prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine whether there exists a submodule $I$ of a module $M$ such that $I$ contains no irreducible submodules of $M$, yet $I$ is not the zero module. Consider the case where $M$ is not semisimple.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $Y$ be an exponentially distributed random variable with rate parameter 3, i.e., $Y \\sim \\text{Exp}(3)$. Given $Y = y$, the random variable $X$ is uniformly distributed over the interval $[y, 3y]$. Compute the variance of $X$.",
"role": "user"
}
] | \dfrac{14}{27} |
[
{
"content": "Consider the sequence of real numbers \\((a_{n})\\) defined by the recurrence relation:\n\\[ \\frac{n(n+2)}{(n+1)}a_{n}-\\frac{n^2-1}{n}a_{n+1}=n(n+1)a_{n}a_{n+1} \\]\nwith \\(a_{1} > 0\\). Let \\((b_{n})\\) be a convergent sequence of real numbers. Calculate:\n\\[ \\lim_{n\\rightarrow\\infty}\\fr... | 0 |
[
{
"content": "Determine whether it is possible to define an inner product on $\\ell^1(\\mathbb{N})$ that induces a norm not equivalent to $\\|\\cdot\\|_1$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the periodicity of the entire function $g(z)$, given that $g(iy)$ and $g(2+iy)$ are real for all $y \\in \\mathbb{R}$. Provide your answer as the period of the function.",
"role": "user"
}
] | 4 |
[
{
"content": "Find the complex number \\( z \\) that satisfies the conditions:\n\\[ |z - \\mathrm{i}| = |z - 1| = |z - 2015|. \\]\nWhat is the value of \\( z \\)?",
"role": "user"
}
] | 1008 + 1008i |
[
{
"content": "Given the transition matrix \\( P = \\begin{pmatrix} 0 & 0.5 & 0.5 & 0 & 0 \\\\\n 0 & 0 & 0 & 0.2 & 0.8 \\\\\n 0 & 0 & 0 & 0.4 & 0.6 \\\\\n 1 & 0 & 0 & 0 & 0 \\\\\n 0.5 & 0 & 0 & 0 & 0.5 \\end{pmatrix} \\), compute the probability that a Markov chain will enter state 5 before it enters state 3, st... | \dfrac{4}{9} |
[
{
"content": "A spotlight is placed at the center of a cube, illuminating a right-angled trihedral corner. Can the spotlight be oriented such that none of the vertices of the cube are illuminated?",
"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... | 590 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... | 356 J |
[
{
"content": "Find a twice differentiable function \\( f: \\Bbb{R} \\to \\Bbb{R} \\) such that \\( f''(x) = (x^2 - 1)f(x) \\) with the initial conditions \\( f(0) = 1 \\) and \\( f'(0) = 0 \\).",
"role": "user"
}
] | e^{-x^2/2} |
[
{
"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... | 196 m/s |
[
{
"content": "Fifty identical six-sided dice are rolled. Determine the number of distinct outcomes where each face (1 through 6) appears an even number of times.",
"role": "user"
}
] | 142506 |
[
{
"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... | Year 2
Sales: $500,000, 100%
Cost of goods sold: $325,000, 65%
Gross profit: $175,000, 35%
Selling expenses: $70,000, 14%
Administrative expenses: $75,000, 15%
Total expenses: $145,000, 29%
Income from operations: $30,000, 6%
Income tax expenses: $10,500, 2%
Net income: $19,500, 4%
Year 1
Sales: $440,000, 100%
Cost of... |
[
{
"content": "Evaluate the contour integral \\( \\oint_{\\gamma}(2z-3\\bar{z}+1)\\,dz \\) where \\( \\gamma \\) is the ellipse \\( \\frac{x^{2}}{4}+\\frac{y^{2}}{9}=1 \\) traversed clockwise.",
"role": "user"
}
] | 36\pi i |
[
{
"content": "Calculate the variance of the expression \\((W_1W_2 - Z_1Z_2)^2\\), where \\(W_1, W_2, Z_1,\\) and \\(Z_2\\) are independent random variables.",
"role": "user"
}
] | 20 |
[
{
"content": "Determine the positive integers \\(k\\) for which the series \\(\\sum_{n=1}^\\infty \\frac{(n!)^2}{(kn)!}\\) is convergent.",
"role": "user"
}
] | k \geq 2 |
[
{
"content": "Evaluate the limit of the sequence as \\( n \\to \\infty \\):\n\n\\[\n\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+2}-\\sqrt[3]{n^{3}+2}}{\\sqrt[7]{n+2}-\\sqrt[5]{n^{5}+2}}\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... | 8.55 * 10^22 molecules |
[
{
"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... | -1/3 |
[
{
"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... | jerk |
[
{
"content": "Let $Z = 4(U^2 - V)$, where $U$ and $V$ are independent uniform random variables on $(0,1)$. Find the probability that $Z > 0$. In other words, calculate $\\mathbb{P}(Z > 0)$. ",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Find a sufficient condition on \\(\\alpha\\) that ensures the integral is finite.",
"role": "user"
}
] | \alpha > -1 |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} n \\int_{0}^{1} x^n p(x) \\, dx \\), where \\( p(x)=a_0+a_1x+a_2x^2+\\cdots+a_kx^k \\) is a polynomial of degree \\( k \\).",
"role": "user"
}
] | p(1) |
[
{
"content": "Find the minimum value of the expression:\n\\[\n\\left| \\log_{x_1} \\left( x_2 - \\frac{1}{4} \\right) + \\log_{x_2} \\left( x_3 - \\frac{1}{4} \\right) + \\cdots + \\log_{x_{2017}} \\left( x_1 - \\frac{1}{4} \\right) \\right|\n\\]\nwhere \\(x_1, x_2, \\ldots, x_{2017} \\in \\left(\\frac{1}{4}, 1... | 4034 |
[
{
"content": "Evaluate the limit \\( \\lim_{h \\to 0} \\frac{\\int_{-h}^h (x^2 + f(x)) \\, dx}{2h} \\), where \\( f(x) \\) is a continuous function, using properties of integrals and the Fundamental Theorem of Calculus.",
"role": "user"
}
] | f(0) |
[
{
"content": "Determine whether the function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) defined by \\[ f(x) = \\begin{cases} x^2 - 2, & x \\in \\mathbb{I}, \\\\ -x^2 + 2, & x \\in \\mathbb{Q} \\end{cases} \\] is continuous at \\( x = \\pm \\sqrt{2} \\) and whether it is discontinuous everywhere else.",
"role":... | \pm \sqrt{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... | 2.74% |
[
{
"content": "Given that \\( a > b > c \\) and \\( n \\in \\mathbf{N} \\), find the maximum value of \\( n \\) such that the inequality \\( \\frac{1}{a-b}+\\frac{1}{b-c} \\geqslant \\frac{n}{a-c} \\) always holds. Choose the correct answer from the following options:\n\nA. 2 \nB. 3 \nC. 4 \nD. 5",
"role"... | C |
[
{
"content": "If the quadratic $x^2+6mx+m$ has exactly one real root, find the positive value of $m$.",
"role": "user"
}
] | \frac19 |
[
{
"content": "Determine the Galois group of the polynomial $x^5 - 6x + 3$ over the field of rational numbers $\\mathbb{Q}$. Assume $\\alpha$ is a root of the polynomial.",
"role": "user"
}
] | S_5 |
[
{
"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... | 3.972 x 10^-2 m |
[
{
"content": "Determine the range of values of $m$ for which there exists a $p > 0$ such that \\( f(m,p) = \\frac{2}{m^2} - \\sum_{k=m}^{m+p} \\frac{1}{(2k-1)^2} < 0 \\).",
"role": "user"
}
] | m \geq 7 |
[
{
"content": "Given that \\( \\sum_{k=-\\infty}^{\\infty}c_ke^{ikx} \\) is a Fourier series for a function \\( f(x) \\), determine the function to which \\( \\sum_{k=-\\infty}^{\\infty}c_ke^{i4kx} \\) converges.",
"role": "user"
}
] | f(4x) |
[
{
"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... | $5,081.45 |
[
{
"content": "What is the remainder when \\(2^{1990}\\) is divided by \\(1990\\)?",
"role": "user"
}
] | 1024 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\infty} t^3 e^{-3t} \\, dt \\) using the Laplace transform.",
"role": "user"
}
] | \dfrac{2}{27} |
[
{
"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... | $171,638,565.92 |
[
{
"content": "Find a prime number $p$ such that $x^2 + y^2 = p$, $u^2 + 2v^2 = p$, and $r^2 + 3s^2 = p$.",
"role": "user"
}
] | 73 |
[
{
"content": "Compute the limit: $$\\lim_{x\\rightarrow 0 }\\biggr ( \\dfrac{1}{x}\\ln (x!)\\biggr )$$",
"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... | k[(3.358e-13)cos(6.981t) + (6.717e-13)sin(6.981t)] |
[
{
"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 isotopic abundance of Ar-38 is 0.588% and the isotopic abundance of Ar-40 is 99.406%. |
[
{
"content": "Determine whether every invertible real matrix can be decomposed into an orthogonal matrix and a symmetric positive-definite matrix, such that $M = OP$, where $O$ is orthogonal and $P$ is symmetric positive-definite.",
"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... | 0.27 mol |
[
{
"content": "Determine whether the statement is true or false: For any square matrix $A$, if $A^8x=b$, then $b$ is in the image of $A^2$.",
"role": "user"
}
] | True |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.9861 cm |
[
{
"content": "Consider the points \\(A=(0,1)\\), \\(B=(0,0)\\), and \\(C=(1,0)\\). A random point \\((X,Y)\\) is chosen on the union of the line segments \\(\\overline{AB}\\) and \\(\\overline{BC}\\). Find the covariance \\(\\text{Cov}(X,Y)\\) of the coordinates of this random point.",
"role": "user"
}
] | -\dfrac{1}{16} |
[
{
"content": "Find the number of positive integer pairs \\((p, n)\\), where \\(p\\) is a prime number, such that \\[ p^2 = \\frac{(2^n + 1)^5 - 1}{2^n} \\].",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the prime powers $q$ for which the map \\( \\varphi: \\Bbb{F}_q^2 \\to \\Bbb{F}_q^2 \\) defined by \\( (x,y) \\mapsto (x^q,y^q) \\) is an \\( \\Bbb{F}_q \\)-linear transformation.",
"role": "user"
}
] | q |
[
{
"content": "Given the constraints:\n\\[ x_1 \\leq 4, \\quad x_1 + x_2 \\leq 13, \\quad x_1 + x_2 + x_3 \\leq 29, \\quad x_1 + x_2 + x_3 + x_4 \\leq 54, \\quad x_1 + x_2 + x_3 + x_4 + x_5 \\leq 90 \\]\nwhere \\( x_1, x_2, x_3, x_4, x_5 \\geq 0 \\), find the maximum value of the expression:\n\\[ \\sqrt{x_1} + \... | 20 |
[
{
"content": "Evaluate the integral \\( \\int\\int\\int_E z \\, dV \\), where \\( E \\) is the region above the paraboloid \\( z = x^2 + y^2 \\) and below the plane \\( z = 6y \\). Express the integral in cylindrical coordinates and compute the volume, rounding your answer to the nearest hundredth.",
"role"... | 1908.52 |
[
{
"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... | 2kQ/(r*sqrt(4r^2+L^2)) |
[
{
"content": "Using Rouché's theorem, determine how many roots of the polynomial \\( p(z) = z^8 + 3z^7 + 6z^2 + 1 \\) are contained in the annulus \\( 1 < |z| < 2 \\).",
"role": "user"
}
] | 5 |
[
{
"content": "Solve for $x$: $$2^x+6=3\\cdot2^x-26$$",
"role": "user"
}
] | 4 |
[
{
"content": "Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$. The point $P$ is on $CN$ and $\\angle CAP = \\angle CBP = 10^\\circ$. If arc $MA$ is equal to $40^\\circ$, then find arc $BN$ (in degrees).\n\n[asy]\nimport geometry;\nimport graph;\n\nunitsize(2 cm);\n\npair A, B... | 20^\circ |
[
{
"content": "Determine the radius of convergence for the series \\(\\sum_{n=1}^{\\infty}{\\frac{(2n)!}{(3n+1)!}x^n}\\).",
"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... | 0.86 kJ/K |
[
{
"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... | 1.60 |
[
{
"content": "Given an uncountable subset $S$ of the unit interval $[0,1]$, determine whether $S$ has an accumulation point that is also a member of $S$. 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... | 1.10 M |
[
{
"content": "Find the natural number \\( n \\) such that the field \\( \\mathbf{Q}(z_{5},z_{7}) \\) is equal to \\( \\mathbf{Q}(z_{n}) \\), where \\( z_{k} = e^{2\\pi i / k} \\) is a primitive \\( k \\)th root of unity.",
"role": "user"
}
] | 35 |
[
{
"content": "Let $X_1, X_2, \\ldots, X_N$ be independent and identically distributed random variables following a normal distribution. Calculate the probability that $X_1 < X_2 < \\ldots < X_N$. Express your answer as a single probability value.",
"role": "user"
}
] | \dfrac{1}{N!} |
[
{
"content": "Find the radius of convergence for the series \\(\\sum_{n=1}^\\infty \\frac{x^{n!}}{n!}\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the points on a given line that maximize the absolute difference between the distances from two fixed points lying on opposite sides of the line.",
"role": "user"
}
] | P |
[
{
"content": "Find the maximum value of the expression \\( x_{1} + 2x_{2} + \\cdots + 49x_{49} \\) given that \\( x_{1}^{2} + 2x_{2}^{2} + \\cdots + 49x_{49}^{2} = 1 \\).",
"role": "user"
}
] | 35 |
[
{
"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... | 6.7e-2 m |
[
{
"content": "Given that \\( \\lim_{n \\to \\infty}(a \\sqrt[n]{n} + b)^{n/\\ln n} = e^{-3} \\), find the value of \\(4b + 3a\\).",
"role": "user"
}
] | 7 |
[
{
"content": "Find the optimal solution to the optimization problem:\n\\[\\begin{aligned}\n&\\min_{\\mathbf{x} \\in \\mathbb{R}^n} && \\|\\mathbf{B}\\mathbf{x}\\|_1 \\\\\n&\\ \\ \\text{s.t.} && \\|\\mathbf{x}\\|_2 \\leq 1\n\\end{aligned}\\]\nwhere \\(\\mathbf{B} \\in \\mathbb{R}^{n \\times n}\\).",
"role": ... | \mathbf{0} |
[
{
"content": "Find the length of the circumference of the unit circle \\((x^2+y^2=1)\\) using the limit formula for arc length:\n\n\\[L = \\lim\\limits_{h\\rightarrow 0} \\sum\\limits_{r=1}^{\\frac{b-a}{h}} \\sqrt{\\left\\{ f\\Big(a+rh\\Big)-f\\Big(a+(r-1)h\\Big)\\right\\}^2 +h^2}\\]",
"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... | 1110.1 Pa |
[
{
"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... | 46682.67 Hz |
[
{
"content": "Find the limit as \\( n \\to \\infty \\) of the following expression:\n\\[\n\\frac{e^{1/n}}{n^2} + 2\\frac{e^{2/n}}{n^2} + 3\\frac{e^{3/n}}{n^2} + \\ldots + n \\frac{e}{n^2}\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... | $27,131.27 |
[
{
"content": "Determine the values of $n$ for which two non-similar invertible matrices, having the same minimal and characteristic polynomials and the same dimension of each eigenspace, are guaranteed to be similar.",
"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... | 2,900 |
[
{
"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... | 196 g |
[
{
"content": "Evaluate the limit of the following integral: \\[ \\lim_{n \\rightarrow \\infty} \\int_{[0,1]} \\left(\\frac{1 - \\sin^2 (x)}{n}\\right)^6 \\, dm(x) \\]",
"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... | 786.039 m/s |
[
{
"content": "A differentiable function \\( f : \\mathbb{R} \\to \\mathbb{R} \\) satisfies the following conditions:\n\n- \\( f(2) = 5 \\)\n- \\( f(3) = 8 \\)\n- \\( f'(x) \\le 4 \\sin^2(\\pi x) \\) for all \\( x \\in \\mathbb{R} \\).\n\nDetermine how many values of \\( s \\ge 2 \\) satisfy \\( f(s) = s^2 \\)."... | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 3} \\left(\\frac{\\sin(x)}{\\sin(3)}\\right)^{\\frac{1}{x-3}} \\] without using L'Hospital's rule or any derivative definition.",
"role": "user"
}
] | e^{\cot 3} |
[
{
"content": "For each natural number \\( n \\), find the possible values of the expression:\n\n\\[ \\gcd(2^{2n+1}+3^{n},2^{2n+2}+3^{n+1}) \\]",
"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... | $3,905.3 |
[
{
"content": "Find the smallest positive integer \\( n \\) such that \\( n/1 \\) generates the ideal \\( \\langle 4/17^8 \\rangle \\) in the ring \\( \\mathbb{Z}[1/17] \\). Explain why the generator of the ideal can be expressed as \\( n/1 \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Let $\\overline{X_n}$ denote the mean of a random sample of size $n$ from a Poisson distribution with mean $\\lambda = 2$. Use the delta method to find the limiting distribution of $\\sqrt{n}\\left( (\\overline{X_n})^2 -4 \\right)$.",
"role": "user"
}
] | \mathcal{N}(0, 32) |
[
{
"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... | 1157 A |
[
{
"content": "If $a,b,c$ are non-negative integers less than $7$ such that \\begin{align*}\na+2b+3c&\\equiv 0\\pmod 7,\\\\\n2a+3b+c&\\equiv 4\\pmod 7,\\\\\n3a+b+2c&\\equiv 4\\pmod 7,\n\\end{align*}then determine the remainder when $abc$ is divided by $7$.",
"role": "user"
}
] | 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... | 5.40% |
[
{
"content": "Can a linear fractional transformation that maps a disk on the Riemann sphere to itself be loxodromic?",
"role": "user"
}
] | No |
[
{
"content": "Find the value of \\( t \\) that satisfies the equation: \\( \\sin^{-1} t - \\sin^{-1} 2t = \\sin^{-1} 4t \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the Frechet derivative of the operator \\( A \\) exists, where \\( A \\colon C[a,b] \\to C[a,b] \\) is defined by \\( [A(x)](s) = \\int_a^b K(s,t) x(t) \\, \\mathrm{d}t \\) for all \\( x \\in C[a,b] \\) and \\( s \\in [a,b] \\). Here, \\( K \\colon [a,b] \\times [a,b] \\to \\math... | A |
[
{
"content": "Let $G$ be a noncyclic finite group of order $27$. Determine the number of elements $x \\in G$ such that $x^9 = e$, where $e$ is the identity element of $G$. Assume you cannot use the Sylow theorems.",
"role": "user"
}
] | 27 |
[
{
"content": "Determine the number of non-similar matrices with the characteristic polynomial \\((\\lambda-1)^4(\\lambda-2)^3\\).",
"role": "user"
}
] | 15 |
[
{
"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... | Increase retirement contributions, contribute to employer-sponsored funds, and donate to charity. |
[
{
"content": "Consider a triangle where instead of trisecting the vertex angles, we bisect the opposite sides and draw lines from the vertices to these points. Determine the area of the resulting triangle in comparison to the original triangle.",
"role": "user"
}
] | \dfrac{1}{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... | 0.368m |
[
{
"content": "If the quadratic equations \\(3x^2+ax+1=0\\) and \\(2x^2+bx+1=0\\) have a common root, find the value of \\(5ab-2a^2-3b^2\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the number of invertible $2\\times2$ matrices over the field $\\mathbb{Z}/p\\mathbb{Z}$ that are unimodular, meaning their determinant is either $1$ or $-1$. Provide your answer as a single integer.",
"role": "user"
}
] | 2p(p^2 - 1) |
[
{
"content": "Evaluate the complex integral \\( \\int_C \\frac{d|z|}{\\overline{z}} \\), where \\( C = \\{ e^{it} : t \\in [0, 2\\pi] \\} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the value of $k$ so that\n\\[3 + \\frac{3 + k}{4} + \\frac{3 + 2k}{4^2} + \\frac{3 + 3k}{4^3} + \\dotsb = 8.\\]",
"role": "user"
}
] | 9 |
[
{
"content": "Determine the values of \\( n \\) for which the limit \\[ \\lim_{(x,y) \\to (0,0)} \\frac{x^2y}{x^{2n}+y^2} \\] exists.",
"role": "user"
}
] | n < 2 |
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