Olympiad problems

2010 Sharygin Geometry Olympiad, 8

Tags: incenter , geometry

Let $AH$ be the altitude of a given triangle $ABC.$ The points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively. $BC$ touches the incircle of the triangle $ABC$ at a point $L.$ Find $\angle LI_bI_c.$

2010 BAMO, 3

Tags: graphing

All vertices of a polygon $P$ lie at points with integer coordinates in the plane, and all sides of $P$ have integer lengths. Prove that the perimeter of $P$ must be an even number.

2024 Bulgarian Winter Tournament, 11.4

Tags: combinatorics

Let $n, k$ be positive integers with $k \geq 3$. The edges of of a complete graph $K_n$ are colored in $k$ colors, such that for any color $i$ and any two vertices, there exists a path between them, consisting only of edges in color $i$. Prove that there exist three vertices $A, B, C$ of $K_n$, such that $AB, BC$ and $CA$ are all distinctly colored.

2002 IMO Shortlist, 8

Tags: geometry , circles , right angle

Let two circles $S_{1}$ and $S_{2}$ meet at the points $A$ and $B$. A line through $A$ meets $S_{1}$ again at $C$ and $S_{2}$ again at $D$. Let $M$, $N$, $K$ be three points on the line segments $CD$, $BC$, $BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$. Let $E$ and $F$ be points on those arcs $BC$ of $S_{1}$ and $BD$ of $S_{2}$ respectively that do not contain $A$. Given that $EN$ is perpendicular to $BC$ and $FK$ is perpendicular to $BD$ prove that $\angle EMF=90^{\circ}$.

2018 District Olympiad, 1

Tags: function

Find all strictly increasing functions $f : \mathbb{N} \to \mathbb{N} $ such that $\frac {f(x) + f(y)}{1 + f(x + y)}$ is a non-zero natural number, for all $x, y\in\mathbb{N}$.

2007 Croatia Team Selection Test, 8

Tags: number theory

Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.

2017 NIMO Problems, 7

Tags:

Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive. [i]Proposed by Mehtaab Sawhney[/i]

2015 Math Prize for Girls Olympiad, 2

Tags: geometry , 3d geometry , tetrahedron

A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.

2006 USA Team Selection Test, 2

Tags: geometry , circumcircle , trigonometry , incenter

In acute triangle $ABC$ , segments $AD; BE$ , and $CF$ are its altitudes, and $H$ is its orthocenter. Circle $\omega$, centered at $O$, passes through $A$ and $H$ and intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. The circumcircle of triangle $OPQ$ is tangent to segment $BC$ at $R$. Prove that $\frac{CR}{BR}=\frac{ED}{FD}.$

Durer Math Competition CD Finals - geometry, 2017.C+1

Tags: geometry , circles , parallel

Given a plane with two circles, one with points $A$ and $B$, and the other with points $C$ and $D$ are shown in the figure. The line $AB$ passes through the center of the first circle and touches the second circle while the line $CD$ passes through the center of the second circle and touches the first circle. Prove that the lines $AD$ and $BC$ are parallel. [img]https://cdn.artofproblemsolving.com/attachments/e/e/92f7b57751e7828a6487a052d4869e27e658b2.png[/img]