Olympiad problems

2011 Tournament of Towns, 5

In the convex quadrilateral $ABCD, BC$ is parallel to $AD$. Two circular arcs $\omega_1$ and $\omega_3$ pass through $A$ and $B$ and are on the same side of $AB$. Two circular arcs $\omega_2$ and $\omega_4$ pass through $C$ and $D$ and are on the same side of $CD$. The measures of $\omega_1, \omega_2, \omega_3$ and $\omega_4$ are $\alpha, \beta,\beta$ and $\alpha$ respectively. If $\omega_1$ and $\omega_2$ are tangent to each other externally, prove that so are $\omega_3$ and $\omega_4$.

2002 Moldova Team Selection Test, 3

Tags: geometry , incenter , similar triangles , arc

A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$, respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.

Ukrainian TYM Qualifying - geometry, XI.4

Tags: geometry , construction , arc , ratio

Chords $AB$ and $CD$, which do not intersect, are drawn in a circle. On the chord $AB$ or on its extension is taken the point $E$. Using a compass and construct the point $F$ on the arc $AB$ , such that $\frac{PE}{EQ} = \frac{m}{n}$, where $m,n$ are given natural numbers, $P$ is the point of intersection of the chord $AB$ with the chord $FC$, $Q$ is the point of intersection of the chord $AB$ with the chord $FD$. Consider cases where $E\in PQ$ and $E \notin PQ$.