prompt list | ground_truth string |
|---|---|
[
{
"content": "Given the cubic equation \\( z^3 + az^2 + bz + c = 0 \\) with \\( z \\in \\mathbb{C} \\), set \\( z = w + h \\) where \\( h = -\\frac{1}{3}a \\) to obtain the equation \\( w^3 + Bw + C = 0 \\). Here, \\( B = -\\frac{1}{3}a^2 + b \\) and \\( C = \\frac{2}{27}a^3 - \\frac{1}{3}ab + c \\). Transform ... | \sqrt[3]{4} |
[
{
"content": "Let \\( f: [0, \\infty) \\to \\mathbb{R} \\) be a continuous function, and define \\( \\tau \\) as the infimum of all \\( t \\geq 0 \\) such that \\( |f(t)| \\geq \\varepsilon \\), where \\( \\varepsilon \\geq 0 \\). If \\( \\tau \\in (0, \\infty) \\), is it necessarily true that \\( |f(\\tau)| = ... | 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... | $93,000 |
[
{
"content": "Given that \\( \\lim\\limits_{x\\to b^-} g(x)=\\infty \\) and \\( g(x)>0 \\) for all \\( x \\in [a,b) \\), and that \\( f \\) is bounded on \\( [a,b) \\), determine whether it must be true that \\( \\lim\\limits_{x\\to b^-}\\dfrac{f(x)}{g(x)}=0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the sum of the solutions of the equation $-32x^2 + 84x + 135=0$.",
"role": "user"
}
] | \frac{21}{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... | 76.73 |
[
{
"content": "Evaluate the limit: $$\\lim _{x\\to 0}\\frac{\\exp(x^2)-\\cos x-x\\sin x}{\\sinh^2x}$$",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to 0}\\frac{\\sqrt{1+2x}-\\sqrt[\\Large3]{1+5x}}{\\sqrt[\\Large5]{1+x}-\\sqrt[\\Large5]{1+2x}} $$",
"role": "user"
}
] | \dfrac{10}{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... | 149 |
[
{
"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... | -332.6N i -110.01N j |
[
{
"content": "Determine the number of distinct $k$-dimensional subspaces in an $n$-dimensional vector space $V$ over a finite field with $q$ elements.",
"role": "user"
}
] | \dbinom{n}{k}_q |
[
{
"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... | 1.8e2 g |
[
{
"content": "In triangle $\\triangle ABC$, the in-centers are $I(0,0)$, $I_1(2,3)$, and $I_2(5,7)$, and the ex-center is $I_3$. Calculate the distance between the orthocenters of triangles $\\triangle II_1I_3$ and $\\triangle I_1I_2I_3$. Provide your answer as a single numerical value.",
"role": "user"
}... | \sqrt{74} |
[
{
"content": "Is a bounded function $f: S \\to \\mathbb{R}$, defined on a Jordan measurable set $S \\subset \\mathbb{R}^n$, Riemann integrable over $S$ if and only if the set $D$ of its discontinuities has Lebesgue measure zero in $\\mathbb{R}^n$?",
"role": "user"
}
] | Yes |
[
{
"content": "Does there exist a function $f$ that is continuous and differentiable on $\\mathbb{R}$, but whose derivative $f'$ is not differentiable at some point?",
"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... | 19.44% |
[
{
"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... | 8.9 |
[
{
"content": "Find a complex function that is holomorphic at $z = -i$ but not conformal at this point.",
"role": "user"
}
] | (z + i)^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6,699.66 |
[
{
"content": "Given two matrices \\( A \\) and \\( C \\) of order \\( n \\times n \\) and \\( m \\times n \\) respectively, consider the block matrix:\n\n\\[ D=\\left( \\begin{matrix} C\\\\ CA\\\\ :\\\\ :\\\\ CA^{q-1} \\end{matrix}\\right) \\]\n\nwhere \\( q \\in \\mathbb{N}^* \\). Determine the smallest value ... | 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... | 2.2e-18 J |
[
{
"content": "Find the remainder when \\( 45^{17^{17}} \\) is divided by 204.",
"role": "user"
}
] | 45 |
[
{
"content": "Evaluate the limit: $$ \\lim_{n\\to\\infty} e^{-n} \\left(1+n+\\frac{n^2}{2}+\\ldots+\\frac{n^n}{n!} \\right) $$",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Given that \\( z \\) is a complex number satisfying \\(|z - 3i| = 3\\) and \\( \\theta = \\arg{z} \\) with \\( 0 < \\theta < \\frac{\\pi}{2} \\), compute all possible values of \\( \\cot{\\theta} - \\frac{6}{z} \\).",
"role": "user"
}
] | i |
[
{
"content": "Consider a triangle $\\triangle ABC$ with centroid $G$, circumcenter $S$, and incenter $I$. Let $R$ and $r$ be the circumradius and inradius, respectively. Determine which of the following statements is incorrect:\n\n(A) $SI^2 = R^2 (1 - \\cos A \\cos B \\cos C)$, where $A, B, C$ are the angles of... | A |
[
{
"content": "Let $f(x) = x^2 - x + 2010$. What is the greatest common divisor of $f(100)$ and $f(101)$?",
"role": "user"
}
] | 10 |
[
{
"content": "Find all positive integers \\( m \\) such that there exists a graph \\( G \\) with chromatic number \\( \\chi(G) \\le m \\) and flowing chromatic number \\( T(G) \\ge 2^m \\), and without any cycle of length less than 2017.",
"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... | 1V |
[
{
"content": "Find the maximum value of the function \\( f(t) = e^{-\\frac{t^2}{2}}\\frac{| e^{-ta}(1+at)+e^{ta}(1-at)|}{\\sqrt{e^{-2ta}+e^{2ta}}} \\) for some \\( a > 0 \\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Determine whether the statement is true: If $P \\geq 5$ is a prime number, then at least one of the numbers $P+6$, $P+12$, $P+18$, or $P+30$ is also prime.",
"role": "user"
}
] | False |
[
{
"content": "Suppose $h(z)$ and $k(z)$ are holomorphic functions on a domain $\\Omega$. If $h(z)k(z)$ is identically zero on $\\Omega$, does it imply that either $h(z)$ or $k(z)$ is identically zero on $\\Omega$? Answer yes or no and justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to1} \\frac{x^{\\frac{1}{13}}-x^{\\frac{1}{7}}}{x^{\\frac{1}{5}}-x^{\\frac{1}{3}}} $$ using L'Hopital's rule or any other appropriate method.",
"role": "user"
}
] | \dfrac{45}{91} |
[
{
"content": "In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\\mbox{cm}^2$ and the area of face $ABD$ is $12 \\mbox { cm}^2$. These two faces meet each other at a $30^\\circ$ angle. Find the volume of the tetrahedron in $\\mbox{cm}^3$.",
"role": "user"
}
] | 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... | $13,311.74 |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\infty} \\frac{\\sin^2 x}{x^2} \\mathrm{d}x \\) using the residue theorem. Consider the function \\( \\frac{e^{2iz}-1-2iz}{z^2} \\) and integrate it within a semicircle in the complex plane.",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Let the real numbers \\( x_{1}, x_{2}, \\cdots, x_{2008} \\) satisfy the condition \\( \\left|x_{1} - x_{2}\\right| + \\left|x_{2} - x_{3}\\right| + \\cdots + \\left|x_{2007} - x_{2008}\\right| = 2008 \\). Define \\( y_{k} = \\frac{1}{k} (x_{1} + x_{2} + \\cdots + x_{k}) \\) for \\( k = 1, 2, \\cd... | 2007 |
[
{
"content": "Given that \\(\\lim_{n \\to \\infty} \\sum^n_{k =0} \\frac{nC_k}{n^k(k+3)} =ae+b\\), find the value of \\(a-b\\).",
"role": "user"
}
] | 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... | 1.23% |
[
{
"content": "Compute the limit \\( \\lim_{a\\to \\infty}\\frac{1}{a}\\int_1^a a^{\\frac{1}{x}} \\, dx \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$. The arc is divided into seven congruent arcs by six equally spaced points $C_1$, $C_2$, $\\dots$, $C_6$. All chords of the form $\\overline {AC_i}$ or $\\overline {BC_i}$ are drawn. Find the product of the lengths of these twelv... | 28672 |
[
{
"content": "Given that $\\tan x \\tan 2x = \\cot 2x \\cot 3x$, find the value of $\\cos 8x$.",
"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... | 210.54 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... | 134862.64J |
[
{
"content": "Let \\( x \\) and \\( y \\) be complex numbers such that \\( |x| = |y| = 1 \\). Find the maximum value of the expression:\n\\[\n|1+x| + |1+xy| + |1+xy^2| + \\cdots + |1+xy^{2013}| - 1007|1+y|\n\\]",
"role": "user"
}
] | 2014\sqrt{2} |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2\\pi} \\cos(\\cos t) \\, dt \\).",
"role": "user"
}
] | 2\pi J_0(1) |
[
{
"content": "Three particles A, B, and C are moving in the plane along parallel lines at constant speeds. Initially, the area of triangle ABC is 2. Five seconds later, the area of triangle ABC is 3. Determine the sum of all possible values of the area of triangle ABC after an additional five seconds.",
"ro... | 12 |
[
{
"content": "Given three numbers such that the sum of the numbers is 6, the sum of their squares is 8, and the sum of their cubes is 5, find the sum of their fifth powers.",
"role": "user"
}
] | \dfrac{118}{3} |
[
{
"content": "Calculate the multiplicity of the intersection of the two plane curves \\(F=2X_0^2X_2-4X_0X_1^2+X_0X_1X_2+X_1^2X_2\\) and \\(G=4X_0^2X_2-4X_0X_1^2+X_0X_1X_2-X_1^2X_2\\) at the point \\((1:0:0)\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0^+}\\left(\\frac{x - \\arctan x}{x^3}\\right)^{\\frac{1}{x^2}}. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "On a circle with center \\( O \\) and radius 1, the point \\( A_{0} \\) is fixed. Points \\( A_{1}, A_{2}, \\ldots, A_{999}, A_{1000} \\) are placed such that \\( \\angle A_{0} OA_k = k \\) radians. If the circle is cut at points \\( A_{0}, A_{1}, \\ldots, A_{1000} \\), how many arcs of different ... | 3 |
[
{
"content": "Determine the number of ring homomorphisms from the ring \\( \\mathbb{Z}[x,y]/(x^3+y^2-1) \\) to \\( \\mathbb{Z}_7 \\).",
"role": "user"
}
] | 11 |
[
{
"content": "Find the largest integral value of $x$ which solves: $\\frac{1}{3}<{\\frac{x}{5}}<{\\frac{5}{8}}$",
"role": "user"
}
] | 3 |
[
{
"content": "Find the limit of \\( \\left\\lfloor \\frac{1}{\\int (\\frac{rn}{n^3 + r}) \\, dn} \\right\\rfloor \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the range of \\( \\alpha \\) for which the function \\( c(u_1, u_2) = 1 + \\alpha(1- 2u_1)(1- 2u_2) \\), where \\( u_1, u_2 \\in (0,1) \\), is a copula density for the random variables \\((U_1, U_2)\\).",
"role": "user"
}
] | [-1, 1] |
[
{
"content": "Let $a$ and $b$ be the roots of $x^2 - 4x + 5 = 0.$ Compute\n\\[a^3 + a^4 b^2 + a^2 b^4 + b^3.\\]",
"role": "user"
}
] | 154 |
[
{
"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... | 374W |
[
{
"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... | $33,777 |
[
{
"content": "The roots of $x^3 + Px^2 + Qx - 19 = 0$ are each one more than the roots of $x^3 - Ax^2 + Bx - C = 0.$ If $A,$ $B,$ $C,$ $P,$ and $Q$ are constants, compute $A + B + C.$",
"role": "user"
}
] | 18 |
[
{
"content": "Evaluate the integral \\( \\int_0^{1/\\sqrt{3}} \\sqrt{x+\\sqrt{x^2+1}}\\,dx \\).",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Evaluate the integral \\( I = \\int_{C(0,2)^+} \\frac{z^3}{z^3+z^2-z-1} \\, dz \\), where \\( C(0,2)^+ \\) denotes the positively oriented circle centered at the origin with radius 2.",
"role": "user"
}
] | -2\pi i |
[
{
"content": "In a race with 20 runners, the time it takes each runner to finish is an independent exponential random variable with a mean of 60 minutes. What is the expected time that you leave if you leave after 3 runners finish?",
"role": "user"
}
] | \dfrac{541}{57} |
[
{
"content": "Evaluate \\( \\cot\\left(\\frac{\\pi}{12}\\right) \\) to the nearest hundredth using a Taylor series approximation.",
"role": "user"
}
] | 3.73 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty} \\int_{0}^{\\pi/4}\\tan^n x \\ dx$$",
"role": "user"
}
] | 0 |
[
{
"content": "Find the minimum value of the expression $PA + PB + PC + PD$, where $P$ is any point in the plane and $A(0, 1)$, $B(1, 1)$, $C(1, -1)$, $D(-1, 0)$ are fixed points.",
"role": "user"
}
] | 2\sqrt{5} |
[
{
"content": "In a complex vector space, if a normal operator \\( N \\) can be expressed as \\( N = UDU^* \\), where \\( U \\) is a unitary matrix and \\( D \\) is a diagonal matrix, do the diagonal entries of \\( D \\) correspond to the eigenvalues of \\( N \\)?",
"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... | 8.0e6 N/C |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty}n\\int_2^e{(\\ln x)^n}dx$$",
"role": "user"
}
] | e |
[
{
"content": "Given that \\( a_1, a_2, \\ldots, a_{12} \\) are positive integers and:\n\n\\[ \\log_8 a_1 + \\log_8 a_2 + \\ldots + \\log_8 a_{12} = 2014 \\]\n\nExpress the above as:\n\n\\[ \\log_8(a_1a_2a_3 \\ldots a_{12}) = 2014 \\]\n\nAnd further simplify to:\n\n\\[ \\log_8(a^{12}r^{66}) = 2014 \\]\n\nwhere \... | 46 |
[
{
"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... | 175984.00 |
[
{
"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... | probabilities rather than frequencies |
[
{
"content": "Let $P$ be a point on the unit circle, and let $A_1, A_2, \\ldots, A_n$ be the vertices of a regular polygon inscribed in the circle. If the sum of the distances from $P$ to the vertices is maximized when $P$ is at a certain point $Q$, find the distance from the center of the circle to $Q$.",
... | 1 |
[
{
"content": "Calculate the remainder when \\( \\prod_{i=1}^{1903} (2^i + 5) \\) is divided by 1000.",
"role": "user"
}
] | 931 |
[
{
"content": "A point P moves inside a triangle formed by vertices A(0,0), B(1,√3), and C(2,0), satisfying the conditions PA ≤ 1, PB ≤ 1, or PC ≤ 1. If the area bounded by the curve traced by P is equal to \\(\\frac{a\\pi}{b}\\), find the minimum value of \\(a+b\\).",
"role": "user"
}
] | 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... | 28.63186029 meters |
[
{
"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... | v_r = -11/7 m/s, v_y = 66/7 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... | 46.95 |
[
{
"content": "Determine whether the set of all $2 \\times 2$ Hermitian matrices forms a 4-dimensional real vector space that is isomorphic to $\\mathbb{R}^4$. 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... | 93.25^oC |
[
{
"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... | 0.54 |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} \\sum_{k=1}^n \\frac{1}{\\sqrt{n+k}}. \\)",
"role": "user"
}
] | \infty |
[
{
"content": "In a road traffic survey, cars passing a marker on a road are counted. The streams of cars in the two directions are modeled as independent Poisson processes with intensities of 2 cars per minute and 3 cars per minute, respectively. Counting stops once 400 cars have passed. Let \\( T_{400} \\) be ... | 85.13 |
[
{
"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... | 0.78 m |
[
{
"content": "Let $V$ be an $n$-dimensional vector space over the field $K$, and let $T: V \\rightarrow V$ be a linear operator. Suppose there exists a vector $v_0 \\in V$ that does not belong to any proper subspace of $V$ invariant under $T$. Can we conclude that the minimal polynomial of $T$ over $K$ has degr... | Yes |
[
{
"content": "Determine the maximum number of edges in a directed graph with 7 vertices, 2 connected components, and 4 strongly-connected components.",
"role": "user"
}
] | 21 |
[
{
"content": "Find the set of all points where the Taylor series of the function \\( f(x)=\\sum_{n=0}^{\\infty} \\frac {x^2}{(1+x^2)^n} \\) around the point \\( x=e \\) converges to \\( f(x) \\).",
"role": "user"
}
] | \mathbb{R} \setminus \{0\} |
[
{
"content": "Determine whether there exist polynomials $f_1, \\dots, f_k \\in \\mathbb{Z}[x]$ such that for any prime $p$, there is an index $i(p) \\leq k$ for which $f_{i(p)}$ is irreducible and of the same degree modulo $p$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $X, Y,$ and $Z$ be independent random variables, and let $\\mathscr{C} = \\sigma(Y, Z)$. If $C \\in \\mathscr{C}$, are $X$ and $1_C$ always independent?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the maximum size of a subset \\( A \\) of the set \\( M = \\{1, 2, \\ldots, 2017\\} \\) such that any two numbers in \\( A \\) are neither relatively prime nor does one divide the other.",
"role": "user"
}
] | 504 |
[
{
"content": "In triangle $\\triangle ABC$, the angle bisectors of $\\angle BAC$ and $\\angle BCA$ intersect sides $BC$ and $AB$ at points $K$ and $P$, respectively. Given that $AC = AP + CK$, determine the measure of $\\angle ABC$. \\(\\text{Provide your answer in degrees.}\\)",
"role": "user"
}
] | 60 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to0}(x\\Gamma(x))^\\frac{1}{x} \\]",
"role": "user"
}
] | e^{-\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... | 2i - j |
[
{
"content": "Determine whether the n-th anti-derivative of $\\log x$ can be expressed using elementary functions.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether there exists a locally integrable function $g$ on the interval $(0,1)$ such that \\( \\int_x^1 g(t) \\, dt = \\ln(x) \\) for all \\( x \\in (0,1) \\).",
"role": "user"
}
] | -\dfrac{1}{x} |
[
{
"content": "Calculate the sum: \\( \\sum_{k=0}^{2n} \\cos \\left(\\alpha+\\frac{2k\\pi}{2n+1}\\right) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Prove that \\( \\left(\\frac{1}{n!}\\right)^{1/n} \\to 0 \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the residue of the function \\( H(z) = \\frac{z^3}{(z^2-1)^2} \\) at the point \\( z=1 \\) using the formula for residues at poles of higher order.",
"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... | C2H4O |
[
{
"content": "In triangle \\( ABC \\), let \\( AL, BM, CN \\) be the medians intersecting at point \\( K \\). Given that a circumcircle can be drawn around quadrilateral \\( CLK M \\) and \\( AB = 2 \\), determine the length of the median \\( CN \\).",
"role": "user"
}
] | \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... | 11.84 m |
[
{
"content": "In a minimal uniquely ergodic system \\((X,T)\\) with measure \\(\\mu\\), does there exist a point \\(p \\in X\\) such that \\(\\mu(\\partial B(p,t)) = 0\\) for all \\(t > 0\\) with respect to some metric \\(d\\) that induces the same topology?",
"role": "user"
}
] | Yes |
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