Olympiad problems

2016 IFYM, Sozopol, 2

On the VI-th International Festival of Young Mathematicians in Sozopol $n$ teams were participating, each of which was with $k$ participants ($n>k>1$). The organizers of the competition separated the $nk$ participants into $n$ groups, each with $k$ people, in such way that no two teammates are in the same group. Prove that there can be found $n$ participants no two of which are in the same team or group.

2017 Dutch IMO TST, 3

Tags: number theory

Let $k > 2$ be an integer. A positive integer $l$ is said to be $k-pable$ if the numbers $1, 3, 5, . . . , 2k - 1$ can be partitioned into two subsets $A$ and $B$ in such a way that the sum of the elements of $A$ is exactly $l$ times as large as the sum of the elements of $B$. Show that the smallest $k-pable$ integer is coprime to $k$.

2014 Contests, 3

Tags: algebra , polynomial , inequalities , induction , trigonometry , calculus , derivative

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

2015 Saudi Arabia BMO TST, 1

Tags: sequence , recurrence relation

Prove that for any integer $n \ge 2$, there exists a unique finite sequence $x_0, x_1,..., x_n$ of real numbers which satisfies $x_0 = x_n = 0$ and $x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0$ for all $i = 1,2,...,n - 1$. Prove moreover that $ |x_i| \le \frac12$ for all $i = 1,2,...,n - 1$. Nguyễn Duy Thái Sơn

2024 Iran Team Selection Test, 2

Tags: geometry

For a right angled triangle $\triangle ABC$ with $\angle A=90$ we have $AC=2AB$. Point $M$ is the midpoint of side $BC$ and $I$ is incenter of triangle $\triangle ABC$. The line passing trough $M$ and perpendicular to $BI$ intersect with lines $BI$ and $AC$ at points $H$ and $K$ respectively. If the semi-line $IK$ cuts circumcircle of triangle $\triangle ABC$ at $F$ and $S$ be the second intersection point of line $FH$ with circumcircle of triangle $\triangle ABC$ , then prove that $SM$ is tangent to the incircle of triangle $\triangle ABC$. [i]Proposed by Mahdi Etesami Fard[/i]

2000 Moldova National Olympiad, Problem 8

Tags: game

Initially the number $2000$ is written down. The following operation is repeatedly performed: the sum of the $10$-th powers of the last number's digits is written down. Prove that in the infinite sequence thus obtained, some two numbers will be equal.

2008 Moldova National Olympiad, 12.7

Tags: hyperbola , geometry , conic

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.

2013 Today's Calculation Of Integral, 861

Tags: calculus , integration , geometry , function , analytic geometry , calculus computations

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

2020 LMT Fall, B5

Tags: algebra

Given the following system of equations $a_1 + a_2 + a_3 = 1$ $a_2 + a_3 + a_4 = 2$ $a_3 + a_4 + a_5 = 3$ $...$ $a_{12} + a_{13} + a_{14} = 12$ $a_{13} + a_{14} + a_1 = 13$ $a_{14 }+ a_1 + a_2 = 14$ find the value of $a_{14}$.

2003 Tournament Of Towns, 5

Tags: modular arithmetic , combinatorics

Prior to the game John selects an integer greater than $100$. Then Mary calls out an integer $d$ greater than $1$. If John's integer is divisible by $d$, then Mary wins. Otherwise, John subtracts $d$ from his number and the game continues (with the new number). Mary is not allowed to call out any number twice. When John's number becomes negative, Mary loses. Does Mary have a winning strategy?

2004 Switzerland Team Selection Test, 11

Tags: functional equation , functional

Find all injective functions $f : R \to R$ such that for all real $x \ne y$ , $f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+ f(y)}{f(x)- f(y)}$

2003 India IMO Training Camp, 8

Tags: inradius , geometry

Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that (a) triangle $ABC$ is acute, (b) $a+b+c>r+r_1+r_2+r_3$.

2023 4th Memorial "Aleksandar Blazhevski-Cane", P4

Tags: number theory , construction

Does the equation $$z(y-x)(x+y)=x^3$$ have finitely many solutions in the set of positive integers? [i]Proposed by Nikola Velov[/i]

2010 NZMOC Camp Selection Problems, 3

Tags: number theory , prime

Find all positive integers n such that $n^5 + n + 1$ is prime.

2025 Malaysian IMO Training Camp, 4

Tags: algebra , function

Find all functions $f:\mathbb R\to\mathbb R$ such that \[f(x^2)+2xf(y)=yf(x)+xf(x+y).\] [i](Proposed by Yeoh Yi Shuen)[/i]

2022 CMIMC Integration Bee, 4

Tags: calculus , integration

\[\int_0^1 \sqrt{x}\log(x)\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2023/2024 Tournament of Towns, 3

Tags: combinatorial geometry

A square was split into several rectangles so that the centers of rectangles form a convex polygon. a) Is it true for sure that each rectangle adjoins to a side of the square? b) Can the number of rectangles equal 23 ? Alexandr Shapovalov

2005 All-Russian Olympiad Regional Round, 8.8

Tags: vector , induction , inequalities , combinatorics

8.8, 9.8, 11.8 a) 99 boxes contain apples and oranges. Prove that we can choose 50 boxes in such a way that they contain at least half of all apples and half of all oranges. b) 100 boxes contain apples and oranges. Prove that we can choose 34 boxes in such a way that they contain at least a third of all apples and a third of all oranges. c) 100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at least half of all apples, and half of all oranges and half of all bananas. ([i]I. Bogdanov, G. Chelnokov, E. Kulikov[/i])

2023 Junior Balkan Team Selection Tests - Moldova, 10

Tags: combinatorics

In a chess tournament participated $ 100 $ players. Every player played one game with every other player. For a win $1$ point is given, for loss $ 0 $ and for a draw both players get $0,5$ points. Ion got more points than every other player. Mihai lost only one game, but got less points than every other player. Find all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.

2016 Junior Regional Olympiad - FBH, 5

Tags: table , combinatorics

In table [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9hLzBjNjFlZWFjM2ZlOTQzMTk2YTdkMzQ2MjJiYzYyMWFlN2Y0ZGZlLnBuZw==&rn=dGFibGljYWEucG5n[/img] $10$ numbers are circled, in every row and every column exactly one. Prove that among them, there are at least two equal

2015 Paraguay Juniors, 3

Tags: geometry

Tadeo draws the rectangle with the largest perimeter that can be divided into $2015$ squares of sidelength $1$ $cm$ and the rectangle with the smallest perimeter that can be divided into $2015$ squares of sidelength $1$ $cm$. What is the difference between the perimeters of the rectangles Tadeo drew?

2007 Stanford Mathematics Tournament, 1

Tags:

How many rational solutions for $x$ are there to the equation $x^4+(2-p)x^3+(2-2p)x^2+(1-2p)x-p=0$ if $p$ is a prime number?

Russian TST 2020, P3

Tags: ceiling function , number theory , perfect square

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

2014 Romania National Olympiad, 3

Tags: function , real analysis

Let $ f:[1,\infty )\longrightarrow (0,\infty ) $ be a continuous function satisfying the following properties: $ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} $ $ \text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}. $ [b]a)[/b] Show that $ \lim_{x\to\infty } \frac{f(x)}{x}=0. $ [b]b)[/b] Prove that $ \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0. $

2011 Preliminary Round - Switzerland, 2

Tags: function , number theory

Find all positive integers $n$ such that $n^3$ is the product of all divisors of $n$.