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 a child select $5$ balls from $5$ red, $4$ black, $3$ white, $2$ green, and $1$ yellow balls, assuming balls of the same color are identi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x_0 \\in [-1,1] \\) and \\( x_{n + 1} = x_n - \\arcsin(\\sin^2 x_n) \\) for \\( n \\ge 0 \\). Compute \\( \\lim_{n \\to \\infty} \\sqrt{n} x_n \\)."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int \\sin^{2} x \\tan x\\ dx$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to evaluate the integral o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ways are there to visit each of 6 cities exactly two times each, with the constraint that you may not visit the same city twice in a row?"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\triangle ABC$, the incircle touches $AB$, $BC$, and $CA$ at points $D$, $E$, and $F$ respectively. Let $H$ be the foot of the altitude from $D$ to $EF$.... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{R}^{+} \\to \\mathbb{R}^{+} \\) such that for every three positive real numbers \\( x, y, z \\), the numbers \\( x + f(y) \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the inequality \\(-1 \\leq x \\leq \\frac{x}{x-2}\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to solve the i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle inscribed in an ellipse \\((E)\\). Let \\(K\\) be any point inside triangle \\(ABC\\). The rays \\(AK\\), \\(BK\\), and \\(CK\\) i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a scalene triangle \\(ABC\\), two of the altitudes have lengths 4 and 12. If the length of the third altitude is also an integer, what is the greatest poss... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be an acute triangle with \\(AB < AC\\). Let \\(O\\) be its circumcenter, \\(D\\) be the foot of the altitude from \\(A\\), \\(M\\) be the midpo... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The smaller of two identically oriented equilateral triangles has each side one unit long. The smaller triangle is centered within the larger triangle so that... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a sequence \\( \\{x_n\\} \\) such that \\( \\lim_{n \\to \\infty} \\frac{x_n}{n} = k \\), find the value of \\( \\lim_{n \\to \\infty} (x_n - x_{n-1}) \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x, y, z > 0 \\) such that \\( x + y + z = \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} \\). Calculate the maximum value of the expression\n\\[\n\\frac{1... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c, d\\) be real numbers satisfying:\n\\[ a + c = b + d = 2 \\]\n\\[ \\sqrt{ab} + \\sqrt{cd} = 1 \\]\n\\[ 16abcd = 1 \\]\nFind \\(a^n + b^n + c^n ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\\%$ of her three-point shots and $30\\%$... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the prime numbers \\( p \\) and \\( q \\) that satisfy the conditions:\n\\[ q = p^2 + 1 \\]\n\\[ p = q^2 - 1 \\]"
},
{
"role": "assistant",
"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the sum\n$$\\sum_{k=1}^{\\infty} \\frac{6^k}{(3^{k+1} - 2^{k+1})(3^k - 2^k)}.$$"
},
{
"role": "assistant",
"content": "<think>\nAlright,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find \\( x, y \\in \\mathbb{N} \\) such that \n\\[ (2+3\\sqrt{5})^x = (3+4\\sqrt{7})^y. \\]"
},
{
"role": "assistant",
"content": "<think>\nOkay, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $$\\int_{0}^{\\frac{\\pi}{2}}\\frac{\\tan x}{4\\ln^{2}\\tan x+ \\pi^{2}} \\, dx.$$"
},
{
"role": "assistant",
"content": "<t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral of \\(\\ln(\\sin x)\\) from \\(0\\) to \\(\\frac{\\pi}{2}\\) using contour analysis/residue method."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( I \\) be the incenter of triangle \\( ABC \\), \\( M \\) the midpoint of side \\( AB \\), and \\( W \\) the midpoint of the arc \\( BC \\) of the circ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How much greater, in square inches, is the area of a circle of radius 20 inches than a circle of diameter 20 inches? Express your answer in terms of $\\pi$."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "An open rectangular box can be formed by cutting identical squares off the four corners of a rectangular piece of cardboard and folding up the four sections t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If \\(a\\) is a natural number and \\(b\\) is a digit, and \\((3a + 2007)^2 = 4b85924\\), what is the value of \\(a - b\\)?"
},
{
"role": "assistant",... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions $f : \\mathbb{N} \\mapsto \\mathbb{N}$ such that\n\\[ af(b) + b^2 = bf(a + b) \\]\nworks for all positive integers $a, b$."
},
{
"r... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Compute the value of the integral \\(\\int_0^{\\pi/4} \\frac{x}{\\cos x (\\cos x + \\sin x)} \\, dx.\\)"
},
{
"role": "assistant",
"content": "<th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c, d\\) be positive real numbers. Find the maximum value of the expression\n\n\\[\n\\frac{\\sqrt[4]{\\dfrac{abc}{ab+ac+bc}}+\\sqrt[4]{\\dfrac{abd... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all values of \\( a \\) such that the system\n\\[\n\\begin{cases}\nx^3 + y^2 + z^2 = a \\\\\nx^2 + y^3 + z^2 = a \\\\\nx^2 + y^2 + z^3 = a\n\\end{cases}\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be an equilateral triangle, and \\(D, E,\\) and \\(F\\) be the midpoints of \\(\\overline{BC}, \\overline{CA},\\) and \\(\\overline{AB}\\), resp... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with altitudes \\(BE\\) and \\(CF\\), and let \\(M\\) be the midpoint of \\(BC\\). Let \\(AM \\cap EF = N\\) and \\(X\\) be the po... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( G = GL_{n}(\\mathbb{R}) \\) and \\( D_{n}(\\mathbb{R}) = \\{ \\text{diag}(a_1, \\ldots, a_n) \\mid a_i \\neq 0 \\} \\). Define the centralizer \\( C_{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the function \\( f \\) defined as\n\\[ f(x) = \\sum_{i=0}^{\\alpha-1} (i+\\beta) \\cdot \\textbf{1}_{\\left(\\lfloor x \\rfloor \\equiv i \\pmod{\\alpha... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many matches ended in a draw in the tournament where each of the 6 football teams played against each other exactly once, and the final scores were 10, 9,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Inscribe a circle within an equilateral triangle of side length 1. Now, inscribe three smaller circles in the corners of the triangle, each tangent to the ori... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the function \\( f(x) = \\cos\\left(\\frac{x}{2}\\right) \\cdot f\\left(\\frac{x}{2}\\right) \\) with \\( f(0) = 1 \\), find \\( f(x) \\)."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Compute the determinant of the following matrix:\n\n\\[\n\\left|\\begin{array}{ccccc}\na_1+1 & a_2 & a_3 & \\cdots & a_n \\\\\na_1 & a_2+1 & a_3 & \\cdots & a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) be an increasing function such that \\( f(f(n)) = 3n \\) for all positive integers \\( n \\). Determine \\... |
[
{
"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{2\\cos(nt) + \\cos((n-1)t)}{5 + 4\\cos(t)} \\, dt \\), where \\( n \\) is a positive integer."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\(A + B + C = \\pi\\) and \\(\\tan\\left(\\frac{A+B-C}{4}\\right) + \\tan\\left(\\frac{B+C-A}{4}\\right) + \\tan\\left(\\frac{A+C-B}{4}\\right) = ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral \\(\\int\\limits_{S_n}\\sum\\limits_{k=1}^{n}x_k \\, dx\\), where \\(S_n = \\left \\{ \\sum\\limits_{k=1}^{n}x_k \\leq a, \\; x_k \\geq ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given an equilateral triangle \\(ABC\\) and a point \\(P\\) inside it such that \\(PA = 3\\), \\(PB = 4\\), and \\(PC = 5\\), find the area of the triangle \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all triples of natural numbers $(x,y,z)$ for which:\n\\[ xyz = x! + y^x + y^z + z! \\]"
},
{
"role": "assistant",
"content": "<think>\nOkay, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all positive integers \\( x, y, z, t \\) such that \\( (x-1)(y-1)(z-1)(t-1) \\) divides \\( xyzt - 1 \\)."
},
{
"role": "assistant",
"content... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the sixth number after the decimal point in the number $(\\sqrt{1978} + \\lfloor \\sqrt{1978} \\rfloor)^{20}$."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For angles \\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\), and \\(\\theta\\), define the expression \\(E(\\theta) = \\sum_{cyc}\\frac{\\cos 2(\\theta+\\alpha) \\s... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the function \\( f \\) such that \\( f(x) \\cdot f(y) = (f((x+y)/2))^2 \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the sum of all $n < 2021$ such that $n - 4$ is a prime number, $n$ is a product of two consecutive primes, and $n + 4$ is a square number."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Graph the first 5 terms of a series where the first term \\( a_1 = 7 \\) and the common difference \\( d = 2 \\)."
},
{
"role": "assistant",
"cont... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $A$ be an $n \\times n$ matrix. Define the matrix $B$ as\n\\[B = \\left| \\begin{array}{llllll}\\mbox{tr}(A) & 1 & 0 & \\ldots & \\ldots & 0 \\\\ \\mbox{t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For positive integers \\( m \\) and \\( n \\) such that \\( m+10 < n+1 \\), both the mean and the median of the set \\(\\{m, m+4, m+10, n+1, n+2, 2n\\}\\) are... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\(\\log 36 = a\\) and \\(\\log 125 = b\\), express \\(\\log \\left(\\frac{1}{12}\\right)\\) in terms of \\(a\\) and \\(b\\)."
},
{
"role": "ass... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all Fermat primes \\( p > 3 \\) such that there exist positive integers \\( m \\) and a prime \\( q \\) satisfying \\( p^m + 1 = 10q \\)."
},
{
"... |
[
{
"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 equilateral triangle \\( \\triangle ABC \\) such that \\( PA = 5 \\), \\( PB = 7 \\), and \\( PC = 8 \\). Find the length o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For what positive values of \\( n \\) does there exist a positive integer \\( m \\) such that \\( m = a_1 + a_2 + \\cdots + a_n \\), where each \\( a_i \\) is... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the length of the parametric curve described by\n\\[(x,y) = (2 \\sin t, 2 \\cos t)\\]from $t = 0$ to $t = \\pi.$"
},
{
"role": "assistant",
"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the exact value of $\\sin{20^\\circ} \\cdot \\sin{40^\\circ} \\cdot \\sin{80^\\circ}$ using complex numbers."
},
{
"role": "assistant",
"cont... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the largest positive integer $n$ for which the inequality\n\\[\n\\frac{a+b+c}{abc+1} + \\sqrt[n]{abc} \\leq \\frac{5}{2}\n\\]\nholds true for all $a, b, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the greatest common factor of $2^{15} + 3^{15}$ and $2^{25} + 3^{25}$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have thi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Two triangles are similar. The ratio of their areas is 1:4. If the height of the smaller triangle is 3 cm, how long is the corresponding height of the larger ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the arithmetic sequence $17, a, b, c, 41$, what is the value of $b$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the abelian ring \\( R \\), let \\( I \\) be a primary ideal. Define \\( J = \\{ x \\in R \\mid \\exists m \\in \\mathbb{N} \\text{ such that } x^m \\in I ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many different ways can we choose 8 points from 24 equally spaced points on a circle such that the arc length between any two chosen points is not 3 or 8?... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find a function \\( f: \\mathbb{Q} \\rightarrow \\mathbb{R} \\) that satisfies the equation\n\\[ f(x) + f\\left(1 - \\frac{1}{x}\\right) = \\log |x| \\quad \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "There are positive integers \\( x \\) and \\( y \\) that satisfy the system of equations\n\\[\n\\log_{10} x + 2 \\log_{10} (\\text{gcd}(x,y)) = 60\n\\]\n\\[\n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate $\\arcsin(\\sin(\\pi^2))$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the value of arcsin(sin(π²)). Let me ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the function \\( f(x) = x^3 + 2x \\) and its inverse function \\( g(x) \\). Use the derivative of \\( f(g(x)) \\) to find the gradient of the tangent... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the limit if it exists:\n\\[\n\\lim_{n\\to \\infty}\\frac{\\left (1+\\frac{1}{n} \\right )^{n^{2}}}{e^{n}}\n\\]"
},
{
"role": "assistant",
"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "On the sides of triangle \\( ABC \\), take points \\( M_1 \\), \\( N_1 \\), and \\( P_1 \\) such that each line \\( MM_1 \\), \\( NN_1 \\), and \\( PP_1 \\) d... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Simplify the expression $\\dfrac{ (1 - x^2) ^{1/2} + x^2(1-x^2)^{- 1/2} }{ (1-x^2)}$."
},
{
"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": "Evaluate the integral $\\int_{0}^{\\frac{\\pi}{3}} \\cos^{6} \\theta \\ d\\theta$."
},
{
"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": "Let \\( S_n \\) be the number of Latin squares of order \\( n \\) and let \\( R_n \\) be the number of reduced Latin squares of order \\( n \\) (Latin squares... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the function \\( f: U \\in M_n(K) \\mapsto \\cos(U) \\in M_n(K) \\) where \\( K = \\mathbb{C} \\) or \\( \\mathbb{R} \\). Let \\( A \\) be a matrix i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Count the number of ways to put four balls into six distinguishable boxes."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( k \\) be a fixed positive integer. Find all functions \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) such that for any distinct positive integers \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the rotation matrix\n\\[\nR = \n\\begin{pmatrix}\n1 + r & 1 - r & 1 \\\\\n1 - r & 1 + r & -1 \\\\\n-1 & 1 & 2r \n\\end{pmatrix}\n\\]\nwhere \\(r = \\fra... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( P \\) be any point inside triangle \\( ABC \\) with altitudes intersecting the opposite sides at \\( D, E, \\) and \\( F \\) respectively. If \\( AD =... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the largest positive integer \\(n\\) for which \\(n^3 + 100\\) is divisible by \\(n + 10\\)?"
},
{
"role": "assistant",
"content": "<think... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "There is a deck of $n$ cards, each of which is independently either red or black with equal probability. You start with $\\$1$, and you make a sequence of $n$... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( W, X, Y, \\) and \\( Z \\) be points on a circumference \\( \\omega \\) with center \\( O \\), in that order, such that \\( WY \\) is perpendicular to... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many integer values of \\( x \\) are possible for an obtuse-angled triangle with sides \\( 15 \\), \\( 8 \\), and \\( x \\)?"
},
{
"role": "assist... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "There are 5 points in the plane, no three of which are collinear. We draw all the segments whose endpoints are these points. What is the minimum number of new... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find $\\mu$ such that $\\mathbb{E}_{X \\sim \\chi^2(4)} \\left[ e^{- \\tfrac{1}{2} \\left( \\frac{X - \\mu}{\\sigma} \\right)^2} \\cdot (X - \\mu) \\right] = ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many 4-letter code words can be formed from the letters in \"CALCULUS\"?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have this ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an acute triangle $ABC$, where $R$ is the circumradius, $r$ is the inradius, and $a$, $b$, and $c$ are the side lengths of the triangle, find the maximum v... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all continuous functions \\( f \\), \\( g \\), and \\( h \\) from \\( (0, +\\infty) \\) to \\( \\mathbb{R} \\) satisfying the equation:\n\\[ h(xy) = g(x)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x_i \\) for \\( i=1,2,\\ldots,48 \\) be the zeros of the polynomial \\( P(x) = 18x^{48} + 3x + 2006 \\). Find \\( \\sum_{i=1}^{48} \\frac{x_i}{1 + x_i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Regular hexagon $A_1A_2A_3A_4A_5A_6$ has side length $1$. For $i = 1, \\ldots, 6$, choose $B_i$ to be a point on the segment $A_iA_{i+1}$ uniformly at random,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "When $\\sqrt[3]{2700}$ is simplified, the result is $a\\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is as small as possible. What is $a+b$?"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If \\(\\dfrac{\\left(\\frac{a}{c}+\\frac{a}{b}+1\\right)}{\\left(\\frac{b}{a}+\\frac{b}{c}+1\\right)}=11\\), where \\(a, b,\\) and \\(c\\) are positive intege... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all pairs \\((a, b)\\) with \\(a, b \\in \\mathbb{N}^{*}\\) such that \\(\\frac{a}{b} + \\frac{21b}{25a} \\in \\mathbb{N}\\)."
},
{
"role": "assi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a trapezoid with bases \\(AB\\) and \\(CD\\). Let \\(O\\) be the intersection point of its diagonals \\(AC\\) and \\(BD\\). If the area of t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given triangle \\(ABC\\) such that \\(AB - BC = \\frac{AC}{\\sqrt{2}}\\). Let \\(M\\) be the midpoint of \\(AC\\), and \\(N\\) be the foot of the angle bisect... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the Diophantine equation $(x^2 + y^2)^2 + x^4y^4 = z^2$ in integers, assuming $x$ and $y$ are coprime."
},
{
"role": "assistant",
"content":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $a, b, c \\in \\mathbb{R_{++}}$ such that $a^2 + b^2 + c^2 = 3$. Calculate the minimum value of the expression\n\\[\n\\frac{1}{a^{11}} + \\frac{1}{b^{11}}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the number of ways to select \\( n \\) objects from \\( 3n \\) objects, where \\( n \\) of the objects are identical and the remaining \\( 2n \\) objects... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle BAC = 60^\\circ$, $\\angle ADC = 150^\\circ$, and $D \\in [BC]$ such that $DC = AB$. Find $\\angle ABC$."
},
{
"role": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $\\triangle ABC$ and $P, Q$ be two isogonal conjugate points. Let $D \\in (BCP)$ such that $DP$ is the diameter, and $E \\in (BCQ)$ such that $QE$ is the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\( x + y = 7 \\) and \\( x^2 - y^2 = 21 \\), find the value of \\( 2x + 3y \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c\\) be nonnegative real numbers such that \\(a \\ge \\frac{4}{3}\\) and \\(\\frac{a+b+c-3}{abc-1} = \\frac{9}{11}\\). Calculate the minimum valu... |
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