Olympiad problems

2019 Singapore Junior Math Olympiad, 1

In the triangle $ABC, AC=BC, \angle C=90^o, D$ is the midpoint of $BC, E$ is the point on $AB$ such that $AD$ is perpendicular to $CE$. Prove that $AE=2EB$.

1995 Tuymaada Olympiad, 5

Tags: combinatorial geometry , isosceles , isosceles triangle , combinatorics

A set consisting of $n$ points of a plane is called an isosceles $n$-point if any three of its points are located in vertices of an isosceles triangle. Find all natural numbers for which there exist isosceles $n$-points.

2018 Yasinsky Geometry Olympiad, 6

Tags: circumcircle , incenter , perpendicularity , perpendicular , isosceles , isosceles triangle , geometry

Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular. (Vyacheslav Yasinsky)

2007 Korea Junior Math Olympiad, 7

Tags: geometry , incenter , circumcenter , isosceles , isosceles triangle

Let the incircle of $\triangle ABC$ meet $BC,CA,AB$ at $J,K,L$. Let $D(\ne B, J),E(\ne C,K), F(\ne A,L)$ be points on $BJ,CK,AL$. If the incenter of $\triangle ABC$ is the circumcenter of $\triangle DEF$ and $\angle BAC = \angle DEF$, prove that $\triangle ABC$ and $\triangle DEF$ are isosceles triangles.

2014 Hanoi Open Mathematics Competitions, 8

Tags: geometry , right triangle , isosceles , isosceles triangle

Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.

2008 Federal Competition For Advanced Students, P1, 4

Tags: geometry , isosceles , isosceles triangle , equal segments , midpoint

In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$. Let $G$ be the foot of the perpendicular from $C$ to $ AB$. Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles.

2007 Sharygin Geometry Olympiad, 2

Tags: diagonal , geometry , quadrilateral , isosceles , isosceles triangle

Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

2010 Sharygin Geometry Olympiad, 8

Tags: geometry , angle bisector , isosceles , isosceles triangle

Bisectrices $AA_1$ and $BB_1$ of triangle $ABC$ meet in $I$. Segments $A_1I$ and $B_1I$ are the bases of isosceles triangles with opposite vertices $A_2$ and $B_2$ lying on line $AB$. It is known that line $CI$ bisects segment $A_2B_2$. Is it true that triangle $ABC$ is isosceles?