Olympiad problems

2014 Sharygin Geometry Olympiad, 23

Let $A, B, C$ and $D$ be a triharmonic quadruple of points, i.e $AB\cdot CD = AC \cdot BD = AD \cdot BC.$ Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1, B, C$ and $D$ is triharmonic. Points $B_1, C_1$ and $D_1$ are defined similarly. Prove that a) $A, B, C_1, D_1$ are concyclic; b) the quadruple $A_1, B_1, C_1, D_1$ is triharmonic.

2016 Germany Team Selection Test, 1

Tags: concyclic , circles , geometry

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

2016 Latvia Baltic Way TST, 15

Tags: geometry , altitude , concyclic , midpoint

Let $ABC$ be a triangle. Let its altitudes $AD$, $BE$ and $CF$ concur at $H$. Let $K, L$ and $M$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Prove that, if $\angle BAC = 60^o$, then the midpoints of the segments $AH$, $DK$, $EL$, $FM$ are concyclic.

2011 Dutch Mathematical Olympiad, 2

Tags: geometry , concyclic , equilateral

Let $ABC$ be a triangle. Points $P$ and $Q$ lie on side $BC$ and satisfy $|BP| =|PQ| = |QC| = \frac13 |BC|$. Points $R$ and $S$ lie on side $CA$ and satisfy $|CR| =|RS| = |SA| = 1 3 |CA|$. Finally, points $T$ and $U$ lie on side $AB$ and satisfy $|AT| = |TU| = |UB| =\frac13 |AB|$. Points $P, Q,R, S, T$ and $U$ turn out to lie on a common circle. Prove that $ABC$ is an equilateral triangle.

2013 Estonia Team Selection Test, 4

Tags: geometry , cyclic quadrilateral , right triangle , concyclic , square

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

Russian TST 2022, P2

Tags: concyclic , incircle , geometry

The quadrilateral $ABCD$ is inscribed in the circle $\Gamma$. Let $I_B$ and $I_D$ be the centers of the circles $\omega_B$ and $\omega_D$ inscribed in the triangles $ABC$ and $ADC$, respectively. A common external tangent to $\omega_B$ and $\omega_D$ intersects $\Gamma$ at $K$ and $L{}$. Prove that $I_B,I_D,K$ and $L{}$ lie on the same circle.

2012 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , cyclic quadrilateral , concyclic , collinear , circumcenter , circumcircle

The quadrilateral $ABCD$ is inscribed in a circle centered at $O$, and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$. Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$. a) Prove that the points $P, Q$, and $R$ are collinear. b) If $U$ and $V$ are the circumcenters of the triangles $ABP$, and $CDP$, respectively, prove that the points $U, R, O, V$ are concyclic.