messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In how many ways can you paint the six vertices of a regular pentagonal pyramid using at most six different colors, such that two vertices connected by an edg... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all polynomials with real coefficients \\( P(x) \\) such that the equality\n\\[ P(2P(x)) = 2P(P(x)) + 2(P(x))^2 \\]\nholds for any real number \\( x \\).... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The numbers $1$ to $6$ are placed in a T-shaped grid so that any two numbers in adjacent cells form a two-digit prime number in some order. Find the product o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the number of natural numbers $n$ such that $n \\leq 2009$ and $[\\sqrt{n}]$ divides $n$."
},
{
"role": "assistant",
"content": "<think>\nAlr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Elizabeth has an infinite grid of squares. (Each square is next to the four squares directly above it, below it, to its left, and to its right.) She colors in... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve in $\\mathbb R$ the equation $\\sqrt[n]{x-a^n} - \\sqrt[n]{x-b^n}=a-b$, where $a, b\\in \\mathbb R$ and $n\\geq 2$."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that $\\frac{d}{dx}f(x)= \\frac{f^2(x)}{f(x)+x+1}$ and $f(0) = \\lambda$, find the value of $\\int \\frac{\\log{f(x)}}{f^2(x)}dx$."
},
{
"role":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a 4 by 4 by 4 3-dimensional tic-tac-toe board, how many different winning rows are there? (A winning row can be horizontal, vertical, up-down, or diagonal.... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Two ceiling fans hang one above the other. The top fan has three blades and makes $1.4$ clockwise revolutions per second, while the bottom fan has four blades... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and for every pair of st... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( H \\) be a normal subgroup of \\( G \\) and let \\( P_H \\) be a Sylow \\( p \\)-subgroup of \\( G/H \\). If \\( H \\) is a \\( p \\)-group, find the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find positive integers \\( m \\) and \\( n \\) such that \\( n(n+1)(n+2)(n+3)(n+4)(n+5) = m^2 \\)."
},
{
"role": "assistant",
"content": "<think>\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that\n\n\\[ 2f(x+y) + f(2x-2y) = f(2x) + f(2y) + 2f(x-y), \\quad \\forall x, y \\in... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the area of the sphere given by $z=\\sqrt{9-x^2-y^2}$ that is cut by the cylinder $x^2+y^2=5$."
},
{
"role": "assistant",
"content": "<t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "There are 4 chambers of distinct colors, each containing one man wearing a jacket of the chamber’s color. Every 15 minutes, the men can either stay in their c... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation in primes $(p, q)$ such that:\n\\[ p^3 - q^7 = p - q \\]"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to sol... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let cevians $AD, BE$, and $CF$ of $\\triangle ABC$ meet at an interior point $P$. If $AB = 3 \\cdot FB$ and $AC = 4 \\cdot AE$, compute the value of $\\frac{P... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( A \\in \\mathcal{M}_n(\\mathbb{R}) \\). Define \\( C_A \\) as the subset of \\( M \\in \\mathcal{M}_n(\\mathbb{R}) \\) such that \\( AM = MA \\) and \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int^{\\frac{\\pi}{2}}_{0} \\frac{\\,dx}{\\left(4\\sin^2{x}+9\\cos^2{x}\\right)^2}$."
},
{
"role": "assistant",
"content":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the game Mastermind, one player selects four pegs, each peg having six possible colors, and places them in a line. The second player knows that the first p... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be an obtuse triangle with \\(\\angle BAC > 90^\\circ\\), and let \\(AH\\) be the height from \\(A\\) to \\(BC\\) with \\(H\\) on \\(BC\\). Let ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x_1, x_2, \\ldots, x_n \\) and \\( y_1, y_2, \\ldots, y_n \\) be non-negative real numbers. Calculate the maximum possible value of \\(\\frac{\\sum_{i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The ratio of the measures of the acute angles of a right triangle is $8:1$. In degrees, what is the measure of the largest angle of the triangle?"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For which value of parameter \\( m \\) in the equation \\( x^2 - 2mx + 3m^2 - 38m + 156 = 0 \\) does one solution exceed the other?"
},
{
"role": "ass... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( n+1 \\) real numbers \\( a_0, a_1, a_2, \\ldots, a_n \\) be given such that \\( a_1, a_2, \\ldots, a_n > 0 \\) and \\( \\sum_{i=1}^n a_i = 1 \\). Calc... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be an obtuse-angled triangle with \\(\\angle ABC > 90^\\circ\\), and let \\((c)\\) be its circumcircle. The internal angle bisector of \\(\\angl... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\(a\\), \\(b\\), and \\(c\\) are the sides of a triangle, calculate the minimum value of the expression\n\\[ \\frac{2(a+b+c)(a^2 + b^2 + c^2) - 3(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the Euro Millions game, you choose 5 numbers between 1 and 50 (no repeats) and 2 stars between 1 and 11 (no repeats). The draw consists of 5 numbers and 2 ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( d \\geq 2 \\) be a positive integer. Define the sequence \\( a_1, a_2, \\dots \\) by\n\\[ a_1 = 1 \\ \\text{and} \\ a_{n+1} = a_n^d + 1 \\ \\text{for ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ways can 10 students be chosen from 67 students standing in a line such that no two chosen students are standing next to each other?"
},
{
"r... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the maximum value of the expression \\(\\frac{x^{\\frac{1}{16}} y^{\\frac{1}{8}} z^{\\frac{1}{4}} t^{\\frac{1}{2}}}{1 + x + 2y + z + t}\\) for \\(x, y, z... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $$\\int_{-\\infty}^{\\infty}\\frac{e^{imx}}{\\cosh(x)}dx, \\quad m\\in\\mathbb{R}.$$"
},
{
"role": "assistant",
"content": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many pairs of integers $(m,n)$ are there such that $1 \\leq m \\leq 2012$ and \\[5^n < 2^m < 2^{m+2} < 5^{n+1}?\\]"
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Which year in the 18th century, when added to the product of its digits, equals 2018?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Can the numbers from 1 to 81 be written in a 9x9 board so that the sum of numbers in each 3x3 square is the same? If yes, how?"
},
{
"role": "assistan... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b > 0\\) and \\(\\frac{1}{a} + \\frac{4}{b} + 2 = a + 2b\\). Find the minimum value of \\(a + 2b\\)."
},
{
"role": "assistant",
"content... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f : [0, +\\infty) \\rightarrow [0, +\\infty) \\) such that\n\n\\[ f(f(x) + f(y)) = xy f(x + y) \\]\n\nfor every two non-negative real n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find a closed form for the product $\\prod_{k=1}^{n-1} \\sin\\left(\\frac{k\\pi}{n}\\right)$."
},
{
"role": "assistant",
"content": "<think>\nOkay... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a regular tetrahedron \\(ABCD\\), point \\(O\\) is the center of the face \\(BCD\\), \\(R\\) is the midpoint of \\(AO\\), and point \\(S\\) is on \\(AC\\) ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the largest values of \\( a \\) and \\( b \\) such that \\( \\tan(ax + b) \\) is continuous on the interval \\( (0, 2\\pi) \\)."
},
{
"role": "as... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the polynomials $x^2 + px + q$ for which $\\max_{x \\in [-1,1]} |p(x)|$ is minimal."
},
{
"role": "assistant",
"content": "<think>\nOkay, so ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the function \\( f \\colon \\mathbb{R^+} \\to \\mathbb{R} \\) defined by\n\\[ f(x) = \\begin{cases} \n0 & \\text{if } x \\in \\mathbb{Q}^c \\\\\n\\fr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the minimum number of $1 \\times k$ dominoes that can be placed on an $n \\times n$ square such that no dominoes overlap and each domino is aligned wi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle $... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all odd integers $n > 1$ such that if $a$ and $b$ are divisors of $n$ and are relatively prime, then $a + b - 1$ is also a divisor of $n$."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A man is walking due east at 2 m.p.h. and to him, the wind appears to be blowing from the north. On doubling his speed to 4 m.p.h. and still walking due east,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find $y$ so that the vectors $\\begin{pmatrix} 1 \\\\ -3 \\\\ -4 \\end{pmatrix}$ and $\\begin{pmatrix} -2 \\\\ y \\\\ -1 \\end{pmatrix}$ are orthogonal."
},... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a triangle $ABC$, let $I$ be the incentre, $O$ be the circumcentre, and $X$ be the X(35) point. If the coordinates of $A$, $B$, and $C$ are $(0,0)$, $(1,0)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\( n+1 \\) points in 2D space, find a function \\( f_n(x) = \\sum_{i=0}^{n} \\beta_i x^i \\) of degree \\( n \\) that passes through all those points. ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ways are there to place 9 different flags on 5 different flagpoles if the order of the flags on the flagpoles is relevant and every flagpole flies at... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\triangle ABC$, points $M$, $I$, and $H$ are the feet of the median, bisector, and height, respectively, drawn from $A$. Given that $BC = 2$, $MI = 2 - \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $ABC$ be a triangle. A circle $(K)$ passing through $B$ and $C$ intersects $CA$ and $AB$ at $E$ and $F$ respectively. Let $P$ be the point such that $PE \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider triangles formed by vertices of a given regular 2008-gon. Determine whether there are more acute-angled triangles or obtuse-angled triangles among th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the greatest common divisor of $1729$ and $1768$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the greatest com... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral\n$$\\int\\frac{\\sin(x)}{\\sin(x)+\\cos(x)}\\,dx$$"
},
{
"role": "assistant",
"content": "<think>\nOkay, I need to evaluate ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the following system of equations in the set of real numbers:\n\\[ x_1 + x_2 + x_3 = 3 \\]\n\\[ x_1^3 + x_2^3 + x_3^3 = x_1^4 + x_2^4 + x_3^4 \\]"
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a regular $n$-gon $A_1A_2\\ldots A_n$ with area $S$. Let lines $l_1, l_2, \\ldots, l_n$ be perpendicular to the plane of the $n$-gon at points $A_1, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integer pairs $(x, y)$ such that $x^{2}(y-1) + y^{2}(x-1) = 1$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), \\(AB = 3\\), \\(AC = 5\\), and \\(BC = 7\\). Let \\(E\\) be the reflection of \\(A\\) over \\(BC\\), and let line \\(BE\\) meet the ci... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If the letters in the word BANANA are rearranged, what is the probability that at least one of the six positions will be occupied by the same letter as in the... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all real-valued functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for any real numbers \\( x \\) and \\( y \\) different from 0, the following ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the minimum value of \\(17\\sqrt{(x-1)^2+(y-3)^2}+39\\sqrt{(x+1)^2+(y-1)^2}+44\\sqrt{(x-3)^2+y^2}\\)."
},
{
"role": "assistant",
"content": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Compute $\\sum_{n=1}^\\infty\\frac{(-1)^n}{n^{2p}}$ for $p \\in \\mathbb{N}.$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to c... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c\\) be positive integers such that \\(a + 2b + 3c \\leq 50\\) and \\(\\frac{a}{b} + \\frac{a}{c} + 1 = 10 \\left( \\frac{b}{c} + \\frac{b}{a} + ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The real number \\( 2.525252... \\) is a rational number, so it can be written in the form \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are integers, \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given triangle \\(ABC\\) with excenters \\(I_a, I_b, I_c\\). Let \\(D, E, F\\) be the feet of the altitudes from \\(I_a, I_b, I_c\\) to \\(BC, CA, AB\\) respe... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "An ellipse has foci at $(9,20)$ and $(49,55)$. Find the length of its major axis if the ellipse is tangent to the x-axis."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What conditions must a function $f:\\mathbb{R}\\to\\mathbb{R}$ satisfy in order for there to exist a function $g:\\mathbb{R}\\to\\mathbb{R}$ such that $g\\cir... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a triangle \\(ABC\\) with three acute angles. Calculate the minimum value of the expression \\((\\sin(A))^{\\sin(B)} + (\\sin(B))^{\\sin(C)} + (\\sin... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\((x_n)\\) and \\((y_n)\\) be two sequences defined as follows:\n\\[ x_1 = 3 \\]\n\\[ x_{n+1} = (x_n^2 - 2)^2 - 2 \\]\n\\[ y_n = \\left(\\frac{3 - \\sqrt... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the sum of all positive integers \\(n\\) such that, given an unlimited supply of stamps of denominations \\(5, n,\\) and \\(n+1\\) cents, \\(91\\) cents ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a partition of the integers $1, 2, \\ldots, 10$ into two disjoint collections $a_1, a_2, \\ldots, a_5$ and $b_1, \\ldots, b_5$ such that $a_1 < a_2 <... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a_n = \\# \\{ f \\mid f : \\{ 1, 2, \\cdots, n \\} \\to \\{ 1, 2, \\cdots, n \\}, f = f \\circ f \\} \\). Find the sum \\( \\sum_{n = 0}^{\\infty} \\f... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation \\( S(n) = 2018 \\cdot S(7n) \\) for natural numbers \\( n \\), where \\( S(n) \\) denotes the sum of the digits of \\( n \\)."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), let \\( M \\) be a point on the bisector \\( AD \\). Lines \\( BM \\) and \\( CM \\) intersect sides \\( AC \\) and \\( AB... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( S_n = 2 + 3 + 5 + \\cdots + p_n \\) be the sum of the first \\( n \\) primes. Find the smallest value of \\( n \\) such that there is a perfect square... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a cyclic quadrilateral and \\(P\\) be a point on the plane. Let \\(l\\) be a line passing through \\(P\\), and let \\(P_1, P_2, P_3, P_4\\) ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given real numbers \\(a, b, c, d, e\\) such that the pairwise differences between any two of them are greater than 1, and the equation \\(2(a^2 + b^2 + c^2 + ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{N}^* \\to \\mathbb{N}^* \\) which satisfy the following conditions:\na) \\( f \\) is strictly increasing.\nb) \\( f(f(n)) =... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b \\in \\mathbb{R}\\) and \\(z \\in \\mathbb{C} \\backslash \\mathbb{R}\\) such that \\(\\left| a - b \\right| = \\left| a + b - 2z \\right|\\). Sol... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a $9 \\times 9$ grid, place several dominoes (each $2 \\times 1$ or $1 \\times 2$ tiles) without overlapping such that each domino shares an edge with at l... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For a prescribed system of nodes \\(a \\leq x_{0}^{(n)} \\leq x_{1}^{(n)} \\leq \\dots \\leq x_{n}^{(n)} \\leq b\\), let \\(f(x)\\) be a continuous function o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral \\(\\int_{0}^{2\\pi}\\frac{d\\theta}{3+\\sin\\theta+\\cos\\theta}\\) using the substitution \\(z = e^{i\\theta}\\), \\(d\\theta = \\frac... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( P \\) be a variable point on a given straight line and \\( A \\) be a fixed point. \\( Q \\) is a point on \\( AP \\) or \\( AP \\) produced such that... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( A \\) be an \\( m \\times n \\) matrix and \\( B \\) be an \\( m \\times 1 \\) matrix. Define \\( C = A^T A \\) and \\( D = A^T B \\). Calculate the d... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many different positive integers are there such that the integer is 19 times the sum of its digits?"
},
{
"role": "assistant",
"content": "<th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions $f:2^\\mathbb{R}\\to2^\\mathbb{R}$ such that $f(A)\\setminus f(B)\\subset f(B\\setminus A)$ for all sets $A,B\\subset\\mathbb{R}.$"
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve for integers \\( n \\) and \\( m \\) the equation\n\\[ 2n^n + (n-1)^n = m^2. \\]"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the functional equation \\( f(x - f(y)) = f(f(y)) + xf(y) + f(x) - 1 \\) for \\( x, y \\in \\mathbb{R} \\)."
},
{
"role": "assistant",
"cont... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve in $\\mathbb{Z}$ the equation $x^4 - y^4 = 2009z^3$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to tackle this equation... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int x \\sin^{-1} x \\ dx$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have this integral to solve: the in... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the smallest positive integer \\( m \\) such that for every positive integer \\( k > m \\), the number \\( 2k \\) can be expressed as the sum of two... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The function $f$ takes nonnegative integers to real numbers, such that $f(1) = 1,$ and\n\\[f(m + n) + f(m - n) = \\frac{f(2m) + f(2n)}{2}\\]for all nonnnegati... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many permutations of the set $\\{1, 2, \\ldots, 14\\}$ are there such that exactly one number is greater than its position?"
},
{
"role": "assista... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the differential equation $y'' = \\frac{y}{y^2 - 1}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to solve the d... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation \\[3x^3-13x^2+30x-4=\\sqrt{(6x+2)(3x-4)^3}\\]"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have this equation to ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the limit \\(\\lim_{x \\rightarrow 0} \\frac{\\sin^3 x + \\sin^3 2x + \\sin^3 3x}{\\sin^4 x + \\sin^4 2x + \\sin^4 3x}\\)."
},
{
"role": "ass... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For a positive integer \\( n \\), define \\( f_n = [2^n \\sqrt{2009}] + [2^n \\sqrt{2010}] \\). Find the smallest positive integer \\( k \\) such that \\( f_k... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The sequence of integers \\( u_{0}, u_{1}, u_{2}, u_{3}, \\ldots \\) satisfies \\( u_{0} = 1 \\) and \\( u_{n+1}u_{n-1} = ku_{n} \\) for each \\( n \\geq 1 \\... |
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