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[
{
"content": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy \\( f(xy) + f(x-y) \\ge f(x+y) \\) for all real numbers \\( x, y \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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{
"content": "\\[\n\\boxed{\\begin{aligned}\n... |
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"content": "Masha has an integer multiple of toys compared to Lena, and Lena has the same multiple of toys compared to Katya. Masha gave 3 toys to Lena, and Katya gave 2 toys to Lena. After that, the number of toys each girl had formed an arithmetic progression. How many toys did each girl originally have? Pr... |
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{
"content": "Given that $a, b, c$ are positive real numbers and $ab + bc + ca = 3$, find the maximum value of the expression $\\dfrac{a+b+c}{\\sqrt{3}}+\\dfrac{\\sqrt{3}}{a+b+c} - \\left(\\dfrac{abc}{\\sqrt{3}}+\\dfrac{\\sqrt{3}}{abc}\\right)$. Let's think step by step and output the final answer within \\boxe... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nIn some other world, today is December D-th.\n\nWrite a program that prints `Christmas` if D = 25, `Christmas Eve` if D = 24, `Christmas Eve Eve` if D = 23 and `C... |
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{
"content": "Find the minimum value of the function \\( f(a) = \\sqrt{5a^2 + 4a + 4} + \\sqrt{5a^2 + 12a + 8} \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "\\[\nf(a)=\\sqrt{5a^{2}+4a+4}+\\sqrt{5a^{2}+12a+8}\n\\]\n\nWrite each radicand ... |
[
{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nSuppose an array sorted in ascending order is rotated at some pivot unknown to you beforehand.\n\n(i.e., [0,1,2,4,5,6,7] might become [4,5,6,7,0,1,2]).\n\nFind ... |
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{
"content": "Given points \\( A = (4,0,0) \\), \\( B = (0,3,0) \\), and \\( C = (0,0,5) \\), find the area of triangle \\( ABC \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "The area of a triangle with vertices \\(\\mathbf{A},\\mathbf{B... |
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"content": "Solve the following system of equations for the components of $\\vec{E}$:\n\n\\[\nE_x=\\frac{D_x}{1+4\\pi\\left(\\chi_1+\\chi_k\\left(E_x^2+E_y^2+E_z^2\\right)\\right)}\n\\]\n\n\\[\nE_y=\\frac{D_y}{1+4\\pi\\left(\\chi_1+\\chi_k\\left(E_x^2+E_y^2+E_z^2\\right)\\right)}\n\\]\n\n\\[\nE_z=\\frac{D_z}{... |
[
{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nThere are N Snuke Cats numbered 1, 2, \\ldots, N, where N is even.\n\nEach Snuke Cat wears a red scarf, on which his favorite non-negative integer is written.\n\n... |
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{
"content": "A circle is divided into 256 sectors. Prove that it is possible to write in each sector an eight-digit number consisting only of ones and twos (for example, 12212111, 22121121) such that: a) any two numbers will be different, b) each pair of numbers in adjacent sectors will differ by only one digi... |
[
{
"content": "Given \\( n \\in \\mathbb{N^{*}} \\) and \\( x_1, x_2, \\ldots, x_n > 0 \\) satisfying \\( \\sum_{i=1}^{n} \\frac{1}{x_i} = n \\), find the minimum value of \\( \\sum_{i=1}^{n} \\frac{x_i^2}{i} \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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... |
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"content": "Each unit square of a 2 by n grid is to be colored with blue or green so that no 2 by 2 green square is obtained. Let \\( c_n \\) be the number of different colorings. Determine the maximum \\( k \\) such that \\( 3^k \\) divides \\( c_{2001} \\). Let's think step by step and output the final answ... |
[
{
"content": "Given $\\triangle ABC$ with an internal point $P$ such that $\\angle PAB=10^\\circ$, $\\angle PBA=20^\\circ$, $\\angle PCA=30^\\circ$, and $\\angle PAC=40^\\circ$, prove that $\\triangle ABC$ is an isosceles triangle.\n(Problem from the 1996 25th USA Olympiad) Let's think step by step and output t... |
[
{
"content": "Does there exist a convex pentagon from which a certain line segments a similar pentagon? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "\\[\n\\boxed{\\text{Yes. For example, any regular pentagon has this property – the pentago... |
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"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nChef has invested his money at an interest rate of X percent per annum while the current inflation rate is Y percent per annum. \n\nAn investment is called *good*... |
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"content": "Let \\(a, b, c\\) be positive real numbers. Calculate the minimum value of the expression\n\\[\n\\frac{a^4}{a^3 + abc + b^3} + \\frac{b^4}{b^3 + abc + c^3} + \\frac{c^4}{c^3 + abc + a^3} - \\frac{a^3 + b^3 + c^3}{a^2 + b^2 + c^2}.\n\\] Let's think step by step and output the final answer within \\... |
[
{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nGiven two binary strings A and B that represent value of two integers, find the product of two strings in Decimal Value.\n \nExample 1:\nInput:\nA = \"1100\" , B ... |
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{
"content": "A circle with center $O$ and radius $15$ is given. Let $P$ be a point such that $|OP|=9$. How many of the chords of the circle pass through $P$ and have integer length? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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{
"content": "Let \n\n\\[... |
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"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nAlice buys a toy with a selling price of 100 rupees. There is a discount of x percent on the toy. Find the amount Alice needs to pay for it.\n\n------ Input Forma... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nGiven a string. Count the number of Camel Case characters in it.\nExample 1:\nInput:\nS = \"ckjkUUYII\"\nOutput: 5\nExplanation: Camel Case characters present:\nU... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nComplete the function which returns the weekday according to the input number:\n\n* `1` returns `\"Sunday\"`\n* `2` returns `\"Monday\"`\n* `3` returns `\"Tuesday... |
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"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nChef has started studying for the upcoming test. The textbook has N pages in total. Chef wants to read at most X pages a day for Y days.\n\nFind out whether it is... |
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"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\n## Story\n\n> \"You have serious coding skillz? You wannabe a [scener](https://en.wikipedia.org/wiki/Demoscene)? Complete this mission and u can get in teh crew!\... |
[
{
"content": "Find all natural numbers $n$ for which there exists an integer $a > 2$ such that $$a^d + 2^d \\mid a^n - 2^n$$ for all positive divisors $d \\ne n$ of $n$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "\\[\n\\boxed{\\;n=2^{k}\\... |
[
{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nGiven a string consisting of only lowercase English alphabets, your task is to find the smallest palindromic substring.\nIn case there are multiple palindromic su... |
[
{
"content": "Prove that \\(1 + x \\geq 2 \\sqrt{x}\\) for \\(x \\geq 0\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "**Proof.** \nLet \\(x\\ge 0\\). Both sides of the inequality are non‑negative, so we may manipulate them safely.\n\n\\... |
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"content": "Given that \\( P(t) \\) is a polynomial of degree \\( n \\) and for any \\( t \\in \\{1, 2, \\ldots, 2^n\\} \\), \\( P(t) = \\frac{1}{t} \\). Find \\( P(0) \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "**Solution.** \n\nL... |
[
{
"content": "What is the flaw in the following process of solving the equation $( -2)^k = 2$?\n\nGiven:\n\\[\n( -2)^{2k} = 2^2 \\implies (( -2)^2)^k = 4 \\implies 4^k \\implies k = 1.\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "The m... |
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"content": "Let \\( A \\) be a Noetherian ring and \\( a, b \\) be ideals in \\( A \\). Suppose \\( M \\) is a finitely generated \\( A \\)-module. Calculate the smallest integer \\( n \\) such that \\((M^a)^b \\simeq M^{n}\\) for all ideals \\( a, b \\) in \\( A \\). Let's think step by step and output the f... |
[
{
"content": "Find the value of x that satisfies\n\\[ \\frac{x-5}{107} + \\frac{x - 7}{105} + \\frac{x - 9}{103} + \\frac{x - 11}{101} = \\frac{x - 104}{4} + \\frac{x - 108}{2} \\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "\\[\n\\frac{x-... |
[
{
"content": "What condition must $p$ satisfy for the equation $x^{2n} + y^{2n} = p$ to have integer solutions?\n\nNote: The original post suggests that the condition $2^{2n} \\mid (p-1)$ is not sufficient, but it does not provide a complete condition. The problem is to determine the necessary condition for $p$... |
[
{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nYou are given a binary tree:\n\n```python\nclass Node:\n def __init__(self, L, R, n):\n self.left = L\n self.right = R\n self.value = n\n`... |
[
{
"content": "Determine whether the sequence \\( a_n = \\frac{(-1)^{n-1} n}{n^2 + 1} \\) converges or diverges. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "The sequence \n\n\\[\na_n=\\frac{(-1)^{\\,n-1}\\,n}{n^{2}+1}\n\\]\n\nhas absolute ... |
[
{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nGiven a string S, remove all consonants and print the modified string that contains vowels only.\nExample 1:\nInput\nS = \"abEkipo\"\nOutput\naEio\nExplanation : ... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nIt's bonus time in the big city! The fatcats are rubbing their paws in anticipation... but who is going to make the most money? \n\nBuild a function that takes in... |
[
{
"content": "What is the slope of the spiral $r = \\theta$ at $\\theta = \\pi/4$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "The spiral is given in polar form \\(r=\\theta\\). Write it in Cartesian coordinates as a parametric curve:\n\n... |
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{
"content": "What is the least positive integer $n$ such that $\\overbrace{f(f(\\dots f}^{21 \\text{ times}}(n)))=2013$ where $f(x)=x+1+\\lfloor \\sqrt x \\rfloor$? ($\\lfloor a \\rfloor$ denotes the greatest integer not exceeding the real number $a$.) Let's think step by step and output the final answer withi... |
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{
"content": "Why did the solution not divide by 2 when calculating the area of the base for the volume of the pyramids in the 1983 AIME Problem 11? The base is a triangle with sides $6$ and $6\\sqrt{2}$, and the height of the pyramid is $3\\sqrt{2}$. The area of the base should be $\\frac{1}{2} \\times 6 \\tim... |
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{
"content": "A horse is tied outside an equilateral triangular garden at one of the vertices. The side length of the triangular fence is 8 m, and the rope is 10 m long. Find the area over which the horse can graze outside the fence. Let's think step by step and output the final answer within \\boxed{}.",
"... |
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{
"content": "Let the sequence \\((a_n)_{n \\geq 1}\\) be defined with \\(a_1 \\geq 0\\) and \\(a_{n+1} = a_n (4 + a_n)\\) for \\(n \\geq 1\\). Find \\(\\lim_{n \\to \\infty} a_n^{1/2^n}\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "**So... |
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{
"content": "Find the sum of the first six terms of an arithmetic sequence in which the sum of any number of terms is equal to four times the square of that number. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "The sum of the first \\(n\\) ... |
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{
"content": "Compute the value of the series \\( y = x + \\frac{2}{3}x^3 + \\frac{2}{3}\\frac{4}{5}x^5 + \\frac{2}{3}\\frac{4}{5}\\frac{6}{7}x^7 + \\cdots \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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{
"content": "The series can be written as \n\n... |
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{
"content": "In triangle $ABC$, the angles are in the ratio $1:3:5$. Find the angle between the bisector of the largest angle and the line containing the altitude drawn to the smallest side of the triangle. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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{
... |
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"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nThese days, chef is very much interested in Mathematics. He has started attending Recitations too! His hunger for problems is increasing day by day!\nToday, chef ... |
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"content": "Find the number that satisfies the equation: \\[\n\\left(2 \\frac{2}{3}+\\square\\right) \\div 3 \\frac{3}{4}-0.4=\\frac{32}{45}, \\square=\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "\\[\n\\boxed{\\frac{3}{2}}\n\\]",
... |
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{
"content": "Calculate the area of the figure bounded by the graphs of the functions:\n\n\\[ y = \\arccos x, \\, y = 0, \\, x = 0 \\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "The region is the part of the first quadrant under the curve... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nFor \"x\", determine how many positive integers less than or equal to \"x\" are odd but not prime. Assume \"x\" is an integer between 1 and 10000.\n\nExample: 5 h... |
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{
"content": "If Person B trades all their chairs for the same number of tables as Person A, Person B needs to pay an additional 320 yuan. If Person B does not pay the extra money, they would receive 5 fewer tables. It is known that the price of 3 tables is 48 yuan less than the price of 5 chairs. How many chai... |
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{
"content": "Let \\(ABC\\) be a triangle and let \\(P\\) be a point in its interior. Let \\(X_1\\), \\(Y_1\\), \\(Z_1\\) be the intersections of \\(AP\\), \\(BP\\), \\(CP\\) with \\(BC\\), \\(CA\\), and \\(AB\\), respectively. Furthermore, let \\(AX_2\\), \\(BY_2\\), \\(CZ_2\\) be the reflections of lines \\(A... |
[
{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\n## A square of squares\nYou like building blocks. You especially like building blocks that are squares. And what you even like more, is to arrange them into a squ... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nThis is related to my other Kata about cats and dogs.\n\n# Kata Task\n\nI have a cat and a dog which I got as kitten / puppy.\n\nI forget when that was, but I do ... |
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{
"content": "For positive real numbers \\(a, b, c\\), calculate the maximum value of the expression\n\\[\n\\frac{a}{3a^2 + b^2 + 2ac} + \\frac{b}{3b^2 + c^2 + 2ab} + \\frac{c}{3c^2 + a^2 + 2bc}.\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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{
"conte... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nIn this problem you will meet the simplified model of game King of Thieves.\n\nIn a new ZeptoLab game called \"King of Thieves\" your aim is to reach a chest with... |
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{
"content": "Let \\( a_1 = 1 \\) and \\( a_n = n(a_{n-1} + 1) \\) for \\( n = 2, 3, 4, \\ldots \\). Define \\( P_n = \\left(1 + \\frac{1}{a_1}\\right)\\left(1 + \\frac{1}{a_2}\\right) \\cdots \\left(1 + \\frac{1}{a_n}\\right) \\). Find \\( \\lim_{n \\to \\infty} P_n \\). Let's think step by step and output the... |
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{
"content": "Consider a fixed circle in the complex plane and a variable circle that intersects the fixed circle at a constant angle. Let $z$ represent the center of the variable circle in the complex plane. If the equation of the fixed circle is $|z - a| = r$, and the variable circle has radius $R$ and is cen... |
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{
"content": "Find all functions \\( f \\) such that \\([f(x)]^2 = 100 + \\int_{0}^{x} [(f(t))^2 + (f'(t))^2] \\, dt\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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{
"content": "\\[\n\\begin{aligned}\n&[f(x)]^{2}=100+\\int_{0}^{x}\\bigl[(f(t))^{2}+(f'(... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nWrite a program to find the remainder when two given numbers are divided.\n\n\nInput\nThe first line contains an integer T, total number of test cases. Then follo... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nIt is September 9 in Japan now.\nYou are given a two-digit integer N. Answer the question: Is 9 contained in the decimal notation of N?\n\n-----Constraints-----\n... |
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{
"content": "Given the equation \\( f(x) = (x - x_0)^{2000}(x - x_1)(x - x_2) \\cdots (x - x_{15})(x^2 + x + p) = 0 \\), where \\( x_i \\) (for \\( i = 0, 1, 2, \\ldots, 15 \\)) are different natural numbers and \\( p \\) is a real number such that \\( x^2 + x + p = 0 \\) has no real solutions. If the sum of a... |
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{
"content": "Let $m$ be a natural number. On the SMO board, the number 1 is written $2^{m}$ times. In one step, we choose two numbers $a$ and $b$ on the board and replace both of them with $a+b$. Show that after $m \\cdot 2^{m-1}$ steps, the sum of the numbers is at least $4^{m}$. Let's think step by step and ... |
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{
"content": "What is the number of ordered pairs \\((x, y)\\), where \\(x, y \\in \\{0, 1, 2, \\ldots, 142\\}\\), such that \\(5x^2 + 7y^2 - 1\\) is a multiple of 143? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "\\[\n5x^{2}+7y^{2}\\equiv1... |
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"content": "In a row of $K$ circles, each circle is assigned a unique number from $1$ to $K$. For each positive divisor $d$ of $K$, the situation of the circles whose assigned numbers are divisible by $d$ is changed $K$ times. Determine the value of $K$ such that after performing this procedure once for all p... |
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{
"content": "Solve the equation $\\left[\\frac{2x^2}{x^2 + 1}\\right] = x$, where $[a]$ denotes the largest integer less than or equal to $a$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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{
"content": "\\[\n\\boxed{\\{0,1\\}}\n\\]",
"role": "assista... |
[
{
"content": "Let \\(a\\) and \\(b\\) be two natural numbers such that \\( p = \\frac{b}{4} \\sqrt{\\frac{2a - b}{2a + b}} \\) is a prime number. How large can \\( p \\) be? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
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"content": "\\[\np=\\frac{b}{4}\\... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nGiven n, a and r as the number of terms, first term and common ratio respectively of an Geometric Progression Series. Find the sum of the series upto nth term.\n ... |
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{
"content": "Write a solution for the following programming challenge. Provide a brief explanation of your approach, followed by the complete code.\nGiven a string s consisting only of letters 'a' and 'b'. In a single step you can remove one palindromic subsequence from s.\nReturn the minimum number of steps t... |
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