Olympiad problems

2018-2019 SDML (High School), 8

Tags: geometry , rhombus

The figure below consists of five isosceles triangles and ten rhombi. The bases of the isosceles triangles are $12$, $13$, $14$, $15$, as shown below. The top rhombus, which is shaded, is actually a square. Find the area of this square. [DIAGRAM NEEDED]

2020 Latvia Baltic Way TST, 9

Tags: geometry

Given $\triangle ABC$, whose all sides have different length. Point $P$ is chosen on altitude $AD$. Lines $BP$ and $CP$ intersect lines $AC, AB$ respectively and point $X, Y$.It is given that $AX=AY$. Prove that there is circle, whose centre lies on $BC$ and is tangent to sides $AC$ and $AB$ at points $X,Y$.

2018 Poland - Second Round, 3

Tags: geometry , plane geometry , bisector , circumcircle

Bisector of side $BC$ intersects circumcircle of triangle $ABC$ in points $P$ and $Q$. Points $A$ and $P$ lie on the same side of line $BC$. Point $R$ is an orthogonal projection of point $P$ on line $AC$. Point $S$ is middle of line segment $AQ$. Show that points $A, B, R, S$ lie on one circle.

2006 Singapore Junior Math Olympiad, 4

Tags: geometry , angle chasing , angle bisector , incenter

In $\vartriangle ABC$, the bisector of $\angle B$ meets $AC$ at $D$ and the bisector of $\angle C$ meets $AB$ at $E$. These bisectors intersect at $O$ and $OD = OE$. If $AD \ne AE$, prove that $\angle A = 60^o$.

2010 May Olympiad, 2

Tags: geometry , rectangle , equal segments

Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.

2016 Dutch IMO TST, 4

Tags: geometry , circles , fixed

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.

2015 Sharygin Geometry Olympiad, P18

Tags: collinear , hexagon , geometry

Let $ABCDEF$ be a cyclic hexagon, points $K, L, M, N$ be the common points of lines $AB$ and $CD$, $AC$ and $BD$, $AF$ and $DE$, $AE$ and $DF$ respectively. Prove that if three of these points are collinear then the fourth point lies on the same line.