messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Petra and Paula take turns rolling a die. Petra rolls first, then Paula rolls twice, then Petra rolls twice, and so on. The first player to roll a 6 wins. Wha... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integral solutions to the equation \\(a^2 + b^2 + c^2 = (ab)^2\\)."
},
{
"role": "assistant",
"content": "<think>\nAlright, so I need to ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a convex quadrilateral with \\(AD = BC\\) and \\(\\angle A + \\angle B = 120^\\circ\\). Let \\(E\\) be the midpoint of side \\(CD\\), and le... |
[
{
"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\\), and \\(AC=15\\). Let \\(D\\) denote the midpoint of \\(\\overline{BC}\\) and let \\(E\\) denote the intersect... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the next number after 2010 that has exactly 16 positive divisors and the sum of its prime factors is 77."
},
{
"role": "assistant",
"content"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the domain and range of the function \\( f(x) = 2\\sqrt[3]{x} \\)."
},
{
"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": "Consider a $100 \\times 100$ board filled with the alphabets $A, B, C, D$ such that each row and each column contains exactly 25 of each alphabet. Define a \"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given positive numbers \\(a\\), \\(b\\), and \\(c\\), calculate the minimum value of the expression\n\\[\n\\frac{a^2}{2a+b+c} + \\frac{b^2}{a+2b+c} + \\frac{c... |
[
{
"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^2 + b^2 + 4c^2 + ab + 3 = 5c(a + b)\\). Find the least value of the expression \\(P = ab + bc + ac\\)."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For a natural number \\( n \\ge 2 \\), find the non-zero numbers \\( a \\) that satisfy the equation\n\\[ \\lfloor x \\rfloor + \\left\\lfloor x + \\frac{1}{a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the area of the domain of the points $(x, y)$ satisfying $3^{|1-x|}(1+|y|) \\leq 3$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all natural triples $(a,b,c)$ such that $2^a = 5^b + 7^c$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this prob... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f \\colon \\mathbb{R}^+ \\rightarrow \\mathbb{R}^+ \\) satisfying, for all \\( x, y \\in \\mathbb{R}^+ \\),\n\n\\[ xf(x+f(y)) + f(xy+1)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all complex-valued functions \\( f \\) of a complex variable such that \n\\[ f(z) + z f(1-z) = 1 + z \\]\nfor all \\( z \\in \\mathbb{C} \\)."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a triangle \\( ABC \\) with altitude feet \\( A', B', C' \\) and reflections \\( Ab, Ac \\) of \\( A' \\) on \\( BB', CC' \\). Let \\( P \\) be any p... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Suppose \\( P(x) \\) is a polynomial of degree 99. For exactly 100 of the 101 integers from 0 to 100, we have \\( P(x) = \\frac{1}{x+1} \\). Also, \\( P(101) ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For all \\(a, b, c > 0\\) such that \\(abc = 1\\), find the maximum value of \n\\[ M = \\frac{1}{\\sqrt{a^5 + b^2 + ab + 6}} + \\frac{1}{\\sqrt{b^5 + c^2 + bc... |
[
{
"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 zeros of the quartic equation \\(x^4 - 18x^3 + kx^2 + 200x - 1984 = 0\\) is \\(-32\\). Find \\(k\\)."
},
{
"role": "ass... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all the roots, real or complex, of the system of simultaneous equations\n\\[\n\\begin{align*}\nx + y + z &= 3, \\\\\nx^2 + y^2 + z^2 &= 3, \\\\\nx^5... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $\\triangle PQR$ be a triangle in the plane, and let $S$ be a point outside the plane of $\\triangle PQR$, so that $SPQR$ is a pyramid whose faces are all... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let triangle \\(ABC\\) with incenter \\(I\\) satisfy \\(AB = 10\\), \\(BC = 21\\), and \\(CA = 17\\). Points \\(D\\) and \\(E\\) lie on side \\(BC\\) such tha... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the maximum value of \\( \\frac{3+\\sqrt{6}\\sin x}{3+\\sqrt{6}\\cos x} \\)."
},
{
"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 each integer $n \\geq 3$, find the least natural number $f(n)$ such that for every subset $A \\subset \\{1, 2, \\ldots, n\\}$ with $f(n)$ elements, there ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Tanya typed a six-digit number, but the two $1$’s she typed did not show. What appeared was $2006$. Find the number of different six-digit numbers she would h... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), the angle bisector of \\(\\angle{ABC}\\) intersects the angle bisector of \\(\\angle{ACB}\\) and the circumcircle of \\(\\triangle{ABC}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f \\) be a function which satisfies:\n\\[ 2f(x) + 3f\\left(\\frac{2x+29}{x-2}\\right) = 100x + 80 \\]\nFind \\( f(3) \\)."
},
{
"role": "assis... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given positive integers $n$ and $m$, find all functions $f: \\mathbb{R}^+ \\to \\mathbb{R}$ such that:\n$$P(x, y): f(x^{n+m}+y^{n+m})=x^nf(x^m)+y^nf(y^m) \\qu... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all real numbers \\(a_1, a_2, a_3, \\ldots, a_{2011}\\) that satisfy the conditions:\n\\[ a_1 + a_2 + \\cdots + a_{2011} = 2011 \\]\nand\n\\[ |a_1 -... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate $6 \\cdot 8\\frac{1}{3}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to calculate 6 multiplied by 8 and one third. ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x, y) = \\max\\{|x+y|, |x-y|, |x-2|\\} \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a set of the first $n$ natural numbers, if one of the numbers is removed, the average of the remaining numbers is $21\\frac{1}{4}$. Specify the number w... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let quadrilateral \\(ABCD\\) be inscribed in circle \\((O)\\). Let \\(AC \\cap BD = E\\). Circle \\((K)\\) is tangent to segments \\(EA\\) and \\(ED\\) and is... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a \\in \\mathbb{N^*} \\) and \\( n \\in \\mathbb{N^*} \\). A box contains \\( a \\) white balls and \\( a + 1 \\) black balls. We take \\( n \\) balls... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the function $f(x) = ||x - 2| - a| - 3$. If the graph of $f$ intersects the $x$-axis at exactly three points, then find $a$."
},
{
"role": "assi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the number of monotone positive integers with at most 1003 digits.\n\n..."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I nee... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a_n \\) be the sequence whose \\( n \\)-th term is the sum of the digits of the natural number \\( 9n \\). For example, \\( a_1 = 9 \\), \\( a_{11} = ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle and let \\(M\\) be the midpoint of \\(\\overline{BC}\\). The lengths \\(AB\\), \\(AM\\), \\(AC\\) form a geometric sequence in tha... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $g$ be an $n$-variable Boolean function defined by \n\\[ g(x_1, x_2, \\cdots, x_n) \\equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \\cdots + x_1 x_2 \\cdots x_n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let the sequence $(a_n)_{n \\in \\mathbb{N}}$ be defined by $a_{n+1}=3^n-2015a_n$ for all $n \\in \\mathbb{N}$. Find $a_0 \\in \\mathbb{R}$ such that the sequ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Simplify $(2x^3)^3$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to simplify (2x³)³. Hmm, let me think about how to do this. A... |
[
{
"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 \\( \\frac{\\sqrt{n}}{2} + \\frac{30}{\\sqrt{n}} \\) is an integer."
},
{
"role": "assistant... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integers \\( k \\) such that the polynomial \\( x^p + k \\) is reducible over the integers \\( \\mathbb{Z} \\), where \\( p \\) is a prime number."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( E \\) be an arbitrary set and for \\( p, q \\in E \\), define \\( d(p, q) = 0 \\) if \\( p = q \\) and \\( d(p, q) = 1 \\) if \\( p \\ne q \\). This i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For how many positive integers $n\\geq 2$ is $1001_n$ a prime number?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Compute $\\sum_{r=1}^{n} r(r-1) \\binom{n}{r}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to compute the sum from r=... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get $5100$. Determine the number of faces of the pyramid."... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Among 100 monkeys dividing 1600 peanuts, let $N$ be the maximum number of peanuts any four monkeys can have the same amount of. Calculate the minimum possible... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f \\) such that \\( f(x+y) = f(x) + f(y) + f(xy) \\) for all \\( x, y \\in \\mathbb{R} \\)."
},
{
"role": "assistant",
"con... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the last three digits of $$2008^{2007^{2006^{\\iddots^{2^{1}}}}}.$$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( P(x) = (1 + x^3)(1 + 2x^9) \\cdots (1 + nx^{3^n}) \\cdots (1 + 1996x^{3^{1996}}) \\). If \\( P(x) = 1 + g(x) \\), and the coefficients of \\( g(x) \\)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the differential equation $y''y+(y')^2=\\ln x$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to solve this differential e... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A machine for numbering pages does not print numbers that are multiples of $3$ or $5$. The numbering sequence is $1, 2, 4, 7, 8, 11, \\ldots$. If the last num... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The arithmetic mean, the geometric mean, and the harmonic mean of two different positive integers \\(a\\) and \\(b\\) are positive integers. Find the minimum ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the parity of \\( N \\) where \\( N = \\lfloor (2023!)^{\\frac{1}{2023}} \\rfloor \\)?"
},
{
"role": "assistant",
"content": "<think>\nOka... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integers $n$ such that the polynomial $p(x) = x^5 - nx - n - 2$ can be written as the product of two non-constant polynomials with integral coefficie... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a < b\\) be positive real numbers and let \\(f_i: [a, b] \\to \\mathbb{R}\\) (\\(i = 1, 2\\)) be continuous functions on \\([a, b]\\) and differentiabl... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider points \\( A, B, P \\) and a line \\( l \\) with a fixed angle \\( \\alpha \\). Let points \\( C, D \\) be variable points on \\( l \\) such that \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the squa... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the total number of integral solutions to the equation $x^2 + y^2 + z^2 + t^2 = 2^{2004}$, where $0 \\leq x \\leq y \\leq z \\leq t$."
},
{
"role... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( ABC \\) be a triangle with area 1. Let \\( A' \\), \\( B' \\), and \\( C' \\) be points on \\( BC \\), \\( CA \\), and \\( AB \\) respectively such th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider an $n^2 \\times n^2$ grid divided into $n^2$ subgrids of size $n \\times n$. Find the number of ways to select $n^2$ cells such that there is exactly... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find a pair of points whose distance is not tripled by the map \\( f(x) = 3x \\mod 1 \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For what value of $\\theta$ in the interval $[0, \\pi]$ does the expression $\\sin\\theta\\sin2\\theta\\sin3\\theta$ achieve its maximum value? Express your a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Bob has B blue cards, O orange cards, and W white cards. He scores points as he plays each card. If he plays a blue card, his score is the number of white car... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = x^8 - 8x^6 + 19x^4 - 12x^3 + 14x^2 - 8x + 9 \\)."
},
{
"role": "assistant",
"content": "<think>\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the set \\( S \\) of all polynomials \\( p(x) \\) with coefficients in the set \\(\\{-1, 0, 1\\}\\). Let \\( b = \\frac{1 + \\sqrt{5}}{2} \\). Calcul... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with a point \\(P\\) inside such that \\(\\angle PBA = \\angle PCA\\), \\(\\angle PCB = 2 \\angle PAB\\), and \\(\\angle PAC = \\a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the lowest possible value of the expression\n\\[ \\frac{1}{a+x} + \\frac{1}{a+y} + \\frac{1}{b+x} + \\frac{1}{b+y} \\]\nwhere \\(a, b, x,\\) and \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the sequence defined by $a_1 = a_2 = 1$ and $a_n = \\frac{a_{n-1}^2 + 2}{a_{n-2}}$ for $n \\ge 3$, find the general term $a_n$."
},
{
"role": "a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\((\\cos 0^\\circ, \\sin 0^\\circ) = (1, 0)\\), find the values of \\((\\cos 1^\\circ, \\sin 1^\\circ)\\)."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Peter picked ten consecutive positive integers and divided each of them by a positive integer \\( n \\). The sum of the ten remainders obtained is equal to 20... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with altitude \\(BE\\) and \\(CF\\) intersecting at \\(H\\). Let \\(K\\) and \\(L\\) be the midpoints of \\(CA\\) and \\(AB\\), re... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For \\( X \\in \\mathbb{N}^* \\), let\n\\[ A(X) = 1657145277365 + 212175710912\\sqrt{X}, \\]\n\\[ B(X) = 107578229802750 + 13773980892672\\sqrt{X}, \\]\n\\[ C... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the polynomial $x^5 + ax^3 + b$, which has a double zero not equal to zero, find the relation between $a$ and $b$."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve in $\\mathbb{R}$ the equation: $[\\sin x] \\cdot \\{\\sin x\\} = \\sin x.$\n\nHere, $[\\sin x]$ denotes the greatest integer less than or equal to $\\si... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A triangle has side lengths \\(a\\), \\(b\\), and \\(c\\), all of which are integers. If the perimeter of the triangle is 2018, find the greatest possible val... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine $n$ for the polynomial $P(x)$ of degree $3n$ such that\n\\[\nP(0) = P(3) = \\cdots = P(3n) = 2, \\quad P(1) = P(4) = \\cdots = P(3n-2) = 1, \\quad P... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the number of combinations of $n$ letters together out of $3n$ letters, where $n$ are $a$'s, $n$ are $b$'s, and the remaining $n$ letters are all differe... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the number of ways to choose \\( n \\) elements from the multiset \\(\\{n \\cdot a, n \\cdot b, 1, 2, 3, \\ldots, n+1\\}\\) of size \\(3n + 1\\)."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f(x), g(x) : \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that \n\\[ f(x) + 2f(1-x) = g(x). \\]\nIf \\( g(x) \\) is strictly increasing, determine th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Marie and Jean are playing a game where they alternately choose numbers from 1 to 9 without repetition. The first player to obtain three numbers that sum up t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a parallelogram \\(ABCD\\), let \\(M\\) be the point on side \\(BC\\) such that \\(MC = 2BM\\) and let \\(N\\) be the point on side \\(CD\\) such that \\(N... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the highest value of $f(k,p)$ for $k \\ge 1$ and $p \\in S$, where $f(k,p) = a_{k} + a_{k+r(p)} + a_{k+2r(p)}$ and $r(p)$ is the integer such that the fu... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Three distinct positive integers are such that the sum of any two is exactly divisible by the third. The largest of these integers is 36. What are the two sma... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c\\) be three positive real numbers such that \\(a(b^2 + c^2) = b + c\\). Find the minimum value of \n\\[ P = \\sum \\frac{1}{(1+a)^2} + \\frac{4... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the probability that there will be at least one Canadian in the top three positions in a race with 20 athletes, given that 4 of the athletes are Can... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A rectangle $HOMF$ has sides $HO=11$ and $OM=5$. A triangle $\\Delta ABC$ has $H$ as the orthocenter, $O$ as the circumcenter, $M$ as the midpoint of $BC$, an... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the remainder when $1^{2023} + 2^{2023} + \\cdots + 2024^{2023}$ is divided by $2026$."
},
{
"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 functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the equation\n\\[ f(2x) = f\\left(\\sin\\left(\\frac{\\pi x}{2} + \\frac{\\pi y}{2}\\r... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If \\( x \\) and \\( y \\) are positive integers and \\( 15x = \\sqrt{735y} \\), find the least possible value of \\( x + y \\)."
},
{
"role": "assist... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{Q} \\to \\mathbb{R} \\) that satisfy the functional equation \\( f(x+y) = f(x) + f(y) + xy(x+y) \\)."
},
{
"role": ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the recurrence relation $a_1 = 1$ and $a_n = 2a_{n-1} + 2^n + n^2$, find a formula for $a_n$."
},
{
"role": "assistant",
"content": "<think>... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a sequence of 2017 consecutive positive integers such that each integer in the sequence has a different number of prime factors. Let \\( N \\) be the... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For a fixed natural number \\( k \\), does there exist a Bidiophantine \\( n \\)-gon (where \\( n \\geq 3 \\)) such that each side or diagonal is equal to \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the solution of the equation $2^x + 2^y = 2^z$ where $x, y, z \\in \\mathbb{N}$ and $x + y + z = 2005$."
},
{
"role": "assistant",
"content":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider triangle \\(ABC\\) with \\(AB = AC\\). Points \\(P\\) and \\(Q\\) are consecutively on the perpendicular bisector of \\(AB\\) and \\(AC\\) respective... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A happiness number is a positive integer whose cube has 13 times more positive divisors than the number itself. How many positive divisors does a happiness nu... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\(x, y,\\) and \\(z\\) are real numbers such that \\(x + y + z = 15\\) and \\(xy + yz + zx = 75\\), what are the possible values of \\(xyz\\)?"
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a_1, a_2, \\ldots, a_n\\) be positive real numbers such that \\(s = \\sum_{k=1}^n a_k = 100\\) and \\(s' = \\sum_{k=1}^n a_k^{1-\\frac{1}{k}}\\). Calcu... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the smallest positive integer that can be expressed as the sum of squares of positive integers in two different ways, such that the difference between th... |
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