Olympiad problems

1978 IMO Longlists, 19

Tags: geometry , triangle

We consider three distinct half-lines $Ox, Oy, Oz$ in a plane. Prove the existence and uniqueness of three points $A \in Ox, B \in Oy, C \in Oz$ such that the perimeters of the triangles $OAB,OBC,OCA$ are all equal to a given number $2p > 0.$

2025 PErA, P3

Tags: geometry

Let $ ABC $ be an equilateral triangle with circumcenter $ O $. Let $ X $ and $ Y $ be two points on segments $ AB $ and $ AC $, respectively, such that $ \angle XOY = 60^\circ $. If $ T $ is the reflection of $ O $ with respect to line $ XY $, prove that lines $ BT $ and $ OY $ are parallel.

MathLinks Contest 2nd, 7.3

Tags: geometry , combinatorics

A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned into $21$ parallelograms.

2013 BMT Spring, 8

Tags: geometry

$ABC$ is an isosceles right triangle with right angle $B$ and $AB = 1$. $ABC$ has an incenter at $E$. The excircle to $ABC$ at side $AC$ is drawn and has center $P$. Let this excircle be tangent to $AB$ at $R$. Draw $T$ on the excircle so that $RT$ is the diameter. Extend line $BC$ and draw point $D$ on $BC$ so that $DT$ is perpendicular to $RT$. Extend $AC$ and let it intersect with $DT$ at $G$. Let $F$ be the incenter of $CDG$. Find the area of $\vartriangle EFP$.

2008 Harvard-MIT Mathematics Tournament, 9

Tags: geometry , 3d geometry , tetrahedron , pyramid , similar triangles

Consider a circular cone with vertex $ V$, and let $ ABC$ be a triangle inscribed in the base of the cone, such that $ AB$ is a diameter and $ AC \equal{} BC$. Let $ L$ be a point on $ BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ ABCL$. Find the value of $ BL/LV$.

2019 Balkan MO Shortlist, G5

Tags: circumcircle , geometry , equal segments

Let $ABC$ ($BC > AC$) be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $AB$ at the point $D$. The circumcircles of triangles $BCD, OCD$ and $AOB$ intersect the ray $CA$ (beyond $A$) at the points $Q, P$ and $K$, respectively, such that $P \in (AK)$ and $K \in (PQ)$. The line $PD$ intersects the circumcircle of triangle $BKQ$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $BQ$. Prove that $TB = TQ$.

2008 Harvard-MIT Mathematics Tournament, 9

Tags: ratio , geometry , incenter , angle bisector

Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.