messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation $x^{y} - x = 17$ over the integers."
},
{
"role": "assistant",
"content": "<think>\nAlright, let's see. I need to solve the equ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $S$ be the set of the $125$ points in three-dimensional space of the form $(x, y, z)$ where $x$, $y$, and $z$ are integers between $1$ and $5$, inclusive.... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the limit $\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{{e^x} - \\sqrt {1 - {x^2}} }}{{{x^2}}}$."
},
{
"role": "assistant",
"content": "<... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many positive integers \\( x \\) are there such that \\( \\sqrt{2017 - \\sqrt{x}} \\) is an integer?"
},
{
"role": "assistant",
"content": "<t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( R_I, R_{II}, R_{III}, R_{IV} \\) be the areas of the elliptical region \\(\\frac{(x-10)^2}{10} + \\frac{(y-31)^2}{31} \\le 2009\\) that lie in the fir... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all triples $(n, x, y)$ of positive integers satisfying $n^x + 1 = (n + 1)^y$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's se... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the smallest number of pieces to cut a cube with edge length 6 so that cubes with edge lengths 3, 4, and 5 can be made from them?"
},
{
"role"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Describe all groups \\( G \\) which have the property that for all subgroups \\( H \\le G \\) and for all \\( x, y \\in G \\), if \\( xy \\in H \\), then eith... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the least and greatest value of \\( P = \\sin x \\sqrt{1 + \\cos x} + \\cos x \\sqrt{1 + \\sin x} \\)."
},
{
"role": "assistant",
"content": ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a subset $S$ of $\\mathbb{R}^2$ such that no three points in $S$ are collinear. Let $S$ be dense in $\\mathbb{R}^2$, meaning that for any point $(x, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\mathbb{Z}/12\\mathbb{Z}$, if $\\overline{a} = 5$ and $\\overline{b} = 8$, why do $\\overline{13}$ and $\\overline{1}$ lie in the same equivalence class ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given an equilateral triangle TVW with a point Y inside such that TY=38, VY=42, and WY=44, find the radius of the circumscribed circle of triangle TVW, rounde... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( p_1, p_2, \\ldots, p_n \\) be prime numbers (\\( n > 2 \\)). For \\( 1 \\le i \\le n-1 \\), let \\( q_i = p_1 \\cdots p_{i-1} p_{i+1} \\cdots p_n \\).... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all entire functions $f(z)$ such that $|f(z)| \\ge e^{-\\text{Re}(z)\\text{Im}(z)}$ for all $z \\in \\mathbb{C}$."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int \\frac{1}{1 + x^4} \\, dx$."
},
{
"role": "assistant",
"content": "<think>\nOkay, I need to evaluate the integral of ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int_{0}^{\\pi} \\frac{d\\theta}{(3-2\\cos(\\theta))^2}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For a positive integer \\( n \\), let \\( f(n) = \\frac{\\log n}{\\omega(n) \\log \\log n} \\). Find the minimum value of \\( f(n) \\) over all positive integ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For \\(a, b, c \\in \\mathbb{R}^+\\) with \\(abc = 1\\), calculate the minimum value of the expression \\(\\frac{1}{a^5(b^2+c^2)} + \\frac{1}{b^5(a^2+c^2)} + ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the acute-angled triangle \\(ABC\\), let \\(H_A, H_B, H_C\\) be the feet of the perpendiculars from the vertices \\(A, B, C\\) to the corresponding sides r... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Give an example of a function $f$ that satisfies $f(x+1) = f(x)$ and $f(x) > 0$ for all $x$."
},
{
"role": "assistant",
"content": "<think>\nOkay,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the expected magnitude of the sum of two unit vectors in the Cartesian plane if the directions of both vectors are randomly chosen from the interval $... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve for real numbers \\( x \\) in the equation \\( 6^{\\log_{10} x} + 8^{\\log_{10} x} = x \\)."
},
{
"role": "assistant",
"content": "<think>\n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation \\((z+1)^2 + 160 + 168i = 0\\) for all complex roots."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to solve ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( A \\) be an arbitrary commutative ring with identity. Find a non-zero homomorphism \\( f: \\mathbb{Z} \\to A \\). When is \\( f \\) injective?"
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c \\in \\mathbb{Q}\\) with \\(ac \\neq 0\\). Given that the equation \\(ax^2 + bxy + cy^2 = 0\\) has a nonzero solution of the form \\((x, y) = (... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve for \\( x \\), \\( y \\), and \\( z \\) in the equation: \\( x^2 + y^2 + z^2 + xy + xz + yz - 2x + 3y - z + 7 = 0 \\)."
},
{
"role": "assistant"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be an equilateral triangle inscribed in circle \\(O\\). \\(M\\) is a point on arc \\(BC\\). Lines \\(\\overline{AM}\\), \\(\\overline{BM}\\), an... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a convex pentagon $ABCDE$ where all sides have length $1$ and $\\angle B + \\angle D = 180^\\circ$, find the distance between vertex $C$ and the intersecti... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a, b, c \\) be positive reals satisfying \\( a + b + c = 1 \\). Calculate the minimum value of the expression:\n\\[ \\sum \\frac{\\sqrt{a + bc}}{\\sqr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Factor completely: $a^{5} + a + 1$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to factor the polynomial a^5 + a + 1 completely... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c, x\\) be real numbers. If \\(|x| \\le 1\\), then \\(|ax^2 + bx + c| \\le 100\\). Find the maximum value of \\(|a| + |b| + |c|\\)."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} \\frac{1 + 2^2 + 3^3 + \\cdots + n^n}{n^n} \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the sum of the greatest common divisor of 30 and 81 and the least common multiple of 36 and 12?"
},
{
"role": "assistant",
"content": "<th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( S = \\{ 1, 2, \\ldots, 2023 \\} \\) be a subset of the first 2023 positive integers. Set \\(\\mathcal{F}\\) consists of some non-empty subsets of \\( ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How do you compute \\(4^{10} + 5^9 + 6^8 + 7^7 \\pmod{13}\\)?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to compute \\(4^{10}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the spectral radius of the operator \\(iI\\), where \\(I\\) is the identity operator?"
},
{
"role": "assistant",
"content": "<think>\nOkay... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In trapezoid \\(ABCD\\) with \\(AB \\parallel CD\\), the diagonals intersect at point \\(O\\). The areas of triangles \\(AOB\\) and \\(COD\\) are 2019 and 202... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the minimum value of the expression \\( x^4y^2 + x^2y^2 + 8x^2y + x^2 + 2008 \\) where \\( x, y \\in \\mathbb{R} \\)."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Suppose the altitudes of a triangle are 10, 12, and 15. What is the semi-perimeter of the triangle?"
},
{
"role": "assistant",
"content": "<think>... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the set of all positive integers \\( x \\) and \\( y \\) that satisfy the equation \\( 7^x + 5^y = 34m \\), where \\( m \\) is also a positive integer."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f, g: \\mathbb{Q^+} \\rightarrow \\mathbb{Q^+} \\) such that\n\\[ f(g(x+y)) = f(x)g(y). \\]"
},
{
"role": "assistant",
"con... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A ladybug is walking at random on a hexagon and starts at vertex A. Every minute, she moves to one of the two adjacent vertices chosen at random. What is the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given real numbers \\(a_1, \\ldots, a_n\\); positive integers \\(b_1, \\ldots, b_n\\); and a polynomial \\(f(x)\\) such that:\n\\[\n(1-x)^n f(x) = 1 + \\sum_{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all 7-element sets of points in the plane such that any three points of the set are either the vertices of an isosceles triangle or are collinear."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given four positive real numbers \\( a, b, c, d \\), calculate the minimum value of the expression:\n\n\\[\n\\dfrac{2a+c+d}{2b+c+d}+\\dfrac{2b+d+a}{2c+d+a}+\\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the limit \\(\\lim_{n \\to \\infty} \\sum_{k=1}^n \\frac{k}{(2k - 1)^2 (2k + 1)^2}\\)."
},
{
"role": "assistant",
"content": "<think>\nO... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a, b, c \\) be real numbers such that \\( a + b + c = 6 \\) and \\( ab + bc + ca = 9 \\). Calculate the maximum possible value of the expression \\( a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A positive integer $k$ is given. Initially, $N$ cells are marked on an infinite checkered plane. The cross of a cell $A$ is defined as the set of all cells ly... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int_{0}^{\\frac{\\pi}{2}} \\dfrac{x}{x+\\cos x} \\, dx$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the inequality given in the image: \\(\\frac{x^2 + 3x + 2}{x^2 - 4} \\geq 0\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\(a, b, c > 0\\) and \\(a^2 + b^2 + c^2 = 1\\), find the minimum value of \\(\\sum \\frac{a^2}{b+c}\\)."
},
{
"role": "assistant",
"content... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that 1000 gardeners have insect infestations every year, one in ten gardeners is an organic gardener, and one third of all insect infestations happen to... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the coefficient of \\( x^{21} \\) in the expansion of \\( (x + x^2 + x^3 + x^4 + x^5 + x^6)^6 \\)."
},
{
"role": "assistant",
"content": "<th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the number of connected components of the set $\\{x \\in \\mathbb{ R} : x^5 + 60x \\ge 15x^3 + 10x^2 + 20\\}$."
},
{
"role": "assistant",
"co... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The areas of three of the faces of a right, rectangular prism are $24 \\hspace{.6mm} \\mathrm{cm}^2$, $32 \\hspace{.6mm} \\mathrm{cm}^2$, and $48 \\hspace{.6m... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( P \\) be a point inside an acute-angled triangle \\( \\triangle ABC \\) such that \\( \\angle BPC = 180^\\circ - \\angle A \\). Denote by \\( A_1, B_1... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all increasing functions $f:\\mathbb {R}\\to \\mathbb {R}$ such that\n\n\\[\n\\frac{2(f(y)-f\\left (\\frac{x+y}{2}\\right))}{f(x)-f(y)}=\\frac{f(x)-f(y)}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "On a circle, there are 2005 non-negative integers with a total sum of 7022. Let $S$ be the sum of the maximum possible number of pairs of consecutive numbers ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f(x): [0,1] \\longrightarrow \\mathbb{R} \\) be a continuously differentiable function with \\( f(1) = 0 \\), \\( \\int_0^1 [f'(x)]^2 \\, dx = \\frac{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Identify the convex polyhedra in which the sum of the angles at each vertex \"C\" from the faces meeting at \"C\" equals 180°."
},
{
"role": "assistan... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the polynomial \\( ax^3 + x^2 + bx + 1 \\) with three real positive roots, find the minimum value of \\( P = \\frac{5a^2 - 3ab + 2}{a^2(a - b)} \\)."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) so that for any integers \\( a_1, a_2, \\ldots, a_{527} \\), the number \\( \\left( \\prod_{j=1}^{527} a_j \\righ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many positive integer solutions $(x,y,z)$ does the equation $xyz=2004$ have?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the number of positive integers less than 10000 such that the sum of its digits is 25."
},
{
"role": "assistant",
"content": "<think>\nOkay, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For a positive integer \\( n \\), let \\( g(n) = \\left[ \\frac{2024}{n} \\right] \\). Find the value of\n\\[\n\\sum_{n = 1}^{2024} \\left(1 - (-1)^{g(n)}\\ri... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In how many ways can $2n$ employees be divided into two departments of $n$ members each, if each department must choose a president and a vice-president?"
}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The vertices of triangle $ABC$ lie on a circle with center $O$. Suppose the measure of angle $ACB$ is $45^\\circ$. If $|AB| = 10$, then what is the distance b... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( p \\) be a natural number such that \\( 1 \\) is the only perfect square that divides \\( p \\). Find all natural numbers \\( x, y, z \\) (in terms of... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "On a 6 by 6 chess board, how many ways are there to place 3 white rooks and 3 black rooks so that no rooks of opposite color can attack each other? A rook can... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Provide an example of a non-empty subset \\( A \\) of a topological space \\( (X, \\mathcal{T}) \\) such that \\( \\text{int}(\\partial A) \\neq \\emptyset \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate $\\lim_{n\\to\\infty}\\sum_{k=0}^{n}\\binom{n}{k}^{-1}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the lim... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many real roots does the equation $x^{5} - 5x + 4 = 0$ have?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to figure out how... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For integers \\( x_1, x_2, \\ldots, x_n \\), if \\( 240 \\mid x_1^9 + x_2^9 + \\cdots + x_n^9 \\), find the smallest positive integer \\( k \\) such that \\( ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that $\\gcd{(a,b)} = 2$ and $\\text{lcm}(a,b) = 174$, compute the value of $a \\cdot b$."
},
{
"role": "assistant",
"content": "<think>\nOka... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Factor the polynomial $n^4 + 7n^3 + 26n^2 + 51n + 45$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to factor the polynomial n⁴... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate $\\int_{0}^{\\pi} \\frac{1}{(1+ \\sin x)^{2}}\\ dx$ without using complex integration."
},
{
"role": "assistant",
"content": "<think>\nOk... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the polynomial \\( f(x) \\) that satisfies the equation \\( (x - 1)f(x + 1) - (x + 2)f(x) = 0 \\) for every \\( x \\in \\mathbb{R} \\), and given that \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the nonzero natural numbers \\( n \\) for which the equality \\(\\{x\\} + \\left\\{x + \\frac{1}{n}\\right\\} = \\{nx\\} + \\frac{1}{n}\\) holds for all ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( l \\) be a line that is not parallel to the coordinate axes. Find the minimal value of \\( d \\) such that there always exists a point \\( A \\) with ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The Deerfield Door weighed 2019 pounds in 1797. Every other year, the door decomposes and loses \\(x\\) pounds. However, in order to maintain its quality, Dee... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the value of the integral $\\int_0^a x^3 \\sqrt{\\frac{x}{a-x}} \\, dx$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\( m, n, k > 0 \\) and \\( mnk = 1 \\), find the maximum value of \\( (mn + n + 1)(kn + k + 1)(km + m + 1) \\)."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Two of the altitudes of a right triangle have lengths 4 and 5. What is the largest possible length of the third altitude?"
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the equation \\((\\sqrt[3]{a} + \\sqrt[3]{b} - 1)^2 = 49 + 20 \\sqrt[3]{6}\\), find the value of \\(a + b\\)."
},
{
"role": "assistant",
"co... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A certain rectangle has an area of $2(\\sqrt{5} - 1)$ and has the property that when a square is removed from one end, the remaining rectangle is similar to t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all possible values of $k$ so that the sequence defined by $a_1 = k$ and $a_{n+1} = \\frac{a_n(a_n - 1)}{2}$ has all terms as odd integers."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a square $Q$ with side length $2$ and a real number $f$ with $f > 4$, let $M$ be the set of all points $P$ such that the sum of the areas of the four tr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( r \\) be the root of the equation \\( x^2 + 2x + 6 = 0 \\). Find the value of \\( (r+2)(r+3)(r+4)(r+5) + 200 \\)."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given an acute triangle $\\triangle ABC$ with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on $BC$ such that $AD \\perp OH$. Let $K$ be a point on... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an acute-angled triangle \\(ABC\\) with \\(\\angle ABC\\) as the largest angle, let \\(O\\) be the circumcenter. The perpendicular bisectors of \\(BC\\) an... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The product of two of the four roots of the quartic equation \\(x^4 - 18x^3 + kx^2 + 200x - 1984 = 0\\) is \\(-32\\). Determine the value of \\(k\\)."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a triangle \\(ABC\\) with \\(M\\) as the midpoint of side \\(BC\\). Given that \\(\\angle MAC = \\angle ABC\\) and \\(\\angle BAM = 105^\\circ\\), fi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the coefficient of $x^4$ in the expansion of $(1+2x+3x^2)^{10}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the insc... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( P, Q, R, S \\) be the midpoints of sides \\( BC, CD, DA, AB \\) of a convex quadrilateral \\( ABCD \\). Calculate the maximum value of the expression ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the value of \\( S(1) - S(2) + S(3) - S(4) + \\ldots + S(2013) - S(2014) \\), where \\( S(n) \\) denotes the sum of the digits of the number \\( n \\)."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A lady has an unlimited supply of quarters, dimes, nickels, and pennies. Considering only the value of the coins involved, find the number of distinct ways th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The probability of having a computer assignment is $\\frac{7}{8}$. The probability of having a textbook assignment is $\\frac{4}{5}$. What is the probability ... |
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