prompt list | ground_truth string |
|---|---|
[
{
"content": "Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the \"base\" $-n+i$ using the integers $1,2,\\ldots,n^2$ as \"digits.\" That is, the equation\\[r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\\cdots +a_1... | 490 |
[
{
"content": "How many prime numbers are divisible by $39$ ?",
"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... | 725 m |
[
{
"content": "Is the monadic fragment of inclusive first order logic decidable?",
"role": "user"
}
] | Yes |
[
{
"content": "Given seven points on a plane, where the distances between them are denoted by $a_1, a_2, \\ldots, a_{21}$, determine the maximum number of times the same distance value can occur among these 21 distances.",
"role": "user"
}
] | 12 |
[
{
"content": "Let \\( f: [0,4] \\to [1,3] \\) be a differentiable function such that \\( f'(x) \\neq 1 \\) for all \\( x \\in [0,4] \\). Determine the number of fixed points of \\( f \\). Choose the correct option:\n\n(A) \\( f \\) has at most one fixed point.\n(B) \\( f \\) has a unique fixed point.\n(C) \\( f... | 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... | 5.9 m/min |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to0}\\frac{\\int_x^{x^2}\\sinh(t)\\sin(t)\\,dt}{\\int_0^x t^3\\csc(t)\\,dt}. \\]",
"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... | 0.0134 Tesla |
[
{
"content": "Find the adjoint of the operator $A: L^2 \\to L^2$ defined by $Ax = (\\lambda_1 x_1, \\lambda_2 x_2, \\ldots)$, where $|\\lambda_n| \\leq 1$ and $\\lambda_n \\in \\mathbb{R}$.",
"role": "user"
}
] | A^* x = (\lambda_1 x_1, \lambda_2 x_2, \ldots) |
[
{
"content": "Is the set \\( U_A = B_{\\frac{1}{\\|A^{-1}\\|}}(A) \\) an open subset of \\( M(n,K) \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Provide an example of a group that has a cyclic subgroup which is not normal.",
"role": "user"
}
] | S_3 |
[
{
"content": "Evaluate the contour integral \\(\\int_{|z-i|=\\frac{1}{9}} \\frac{1}{z^4+16} \\, dz\\).",
"role": "user"
}
] | 0 |
[
{
"content": "What is the maximum possible length of a geometric progression of prime numbers where the common ratio is a non-integer rational number?",
"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... | 4.2e6 m/s |
[
{
"content": "Compute the contour integral \\( \\int_{\\mathbb{R} - 10i} dz \\frac{e^{iz}}{1 + z^2} \\). The contour \\( \\mathbb{R} - 10i \\) is a horizontal line in the complex plane, parallel to the real axis, and shifted down by 10 units. The integral is oriented from left to right. Express your answer in t... | \pi \left( \dfrac{1}{e} - e \right) |
[
{
"content": "Determine whether a polynomial $P(x) \\in \\mathbb{R}[x]$ that satisfies $P(x) = 2^n$ for at least one rational root for every $n \\in \\mathbb{N}$ must necessarily be linear.",
"role": "user"
}
] | Yes |
[
{
"content": "Suppose the function $f(x,y,z)=xyz$ is defined for $x+y+z=7$, $x,y,z \\ge 0$. What is the range of $f$?",
"role": "user"
}
] | \left[0,\frac{343}{27}\right] |
[
{
"content": "Find the constant \\( c \\) such that the function \\( f_{X,Y}(x,y) = \\begin{cases} cx, & \\text{if } x > 0, y > 0, 1 \\leq x+y \\leq 2, \\\\ 0, & \\text{elsewhere} \\end{cases} \\) is a valid probability density function.",
"role": "user"
}
] | \dfrac{6}{7} |
[
{
"content": "Evaluate the integral: \\[ \\iint_{[0,1]\\times [0,1]} \\max\\{x,y\\}\\,dx dy \\]",
"role": "user"
}
] | \dfrac{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... | 69.62 mT |
[
{
"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% |
[
{
"content": "Determine the number of distinct spanning trees in the complete graph $K_9$ such that vertex $1$ has a degree of $4$. Use Prufer codes to find the solution.",
"role": "user"
}
] | 143360 |
[
{
"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... | 121000 |
[
{
"content": "Determine the radius of convergence of the series \\( \\sum_{n=0}^{\\infty} z^{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... | sin t = \frac{Ru}{(k+R)w} |
[
{
"content": "Find the volume of the region bounded above by the cylinder \\( z = x^2 \\) and below by the region enclosed by the parabola \\( y = 2 - x^2 \\) and the line \\( y = x \\).",
"role": "user"
}
] | \dfrac{63}{20} |
[
{
"content": "Calculate the work done by the force field \\( F = (z, y^2, -x) \\) on a particle moving along the curve defined by the intersection of the surfaces \\( y = z^2 \\) and \\( x^2 + z^2 = 1 \\).",
"role": "user"
}
] | -2\pi |
[
{
"content": "Find the coefficient of \\(s^{14}\\) in the expansion of \\((s+s^2+s^3+s^4+s^5+s^6)^7\\).",
"role": "user"
}
] | 1667 |
[
{
"content": "Determine how many equilateral triangles can be formed with pearls, starting with 2008 pearls and increasing the number of pearls by an arithmetic sequence where each term increases by 1 more than the previous increase. Specifically, after n rounds, the number of pearls is given by \\( 2008 + \\fr... | 2 |
[
{
"content": "Which of the following sets is not a null set as defined by the given criteria? A set $X \\subseteq \\mathbb{R}$ is a null set if for every $\\epsilon > 0$, there exists a countable collection $\\{(a_k, b_k)\\}_{k=1}^\\infty$ of open intervals such that $X \\subseteq \\cup_{k=1}^\\infty (a_k, b_k)... | B |
[
{
"content": "Evaluate the fourth finite difference \\(\\bigtriangleup ^{4}\\) of the function \\((x^2 + 2x)^3\\) at \\(x=1\\).",
"role": "user"
}
] | 5808 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow 0}\\frac{1}{x}\\int_{0}^{x}\\sin(t^3){\\rm d}t$$ using the fundamental theorem of calculus and L'Hopital's rule.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_2^\\infty \\frac{\\ln(t-1)}{(t-1)^5} \\, dt \\) using substitution and integration by parts.",
"role": "user"
}
] | \dfrac{1}{16} |
[
{
"content": "Consider a bipartite graph that has a Hamiltonian path but does not have a Hamiltonian cycle. Is it possible for the partitions X and Y of this graph to have different sizes?",
"role": "user"
}
] | Yes |
[
{
"content": "Solve the equation \\(\\ln x = \\frac{1}{x} - 1\\) analytically, without using graphing or guess-and-check methods.",
"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... | 4.219 |
[
{
"content": "Which of the following subsets of natural numbers \\(\\mathbb{N}\\) intersects every set of the form \\(\\{a + nd : n \\in \\mathbb{N}\\}\\), where \\(a, d \\in \\mathbb{N}\\)?\n\n(a) \\(\\{2k : k \\in \\mathbb{N}\\}\\) \n(b) \\(\\{k^2: k \\in\\mathbb{N}\\}\\) \n(c) \\(\\{k + k! : k \\in \\mathb... | 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... | (2.75 m/s)i + (2.04 m/s)j |
[
{
"content": "Let $X$ and $Y$ be independent random variables, each uniformly distributed over the interval (0,1). Find the probability density function of the random variable $Z = XY$. Provide your answer in terms of $z$ for $0 < z < 1$. ",
"role": "user"
}
] | -\ln z |
[
{
"content": "Find all values of $t \\geq 1$ such that the recursively defined sequence $(a_n)$, where $a_1 = t$ and $a_{n+1}a_n = 3a_n - 2$, is increasing.",
"role": "user"
}
] | (1, 2) |
[
{
"content": "Determine whether the following inequality holds for continuous functions $f(x)$ and $g(x)$, given that $\\int_0^1 f(x) \\, dx = 0$:\n\n$$\\left( \\int_0^1 f(x)g(x) \\, dx \\right)^2 \\leq \\left( \\int_0^1 g(x)^2 \\, dx - \\left( \\int_0^1 g(x) \\, dx \\right)^2 \\right) \\left( \\int_0^1 f(x)^2 ... | 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... | [N2O4] = 0.014 M, [NO2] = 0.082 M |
[
{
"content": "Let $X$ and $Y$ be independent random variables with standard normal distributions. Define $Z = X^2 + Y$. Find the expected value $E(Z/X)$. Express your answer in terms of known constants or functions.",
"role": "user"
}
] | 0 |
[
{
"content": "Consider an ellipse given by \\(\\frac{x^2}{4} + y^2 = 1\\). A point \\(P\\) on the ellipse has an eccentric angle of \\(45^\\circ\\). Through this point, variable pairs of chords are drawn such that they are perpendicular to each other and meet the ellipse at points \\(A\\) and \\(B\\). If the li... | \dfrac{9}{10} |
[
{
"content": "Find the sum of the squares of all elements in the solution set \\( A \\) of the equation \\( x^{2}+x-1=x \\pi^{x^{2}-1}+\\left(x^{2}-1\\right) \\pi^{x} \\), where \\( \\pi \\) is an irrational number and \\( x \\) is a real number. Choose the correct answer from the following options:\n\nA. 0 \... | C |
[
{
"content": "Let $X_1$ and $X_2$ be independent and identically distributed continuous random variables with probability density function $f(x) = 6x(1-x)$ for $0 < x < 1$, and $f(x) = 0$ otherwise. Using Chebyshev's inequality, determine the lower bound for the probability $P\\left(|X_1 + X_2 - 1| \\le \\frac{... | \dfrac{3}{5} |
[
{
"content": "Determine whether there exists a subfield of the real numbers that is both uncountable and locally compact.",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "Determine the residue of the function \\( f(z) = \\frac{1}{z - \\sin z} \\) at \\( z = 0 \\).",
"role": "user"
}
] | \dfrac{3}{10} |
[
{
"content": "Determine the range of values for $a$ such that for any $x > 0$, the inequality $f(x) = a - x - |\\log x| \\leq 0$ holds.",
"role": "user"
}
] | (-\infty, 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... | 1.02e-3 cm^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... | $4,065.57 |
[
{
"content": "Evaluate the line integral \\( \\int_\\gamma x^2\\,dx + y^2\\,dy \\), where \\( \\gamma \\) is the circle defined by \\((x-c)^2+(y-d)^2=s^2\\), traversed one revolution clockwise.",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\( \\Omega \\) be a connected and simply-connected open subset of \\( \\mathbf{C} \\) that does not contain the origin. Consider a simple smooth curve \\( \\gamma :[0,1] \\to \\Omega \\) with \\( \\zeta := \\gamma(0) \\). Let \\( \\phi:[0,1] \\to \\mathbf{C} \\) be a smooth map such that \\( ... | Yes |
[
{
"content": "Evaluate the integral \\( \\int_{-1}^{0}\\frac{x^2+2x}{\\ln(x+1)}dx \\).",
"role": "user"
}
] | \ln 3 |
[
{
"content": "Suppose $f$ and $g$ are two analytic functions on an open set containing the closed disc $\\overline{D}$, where $D$ is the open disc in $\\mathbb{C}$ with radius 1. Given that $|f(z)| = |g(z)|$ for all $z$ on the boundary of $D$, and that $f$ has a zero of order $n$ at the origin, determine the or... | n |
[
{
"content": "Let $A$ be a symmetric positive definite matrix and $D$ be a diagonal matrix with positive elements. Determine if the induced $2$-norm of the matrix $K = (A + D)^{-1}A$ satisfies $\\|K\\| \\leq 1$. Provide a justification for your answer.",
"role": "user"
}
] | \|K\| \leq 1 |
[
{
"content": "Given a vector $x$ and a subspace $A$ spanned by vectors $a_1$ and $a_2$, if the projection of $x$ onto $A$ is denoted by $x^* = Px$, where $P = A(A^TA)^{-1}A^T$ is the projection matrix, and there exists a vector $y$ in $A$ that is orthogonal to $x^*$, is $y$ also orthogonal to the original vecto... | Yes |
[
{
"content": "Consider a system of polynomials $f_1, \\ldots, f_m \\in \\mathbb{F}[x_1, \\ldots, x_n]$ each of degree $d$, where $n = O(1)$. If this system has a solution in the algebraic closure $\\bar{\\mathbb{F}}$, what is the minimum degree of extension of $\\mathbb{F}$ required to guarantee a solution? Exp... | d^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... | 13.4% |
[
{
"content": "Determine the minimum possible size of the group of isometries that preserve a genus $g \\geq 2$ surface smoothly embedded in $\\mathbb{R}^3$, considering both orientation-preserving and orientation-reversing isometries.",
"role": "user"
}
] | 1 |
[
{
"content": "A sequence $\\{a_n\\}$ is defined by the recurrence relation $a_n = a_{n-1} + 2a_{n-2} - a_{n-3}$ with initial conditions $a_1 = a_2 = \\frac{a_3}{3} = 1$. Find the value of the infinite series $$a_1 + \\frac{a_2}{2} + \\frac{a_3}{2^2} + \\cdots$$",
"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... | 25 cm |
[
{
"content": "Determine the measure of the set of all irrational numbers $x$ in the interval $[0,1]$ such that, in its decimal representation, if a $1$ appears, it is immediately followed by a $2$. Assume the decimal representation does not terminate.",
"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... | 5.04e-4 mol/L |
[
{
"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... | 8.73% |
[
{
"content": "In triangle \\(KLM\\), the base \\(KM\\) is \\(\\frac{\\sqrt{3}}{2}\\) and side \\(KL\\) is 1. A circle passes through points \\(K\\) and \\(L\\) with its center on the altitude \\(LF\\) dropped to the base \\(KM\\). Given that \\(FM = \\frac{\\sqrt{3}}{6}\\) and point \\(F\\) lies on \\(KM\\), fi... | \dfrac{3\pi}{8} |
[
{
"content": "Determine the possible values of the positive integer $n$ such that $2018^n = a^4 + b^4 + (b^2 + c^2)^2$, where $a$, $b$, and $c$ are positive integers.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the line integral \\( \\int_{\\gamma} \\frac{e^{iz}}{z^2} \\, dz \\), where \\( \\gamma(t) = e^{it} \\) for \\( 0 \\leq t \\leq 2\\pi \\).",
"role": "user"
}
] | -2\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... | 3.24 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... | $181,400 |
[
{
"content": "Through the focus $F$ of the parabola $y^2 = 8(x+2)$, a line with an inclination angle of $60^{\\circ}$ is drawn. If this line intersects the parabola at points $A$ and $B$, and the perpendicular bisector of segment $AB$ intersects the $x$-axis at point $P$, find the length of segment $PF$. Choose... | 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... | 0.0708 moles |
[
{
"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 angle alpha lies between the two values when x = 7mm and x = 7.3mm |
[
{
"content": "Determine whether there exists a bijection \\( f \\) on \\( n \\) elements such that \\( f(1) = 1 \\) and \\( f(i) + j \\equiv f(j) + i \\pmod{n} \\) for all \\( 1 \\leq i < j \\leq n \\), given that \\( n \\) is odd.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the following statement is true or false: In any metric space, a non-empty open set without any isolated points is also closed.",
"role": "user"
}
] | B |
[
{
"content": "Verify the three-dimensional delta function property: \\[ \\int_{a}^{b} \\int_{c}^{d} \\int_{e}^{f} \\delta(x-X)\\delta(y-Y)\\delta(z-Z) f(x)f(y)f(z) \\, dx \\, dy \\, dz = f(X)f(Y)f(Z) \\] where \\( a<X<b, c<Y<d, e<Z<f \\) define the volume where the delta function is located.",
"role": "user... | f(X)f(Y)f(Z) |
[
{
"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... | 902 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... | 13.1 m/s |
[
{
"content": "Determine the maximum area of a rectangle DACB, where points C and B lie on the curve \\( y = \\frac{8}{1+x^2} \\) and points A and D are on the x-axis.",
"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... | 3.44 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... | 1954 m/s^2 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\left(\\frac{\\sin^n1+\\cos^n1}{2}\\right)^n.$$",
"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... | 103 kJ |
[
{
"content": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a strictly increasing function that is a one-to-one mapping. Define $P$ as the solution set of the equation $x = f(x)$ and $Q$ as the solution set of the equation $x = f(f(x))$. Determine the correct relationship between the sets $P$ and $Q$:\n\n(A)... | B |
[
{
"content": "Is the base field $F$ uniquely determined in the definition of a CM-field?",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the volume under the surface defined by the function \\( f(x,y) = |x-y| \\) over the triangular region with vertices at \\((0,0)\\), \\((l,0)\\), and \\((0,l)\\).",
"role": "user"
}
] | \dfrac{l^3}{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... | 1.65e13 electrons |
[
{
"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... | 1.485e-20 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... | 3 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... | 0.27 s |
[
{
"content": "Find an irreducible polynomial in \\(\\mathbb{Z}[x]\\) that is reducible modulo 2, 3, and 5.",
"role": "user"
}
] | x^4 + 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... | $6,328 |
[
{
"content": "Seven sticks with lengths 2, 3, 5, 7, 11, 13 and 17 inches are placed in a box. Three of the sticks are randomly selected. What is the probability that a triangle can be formed by joining the endpoints of the sticks? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{9}{35} |
[
{
"content": "Determine whether the coefficient \\( m_{a,b,c} \\) of the monomial \\( x_1^{a+c+1}x_2^{a+b+1}x_3^{b+c+1} \\) in the expansion of the polynomial \\( f_{a,b,c}(x_1,x_2,x_3) = (x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1} \\) is zero when \\( a, b, c \\) are pairwise unequal nonnegative integers.... | 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... | 8.49% |
[
{
"content": "Find the limit of the sequence using the Stolz-Cesaro theorem: $$\\lim _{n\\to \\infty }\\left(\\sum _{i=1}^n\\:\\frac{i^2}{2^i}\\right)$$",
"role": "user"
}
] | 6 |
[
{
"content": "Find the value of the limit: $$\\lim_{n\\to\\infty} \\sum_{k=0}^n \\frac{{k!}^{2} {2}^{k}}{(2k+1)!}$$",
"role": "user"
}
] | \dfrac{\pi}{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... | $17.67 |
[
{
"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... | 0.013 mol |
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