf{N} \\) such that for every \\( n \\in \\mathbf{N} \\), \\[ \\frac{1}{f(1) f(2)}+\\frac{1}{f(2) f(3)}+\\cdots+\\frac{1}{f(n) f(n+1)}=\\frac{f(f(n))}{f(n+1)} \\]",
"role": "user"
}
] | f(n) = n |
[
{
"content": "Evaluate the surface integral \\( \\int\\int_S \\vec{F} \\cdot \\hat{n} \\, dS \\), where \\( S = \\{(x,y,z) \\in \\mathbb{R}^3 : x^2+y^2+z^2=\\frac{1}{4}\\} \\) and \\( \\vec{F} = \\frac{x\\hat{i}+y\\hat{j}+z\\hat{k}}{(x^2+y^2+z^2)^2} \\). Here, \\( \\hat{n} \\) is the outward unit normal vector ... | 8\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... | 8 sec |
[
{
"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... | 76.0 |
[
{
"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... | 19.795 cm |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\sqrt[n]{\\frac{|\\cos 1|}{1}+\\cdots+\\frac{|\\cos n|}{n}\\ }$$",
"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... | First-level preconventional stage |
[
{
"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... | 2.067*10^46 gm |
[
{
"content": "If points $A$ and $B$ are such that for any point $P$ on a given circle, the ratio $\\frac{PA}{PB} = k$ is constant, determine the value of $k$ that cannot occur.",
"role": "user"
}
] | 1 |
[
{
"content": "Given that $y_1$ and $y_2$ are linearly independent solutions of the differential equation $ty'' + 2y' + te^ty = 0$ and $W(y_1,y_2)(1) = 2$, find the value of the Wronskian $W(y_1,y_2)(5)$. ($W(y_1,y_2)(t_0)$ denotes the value of the Wronskian at $t = t_0$.)",
"role": "user"
}
] | \dfrac{2}{25} |
[
{
"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... | 11.25 m |
[
{
"content": "Given that the point $(9,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=\\frac{f(2x)}2+2$. What is the sum of coordinates of that point?",
"role": "user"
}
] | \frac{29}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... | 8.35 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} \\frac{1}{\\sqrt{n}} \\left( 1 - e^{i\\varphi} + e^{2i\\varphi} - \\cdots + (-1)^n e^{ni\\varphi} \\right) \\]",
"role": "user"
}
] | 0 |
[
{
"content": "If $a \\div b = 2$ and $b \\div c = \\frac{3}{4}$, what is the value of $c \\div a$? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{2}{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... | 25.77 kJ |
[
{
"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... | P = 0.00416T |
[
{
"content": "Calculate the limit: $$\\lim_{n\\to\\infty}\\frac{[x] + [2x]+ ... +[nx]}{n^2}$$ where \\([x]\\) denotes the floor function.",
"role": "user"
}
] | \dfrac{x}{2} |
[
{
"content": "Given the initial conditions \\( a_{0} = 1 \\) and \\( a_{1} = 2 \\), and the recurrence relation:\n\\[\nn(n+1) a_{n+1} = n(n-1) a_{n} - (n-2) a_{n-1}, \\quad n = 1, 2, 3, \\cdots\n\\]\nfind the value of \\( \\frac{a_{0}}{a_{1}} + \\frac{a_{1}}{a_{2}} + \\frac{a_{2}}{a_{3}} + \\cdots + \\frac{a_{5... | \dfrac{2655}{2} |
[
{
"content": "Determine if the n-dimensional sphere $S^n$ and the n-dimensional real projective space $\\mathbb{RP}^n$ can be homeomorphic for any dimension other than 1.",
"role": "user"
}
] | No |
[
{
"content": "Determine whether there exist three quadratic polynomials such that each polynomial has at least one root, but the sum of any two of these polynomials does not have any roots.",
"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... | $16,100.67 |
[
{
"content": "Evaluate the integral \\( \\int_0^{a} f(z) \\delta(z) \\, dz \\) for \\( a > 0 \\). Does it hold that \\( \\int_0^{a} f(z) \\delta(z) \\, dz = f(0) \\)?",
"role": "user"
}
] | f(0) |
[
{
"content": "Determine the number of 11-digit strings that use each of the digits 0 through 9 at least once and do not have any consecutive identical digits.",
"role": "user"
}
] | 163296000 |
[
{
"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 velocity of 4kg ball is -3.33m/s. The velocity of 8kg ball is 6.67m/s |
[
{
"content": "In square \\(ABCD\\), a parallelogram \\(AECF\\) is inscribed such that \\(AF:AD = 1:4\\). This parallelogram is reflected across diagonal \\(AC\\) to form another parallelogram \\(AE'CF'\\). The area of the region common to both parallelograms is \\(m\\) and its perimeter is \\(n\\). Calculate \\... | \dfrac{7}{400} |
[
{
"content": "If $x$ is a positive number such that \\[\\sqrt{8x}\\cdot\\sqrt{10x}\\cdot\\sqrt{3x}\\cdot\\sqrt{15x}=15,\\]find all possible values for $x$.",
"role": "user"
}
] | \frac{1}{2} |
[
{
"content": "Let $P$, $A$, $B$, $C$, $D$ be points on a plane such that $PA = 9$, $PB = 19$, $PC = 9$, $PD = 5$, $\\angle APB = 120^\\circ$, $\\angle BPC = 45^\\circ$, $\\angle CPD = 60^\\circ$, and $\\angle DPA = 135^\\circ$. Let $G_1$, $G_2$, $G_3$, and $G_4$ be the centroids of triangles $PAB$, $PBC$, $PCD$... | 29 |
[
{
"content": "Given that $x$ is a prime number and $y$ is an integer, find the maximum possible value of $y$ such that the expression \\( \\sum_{i=1}^{55}\\frac{i}{xy} \\) is an integer.",
"role": "user"
}
] | 770 |
[
{
"content": "Find all functions \\(f: \\mathbb{Q}^{+} \\rightarrow \\mathbb{Q}^{+}\\) such that for all positive rational numbers \\(x\\) and \\(y\\), the following equation holds:\n\\[\nf(x) = f(x + y) + f\\left(x + x^2 f(y)\\right).\n\\] Provide a proof for your answer.",
"role": "user"
}
] | f(x) = \dfrac{1}{x} |
[
{
"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... | 9.18% |
[
{
"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... | 37.8 J |
[
{
"content": "Determine the discrete logarithm of $x^2 + 1$ in the quotient ring $\\Bbb Z_5[x]/\\langle x^3 + x + 1 \\rangle$. Express your answer in terms of the basis elements of the ring.",
"role": "user"
}
] | 30 |
[
{
"content": "Find all triples \\((m, p, q)\\) where \\(m\\) is a positive integer and \\(p, q\\) are primes such that \\(2^m p^2 + 1 = q^5\\).",
"role": "user"
}
] | (1, 11, 3) |
[
{
"content": "In a vector space $V$, given two vectors $u$ and $v$, determine if the equation $\\|u+v\\|^2 = 4\\langle u,v \\rangle$ necessarily implies that $u = v$. Provide a justification for your answer.",
"role": "user"
}
] | u = 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... | $59726.97 |
[
{
"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... | 10460.86 m/s |
[
{
"content": "Let \\( f(x) = ax^2 + bx + c \\) be a quadratic function where \\( a, b, c \\in \\mathbf{R} \\) and \\( a \\neq 0 \\). The function satisfies the following conditions:\n1. For all \\( x \\in \\mathbf{R} \\), \\( f(x-4) = f(2-x) \\) and \\( f(x) \\geq x \\).\n2. For \\( x \\in (0, 2) \\), \\( f(x) ... | 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... | 5.2m |
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