Olympiad problems

2024 Germany Team Selection Test, 1

Tags: algebra , function

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

2019 ELMO Shortlist, G5

Tags: geometry

Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$. [i]Proposed by Max Jiang[/i]

1967 All Soviet Union Mathematical Olympiad, 084

Tags: geometry , altitude , median , equilateral

a) The maximal height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$. Prove that the angle $ABC$ isn't greater than $60$ degrees. b) The height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$ and bisectrix $|CD|$. Prove that the angle $ABC$ is equilateral.

2016 Romania National Olympiad, 4

Tags: number theory

For $n \in N^*$ we will say that the non-negative integers $x_1, x_2, ... , x_n$ have property $(P)$ if $$x_1x_2 ...x_n = x_1 + 2x_2 + 3x_3 + ...+ nx_n.$$ a) Show that for every $n \in N^*$ there exists $n$ positive integers with property $(P)$. b) Find all integers $n \ge 2$ so that there exists $n$ positive integers $x_1, x_2, ... , x_n$ with $x_1< x_2<x_3< ... <x_n$, having property $(P)$.

2014 Turkey Junior National Olympiad, 1

Tags: inequalities , algebra

Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.

2022 Cyprus JBMO TST, 2

Tags: number theory , prime number , prime

Determine all pairs of prime numbers $(p, q)$ which satisfy the equation \[ p^3+q^3+1=p^2q^2 \]

2018 Moscow Mathematical Olympiad, 6

Tags: combinatorics

There is house with $2^n$ rooms and every room has one light bulb and light switch. But wiring was connected wrong, so light switch can turn on light in some another room. Master want to find what switch connected to every light bulb. He use next practice: he send some workers in the some rooms, then they turn on switches in same time, then they go to master and tell him, in what rooms light bulb was turned on. a) Prove that $2n$ moves is enough to find, how switches are connected to bulbs. b) Is $2n-1$ moves always enough ?

2005 Georgia Team Selection Test, 6

Tags: induction , modular arithmetic , number theory , combinatorics

Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties: 1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$; 2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$; Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.

2012 Middle European Mathematical Olympiad, 2

Tags: geometry , function , algebra , polynomial , inequalities

Let $ a,b$ and $ c $ be positive real numbers with $ abc = 1 $. Prove that \[ \sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)\]

MOAA Individual Speed General Rounds, 2023.10

Tags:

If $x,y,z$ satisfy the system of equations \[xy+yz+zx=23\] \[\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x}=-1\] \[\frac{z^2x}{x+y}+\frac{x^2y}{y+z}+\frac{y^2z}{z+x}=202\] Find the value of $x^2+y^2+z^2$. [i]Proposed by Harry Kim[/i]

2011 All-Russian Olympiad Regional Round, 9.8

Tags: geometry , rectangle , combinatorics

Straight rod of 2 meter length is cut into $N$ sticks. The length of each piece is an integer number of centimeters. For which smallest $N$ can one guarantee that it is possible to form the contour of some rectangle, while using all sticks and not breaking them further? (Author: A. Magazinov)

1994 Poland - First Round, 9

Tags:

Let $a$ and $b$ be positive real numbers with the sum equal to $1$. Prove that if $a^3$ and $b^3$ are rational, so are $a$ and $b$.

2025 EGMO, 1

Tags: number theory

For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$ for all $1 \leqslant i \leqslant m-1$ \\[i]Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.[/i] \\ [i]Proposed by Paulius Aleknavičius, Lithuania[/i]

2006 Moldova MO 11-12, 7

Tags: geometry

Let $n\in\mathbb{N}^*$. $2n+3$ points on the plane are given so that no 3 lie on a line and no 4 lie on a circle. Is it possible to find 3 points so that the interior of the circle passing through them would contain exactly $n$ of the remaining points.

2010 Thailand Mathematical Olympiad, 10

Tags: number theory , binomial , divisible , prime

Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.

2016 AIME Problems, 12

Tags: lotr , ogre

The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and will paint each of the six sections a solid color. Find the number of ways you can choose to paint each of the six sections if no two adjacent section can be painted with the same color. [asy] size(3cm); draw(unitcircle); draw(scale(0.6)*unitcircle); for(int i = 0; i < 6; ++i){ draw(dir(60*i)--0.6*dir(60*i)); } [/asy]

2014 Romania National Olympiad, 4

Tags: geometry , combinatorial geometry

Prove that three discs of radius $1$ cannot cover entirely a square surface of side $2$, but they can cover more than $99.75\%$ of it.

2009 China Team Selection Test, 2

Tags: modular arithmetic , polynomial , number theory , combinatorics , additive combinatorics , algebra

Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.

2023 AIME, 10

Tags:

Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$. One way to do this is shown below. Find the number of positive integer divisors of $N$. [asy] size(160); defaultpen(linewidth(0.6)); for(int j=0;j<=6;j=j+1) { draw((j,0)--(j,2)); } for(int i=0;i<=2;i=i+1) { draw((0,i)--(6,i)); } for(int k=1;k<=12;k=k+1) { label("$"+((string) k)+"$",(floor((k-1)/2)+0.5,k%2+0.5)); } [/asy]

1959 Poland - Second Round, 3

Tags: trigonometry , inequalities

Prove that if $ 0 \leq \alpha < \frac{\pi}{2} $ and $ 0 \leq \beta < \frac{\pi}{2} $, then $$ tg \frac{\alpha + \beta}{2} \leq \frac{tg \alpha + tg \beta}{2}.$$

2022 BAMO, B

Tags: percentage , algebra

You are bargaining with a salesperson for the price of an item. Your first offer is $a$ dollars and theirs is $b$ dollars. After you raise your offer by a certain percentage and they lower their offer by the same percentage, you arrive at an agreed price. What is that price, in terms of $a$ and $b$?

2023 ELMO Shortlist, A5

Tags: algebra

Find the least positive integer $M$ for which there exist a positive integer $n$ and polynomials $P_1(x)$, $P_2(x)$, $\ldots$, $P_n(x)$ with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\] [i]Proposed by Karthik Vedula[/i]

1996 Austrian-Polish Competition, 1

Tags: number theory , divide , divisible , digit

Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits

2013 JBMO TST - Macedonia, 1

Tags: algebra

Let $ x $ be a real number such that $ x^3 $ and $ x^2+x $ are rational numbers. Prove that $ x $ is rational.

2007 Germany Team Selection Test, 3

Tags: ratio , geometry , projective geometry , homothety

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.