Olympiad problems

2010 Sharygin Geometry Olympiad, 3

Let $ABCD$ be a convex quadrilateral and $K$ be the common point of rays $AB$ and $DC$. There exists a point $P$ on the bisectrix of angle $AKD$ such that lines $BP$ and $CP$ bisect segments $AC$ and $BD$ respectively. Prove that $AB = CD$.

2016 Azerbaijan BMO TST, 1

Tags: geometry , bisection

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.

2015 Sharygin Geometry Olympiad, 5

Tags: geometry , bisection , angle bisector

Two equal hard triangles are given. One of their angles is equal to $ \alpha$ (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to $ \alpha / 2$. [i](No instruments are allowed, even a pencil.)[/i] (E. Bakayev, A. Zaslavsky)

2012 Sharygin Geometry Olympiad, 7

Tags: geometry , circles , bisection

Consider a triangle $ABC$. The tangent line to its circumcircle at point $C$ meets line $AB$ at point $D$. The tangent lines to the circumcircle of triangle $ACD$ at points $A$ and $C$ meet at point $K$. Prove that line $DK$ bisects segment $BC$. (F.Ivlev)

2014 Sharygin Geometry Olympiad, 8

Tags: geometry , bisection

Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$. (Tran Quang Hung)

2020 Kosovo National Mathematical Olympiad, 4

Tags: geometry , angle bisector , bisection , national olympiad , Olympiad , Kosovo

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The exterior angle bisector of $\angle BAC$ intersects $\omega$ at point $D$. Let $X$ be the foot of the altitude from $C$ to $AD$ and let $F$ be the intersection of the internal angle bisector of $\angle BAC$ and $BC$. Show that $BX$ bisects segment $AF$.

2016 Azerbaijan Balkan MO TST, 1

Tags: geometry , bisection

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.