prompt list | ground_truth string |
|---|---|
[
{
"content": "Under what conditions is the expression \\( \\frac{aA+b}{cA+d} \\) rational, given that \\( a, b, c, \\) and \\( d \\) are rational numbers and \\( A \\) is irrational?",
"role": "user"
}
] | ad = bc |
[
{
"content": "Given a line \\(e\\), and points \\(A\\) and \\(B\\) not on the line, construct a segment on the line that is seen from both \\(A\\) and \\(B\\) at an angle \\(\\varphi\\).",
"role": "user"
}
] | PQ |
[
{
"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... | 3333 lbs |
[
{
"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... | $251,678,967 |
[
{
"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... | reverse direction |
[
{
"content": "Let \\( f(x) \\) be a polynomial of degree 1007 such that \\( f(k) = 2^k \\) for \\( 0 \\le k \\le 1007 \\). Determine the value of \\( f(2015) \\).",
"role": "user"
}
] | 2^{2014} |
[
{
"content": "Calculate the arc length of the astroid defined by the equation \\( \\sqrt[3]{x^2} + \\sqrt[3]{y^2} = \\sqrt[3]{a^2} \\), where \\( a \\) is a constant.",
"role": "user"
}
] | 6a |
[
{
"content": "Given a polynomial \\(P(x) = x^n + a_{n-1}x^{n-1} + \\dots + a_0\\) of degree \\(n \\geq 3\\), where \\(n\\) is even, and \\(a_{n-1} = -\\binom{n}{1}\\) and \\(a_{n-2} = \\binom{n}{2}\\). If all the roots of \\(P(x)\\) are real, find the remaining coefficients. Note that the product of the roots i... | (x - 1)^n |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\sum_{k=1}^{n}\\frac{6}{k(k+1)(k+3)} \\]",
"role": "user"
}
] | \dfrac{7}{6} |
[
{
"content": "Let $V$ be a 4-dimensional vector space, and let $T: V \\to V$ be a linear map with characteristic polynomial $\\chi_T = x^4 - x^3$. Prove that $V$ has $T$-invariant subspaces of dimensions 1, 2, and 3.",
"role": "user"
}
] | V |
[
{
"content": "Given a 3-digit number $a_2 a_1 a_0$, where $a_2$, $a_1$, and $a_0$ are digits, it is observed that when the number is converted to decimal from different bases, the results are 254, 330, and 416. Specifically, the conversions are as follows:\n\n1. In base $b$, the number is $a_2 b^2 + a_1 b + a_0... | 512 |
[
{
"content": "Determine the number of real solutions for the equation \\(3x^{\\frac{\\log(2)}{\\log(5)}} + 2^{\\frac{\\log(x)}{\\log(5)}} = 64\\), given that \\(x\\) is a positive real number.",
"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... | 50.8235 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... | 5.2 degrees |
[
{
"content": "Given the cubic equation \\(z^3 + Az^2 + Bz + C = 0\\) in the complex numbers, determine the value of \\(\\omega_0\\) such that the substitution \\(z = \\omega - \\omega_0\\) transforms the equation into the form \\(\\omega^3 - m\\omega - n = 0\\), where \\(m\\) and \\(n\\) are real constants.",
... | \dfrac{A}{3} |
[
{
"content": "In a regular pentagon $ABCDE$, point $M$ is located inside such that $\\angle MBA = \\angle MEA = 42^{\\circ}$. An equilateral triangle $\\triangle CDM$ is constructed inside the pentagon. Determine the measure of $\\angle MED$. \\( \\text{(Express your answer in degrees.)} \\)",
"role": "user... | 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... | 970K |
[
{
"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... | 493.3 m/s |
[
{
"content": "If $\\Omega$ is a simply-connected domain in $\\mathbb{R}^n$ and $f$ is an injective continuous map from $\\Omega$ to $\\mathbb{R}^n$, is it necessary that $f(\\Omega)$ is a simply-connected domain?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\int_0^{\\frac{\\pi}{2}} \\sum_{k=1}^n (\\sin{x})^k \\, dx.$$",
"role": "user"
}
] | \infty |
[
{
"content": "Determine the number of elements in the centralizer of the permutation \\((1234)\\) in the alternating group \\(A_4\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the positive integer $n$ that satisfies the equation $2(2^2) + 3(2^3) + 4(2^4) + \\ldots + n(2^n) = 2^{n+10}$. Provide your answer as a single integer.",
"role": "user"
}
] | 513 |
[
{
"content": "The line \\(\\frac{x}{4}+\\frac{y}{3}=1\\) intersects the ellipse \\(\\frac{x^{2}}{16}+\\frac{y^{2}}{9}=1\\) at points \\(A\\) and \\(B\\). Determine the number of points \\(P\\) on the ellipse such that the area of \\(\\triangle PAB\\) is equal to 3. How many such points \\(P\\) exist? Choose the... | 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... | (-10.8i - 5j + 4k) N/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... | 0.33 cm |
[
{
"content": "Determine the number of common tangents to the curves \\( y^2 = 2012x \\) and \\( xy = (2013)^2 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Given a set $A\\subset\\mathbb{R}$ with positive Lebesgue measure $\\mu(A)=a\\in\\mathbb{R}_+$, and a value $b\\in(0,a)$, determine whether there always exists a subset $B\\subset A$ such that $\\mu(B)=b$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find a function \\( f: \\mathbb{R} \\to S \\subset \\mathbb{R} \\) such that for any two real numbers \\( x \\) and \\( y \\), if \\( x < y \\), then \\( f(x) > f(y) \\), and both \\( f(x) \\) and \\( f(y) \\) are positive.",
"role": "user"
}
] | f(x) = e^{-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... | 4.73 |
[
{
"content": "A parallelogram has three of its vertices at $(-1,0)$, $(2,4)$ and $(2,-4)$. What is the positive difference between the greatest possible perimeter and the least possible perimeter of the parallelogram?",
"role": "user"
}
] | 6 |
[
{
"content": "In how many ways can you transform the number 0 into the number 100 in exactly 4 seconds, if each second you can add or subtract any factorial number (e.g., 1!, 2!, 3!, etc.) to your current number?",
"role": "user"
}
] | 36 |
[
{
"content": "Let $A$ be a local Artin $k$-algebra with residue field $k$, where $k$ is an algebraically closed field. Determine whether all vector bundles on $Spec(A)$ are trivial, i.e., whether all locally free modules over $A$ are globally free.",
"role": "user"
}
] | Yes |
[
{
"content": "Is any differential graded (dg) coalgebra weakly equivalent to its cohomology as an $A_\\infty$ coalgebra?",
"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... | {\rm K}=\frac{\left[CaSO_4 \right ]\cdot P_{CO_2}}{\left[H_2SO_4 \right ]} |
[
{
"content": "The sequence 1,000,000; 500,000; 250,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?",
"role": "user"
}
] | 15625 |
[
{
"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 = ka
d = kb |
[
{
"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... | Advertisement costs and recruiter's salary and benefits costs |
[
{
"content": "The set of vectors $\\mathbf{v}$ such that\n\\[\\mathbf{v} \\cdot \\mathbf{v} = \\mathbf{v} \\cdot \\begin{pmatrix} 2 \\\\ 0 \\end{pmatrix}\\]form a curve in the plane. Find the area of the region contained in the curve.",
"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... | 2.65% |
[
{
"content": "Determine whether there exists a continuous positive function defined on the real numbers \\(\\mathbb{R}\\) that converges to zero as \\(x\\) approaches both positive and negative infinity, but for which no power \\(f^p\\) (\\(p>0\\)) is integrable.",
"role": "user"
}
] | Yes |
[
{
"content": "Does there exist a positive continuous function $f(x)$ on $\\mathbb{R}$ such that $f(x) \\to 0$ as $x \\to \\pm \\infty$, but $f^p$ is not integrable for any $p > 0$?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( p, q, r \\) be prime numbers such that \\( \\frac{pqr}{p+q+r} \\) is an integer. Determine which of the following statements is true:\n\n(A) At least one of \\( p, q, r \\) is 2\n\n(B) At least one of \\( p, q, r \\) is 3\n\n(C) At least two of \\( p, q, r \\) are equal\n\n(D) \\( \\frac{p... | D |
[
{
"content": "Find all values of $k$ for which the system\n\\begin{align*}\nx + ky - z &= 0, \\\\\nkx - y - z &= 0, \\\\\nx + y - kz &= 0\n\\end{align*}has a non-trivial solution. (In other words, find all values of $k$ for which the system has a solution other than $(x,y,z) = (0,0,0).$)",
"role": "user"
... | -1,0,1 |
[
{
"content": "Suppose we have a collection of non-empty bounded open sets, $A_{0}, A_{1}, \\dots$ that are subsets of $\\mathbb{R}^n$, such that $A_{i} \\supseteq A_{i+1}$, and the intersection $A = \\bigcap_{i=0}^{\\infty} A_{i}$ is a non-empty closed set. If there is a convergent sequence $a_{0}, a_{1}, \\dot... | 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... | $39,415.20 |
[
{
"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... | 4.67e-4 |
[
{
"content": "Determine the function to which the sequence of functions \\( f_n(x) \\) converges, where\n\\[\nf_n(x)=\\begin{cases}\n n^2x& \\text{ if }\\; 0\\leq x \\leq \\frac{1}{n} \\\\\n -n^2x+2n& \\text{ if }\\;\\; \\frac{1}{n} < x <\\frac{2}{n} \\\\\n 0& \\text{ if }\\;\\;\\frac{2}{n}\\leq x \\leq 1\n\\en... | 0 |
[
{
"content": "Compute the Jacobi symbol \\(\\left(\\frac{77}{257}\\right)\\). Show all steps in your calculation.",
"role": "user"
}
] | -1 |
[
{
"content": "Find an upper bound for the expression \\( \\sup_{a,b \\in B(0,r)} \\left | \\frac{e^a - e^b}{a-b} \\right | \\), where \\( B(0,r) \\) is the closed convex ball of radius \\( r \\) in the complex plane.",
"role": "user"
}
] | e^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... | 8.40% |
[
{
"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... | 682,902.73 |
[
{
"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... | 163.12 nm |
[
{
"content": "Determine whether the function \\( H(x) = \\int_0^x \\left|\\sin\\left(\\frac{1}{t}\\right)\\right| \\, dt \\) is differentiable at \\( x = 0 \\).",
"role": "user"
}
] | \frac{2}{\pi} |
[
{
"content": "Evaluate the limit of the series:\n\n\\[ \\lim_{n \\to \\infty}\\left(\\frac{1}{1+a_1}+\\frac{1}{1+a_2}+\\cdots+\\frac{1}{1+a_n}\\right) \\]\n\nwhere \\( a_1=3 \\) and \\( a_{n+1}=a_n^2+a_n \\) for \\( n=1,2,\\cdots \\).",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Determine which of the following integral representations correctly describes the convolution of a function $f$ with its scaled version $f(\\frac{x}{a})$:\n\n1. \\( f * f(\\frac{x}{a}) = \\int_\\mathbb{R} f(u)f(\\frac{x}{a}-u)du \\)\n2. \\( f * f(\\frac{x}{a}) = \\int_\\mathbb{R} f(\\frac{u}{a})f(... | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{(x,y)\\to(1,-1)}{\\frac{e^{x-y}\\tan{(x+y)}}{x^2-y^2}}. \\]",
"role": "user"
}
] | \dfrac{e^2}{2} |
[
{
"content": "Find all natural numbers $M$ that have exactly 6 divisors and for which the sum of these divisors is 3500.",
"role": "user"
}
] | 1996 |
[
{
"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... | 4/15 meters/second |
[
{
"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... | 0.3 kg |
[
{
"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... | $10,795.54 |
[
{
"content": "Let $f:\\mathbb{R} \\to \\mathbb{R}$ be a continuous function. Determine which of the following statements implies that $f(0) = 0$:\n\n(A) $\\lim_{n \\to \\infty} \\int_{0}^{1} f(x)^n \\, dx = 0$\n\n(B) $\\lim_{n \\to \\infty} \\int_{0}^{1} f(x/n) \\, dx = 0$\n\n(C) $\\lim_{n \\to \\infty} \\int_{... | 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... | 11e6 Pa |
[
{
"content": "Find a polynomial $p(x)$ in $\\mathbb{Q}[x]$ such that $\\mathbb{Q}(\\sqrt{1+\\sqrt{5}})$ is ring isomorphic to $\\mathbb{Q}[x]/\\langle p(x)\\rangle$. Provide the polynomial $p(x)$. ",
"role": "user"
}
] | x^4 - 2x^2 - 4 |
[
{
"content": "Compute the line integral \\(\\int_C (y^2z^3\\,dx + 2xyz^3\\,dy + 3xy^2z^2\\,dz)\\) where \\(C\\) is the straight line path from \\(a = (1,1,1)\\) to \\(b = (2,2,2)\\).",
"role": "user"
}
] | 63 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\left[1- p(n) \\right]^{n-1} \\] where \\( p(n) = o \\left( \\frac{\\log n}{n} \\right) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "What is the remainder when 5 to the 999,999th power is divided by 7?",
"role": "user"
}
] | 6 |
[
{
"content": "Calculate the flux of the vector field \\( F(x,y,z) = (x,y,z) \\) across the surface defined by \\( x^2 - y^2 + z^2 = 0 \\) with \\( y \\in [0, 1] \\).",
"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... | 3.90e+03 N/C |
[
{
"content": "Calculate the limit of the sequence defined by:\n\\[ x_n = \\frac{1}{n}\\left(n + \\frac{n-1}{2} + \\frac{n-2}{3} + \\ldots + \\frac{2}{n-1} + \\frac{1}{n} - \\log(n!)\\right) \\]\nUse asymptotic equivalences to find the limit, considering that \\( \\log(n!) \\sim n\\log(n) \\).",
"role": "use... | \gamma |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} (\\sqrt{2} - \\sqrt[3]{2})(\\sqrt{2} - \\sqrt[4]{2})...(\\sqrt{2} - \\sqrt[n]{2})$$",
"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... | 56.5 A |
[
{
"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... | 452.60-degree Celcius |
[
{
"content": "Derive the identity \\(\\sum_{k=0}^{\\frac{n}{2}} 2k\\binom{n}{2k}=n\\cdot2^{n-2}\\) for even \\(n\\geq 2\\) by considering the number of ways to choose an element \\(x\\) from a set \\([n]\\) and a subset \\(S'\\) of \\([n] / \\{x\\}\\) such that the cardinality of \\(S'\\) is odd.",
"role": ... | n \cdot 2^{n-2} |
[
{
"content": "Evaluate the limit: \\( \\lim_{n\\rightarrow\\infty}\\left(\\lim_{x\\rightarrow\\infty}x-n\\right) \\).",
"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... | $28,249.23 |
[
{
"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... | $193 million |
[
{
"content": "Suppose $f:[0,\\infty)\\rightarrow \\mathbb{R}$ is a twice differentiable function and $f''$ is bounded. Prove that if $\\lim_{x\\rightarrow \\infty}f(x)$ exists and is finite, then $\\lim_{x\\rightarrow \\infty}f'(x)=0$. ",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether it is possible to construct a Boolean algebra with exactly two atoms.",
"role": "user"
}
] | Yes |
[
{
"content": "Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$. Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The area of $\\triangle ABC$ is $3$. What is the slope of $l_3$?",
"rol... | \tfrac34 |
[
{
"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... | 574.3 K |
[
{
"content": "Solve the equation \\(x^{\\log(x)-1 } = \\frac{1}{\\sqrt[4]{10}}\\).",
"role": "user"
}
] | \sqrt{10} |
[
{
"content": "Expand the product $(x+1)^2 \\cdot x$.",
"role": "user"
}
] | x^3 + 2x^2 + x |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0}\\left(\\frac{\\tan\\left(\\frac{\\pi}{2}\\sin^2x\\right)}{x^4}\\right)$$ without using L'Hopital's rule.",
"role": "user"
}
] | \infty |
[
{
"content": "Calculate the volume of the region defined by the set \\( T = \\{(x,y,z) \\in \\mathbb{R}^3 \\mid 0 \\leq x, x^2 \\leq y \\leq \\sqrt{x}, 0 \\leq z \\leq xy \\} \\).",
"role": "user"
}
] | \dfrac{1}{12} |
[
{
"content": "Let $A$ and $P$ be $n \\times n$ matrices with $P$ a self-adjoint projection (i.e., $P = P^* = P^2$) and $\\Vert A \\Vert = 1$, where the norm is the operator norm. Suppose that $A$ and $P$ almost commute, meaning the norm of their commutator $[A, P]$ is small. Is there a matrix $B$ close to $A$ i... | Yes |
[
{
"content": "Is it possible for a model to be $|\\mathcal M|^+$-saturated, where $|\\mathcal M|^+$ denotes the successor cardinal of the cardinality of the model $\\mathcal M$?",
"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... | 6.786% |
[
{
"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... | 3.08 |
[
{
"content": "Given the differentiable function $f(x, y)$, the maximum value of the directional derivative $D_uf(0, 2)$ is $2$ and occurs along the direction vector $u = \\left(\\frac{\\sqrt{2}}{2}, \\frac{\\sqrt{2}}{2}\\right)$. Let $x(s, t) = s^2t$ and $y(s, t) = 2se^t$. Define $F(s, t) = f(x(s, t), y(s, t))$... | 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... | 5.2e4 W |
[
{
"content": "Define a function \\(B(x)\\) for \\(x>0\\) as the maximum integer \\(n\\) satisfying \\(n+ \\left \\lfloor \\tfrac{n}{5} \\right \\rfloor \\le x\\). Determine the value of \\( \\lim_{x \\to \\infty} \\dfrac{B(x)}{x} \\).",
"role": "user"
}
] | \dfrac{5}{6} |
[
{
"content": "Given that \\(a, b, c \\in \\mathbb{R}\\) are distinct and satisfy the equation \\(\\det\\begin{bmatrix}2a&b&c\\\\b&c&2a\\\\c&2a&b\\end{bmatrix} = 0\\), determine the location of the roots of the quadratic equation \\(24ax^2 + 4bx + c = 0\\). Choose the correct option:\n\n(a) At least one root in ... | A |
[
{
"content": "Using strict natural deduction, derive the conclusion \\( T \\lor S \\) from the premises \\( \\lnot(P \\land \\lnot Q) \\) and \\( (\\lnot P \\to S) \\land \\lnot Q \\). The allowed rules are implication, and, or, modus tollens, modus ponens, copy rule, negation, double negation, contradictions, ... | T \lor S |
[
{
"content": "Find the sum of the series \\( \\sum_{k=1}^{\\infty} \\frac{k^2}{(k+1)!} \\).",
"role": "user"
}
] | e - 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... | Purchases Budget in Units
| Months | Units to be sold | Ending Inventory | Total Required Units | Beginning Inventory | Units to be purchased |
|---|---|---|---|---|---|
| January | 700,000 | 205,000 | 905,000 | 200,000 | 705,000 |
| February | 820,000 | 190,000 | 1,010,000 | 205,000 | 805,000 |
| March | 760,000 | 180,000 | 940,000 | 190,000 | 750,000 |
| April | 720,000 | 320,000 | 1,040,000 | 180,000 | 860,000 |
| May | 1,280,000 | 375,000 | 1,655,000 | 320,000 | 1,335,000 |
Purchases Budget in Dollars
| Months | Units to be purchased | Unit cost | Cost of Purchases |
|---|---|---|---|
| January | 705,000 | $7 | $4,935,000 |
| February | 805,000 | $7 | $5,635,000 |
| March | 750,000 | $7 | $5,250,000 |
| April | 860,000 | $7 | $6,020,000 |
| May | 1,335,000 | $7 | $9,345,000 | |
[
{
"content": "Given the quadratic equation \\(x^2 + px + q + 1 = 0\\) with two distinct roots \\(x_1\\) and \\(x_2\\), where \\(p\\) and \\(p^2 + q^2\\) are prime numbers, find the largest possible value of \\(x_1^{2014} + x_2^{2014}\\). Assume \\(p\\) and \\(q\\) are integers, and the roots are whole numbers."... | 2^{2014} |
[
{
"content": "Determine whether the statement is true or false: If a matrix $I+A$ is invertible, then the matrix $I-A$ is also invertible.",
"role": "user"
}
] | B |
[
{
"content": "Determine the element of the form \\((x, y)\\) that generates the group \\(\\mathbb{Z}/ m\\mathbb{Z} \\times \\mathbb{Z}/ n\\mathbb{Z}\\), given that \\(m\\) and \\(n\\) are coprime and the group is isomorphic to the cyclic group \\(\\mathbb{Z}/mn\\mathbb{Z}\\). Use the fact that \\(\\mathbb{Z}/mn... | (1, 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... | 8.94 |
[
{
"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... | WholeMart > Fruit Fly > Night Ryder |
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