Olympiad problems

2016 Bosnia And Herzegovina - Regional Olympiad, 3

Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$ is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$ $a)$ Prove that points $A$, $S$ and $T$ are collinear $b)$ Prove that $AC=AE$

2024 Austrian MO National Competition, 2

Tags: touching circles , altitude , geometry

Let $ABC$ be an acute triangle with $AB>AC$. Let $D,E,F$ denote the feet of its altitudes on $BC,AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$. Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$. [i](Karl Czakler)[/i]

2025 Bundeswettbewerb Mathematik, 3

Tags: touching circles , perpendicular lines , geometry , circles

Let $k$ be a semicircle with diameter $AB$ and midpoint $M$. Let $P$ be a point on $k$ different from $A$ and $B$. The circle $k_A$ touches $k$ in a point $C$, the segment $MA$ in a point $D$, and additionally the segment $MP$. The circle $k_B$ touches $k$ in a point $E$ and additionally the segments $MB$ and $MP$. Show that the lines $AE$ and $CD$ are perpendicular.

2022 German National Olympiad, 3

Tags: incircle , circumcircle , touching circles , sidelengths , geometry

Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$. Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$. Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.

2024 Austrian MO National Competition, 2

Tags: semicircle , touching circles , geometry , cyclic quadrilateral

Let $h$ be a semicircle with diameter $AB$. The two circles $k_1$ and $k_2$, $k_1 \ne k_2$, touch the segment $AB$ at the points $C$ and $D$, respectively, and the semicircle $h$ fom the inside at the points $E$ and $F$, respectively. Prove that the four points $C$, $D$, $E$ and $F$ lie on a circle. [i](Walther Janous)[/i]

2008 Bulgarian Autumn Math Competition, Problem 9.2

Tags: circumcircle , geometry , touching circles , ratio

Given a $\triangle ABC$ and the altitude $CH$ ($H$ lies on the segment $AB$) and let $M$ be the midpoint of $AC$. Prove that if the circumcircle of $\triangle ABC$, $k$ and the circumcircle of $\triangle MHC$, $k_{1}$ touch, then the radius of $k$ is twice the radius of $k_{1}$.

2022 German National Olympiad, 5

Tags: geometry , tangent , touching circles , tangent line

Let $ABC$ be an equilateral triangle with circumcircle $k$. A circle $q$ touches $k$ from outside in a point $D$, where the point $D$ on $k$ is chosen so that $D$ and $C$ lie on different sides of the line $AB$. We now draw tangent lines from $A,B$ and $C$ to the circle $q$ and denote the lengths of the respective tangent line segments by $a,b$ and $c$. Prove that $a+b=c$.