Problems From Around the World

Problem 37 (IMO Proposal)   The incircle of $\triangle ABC$ touches $BC,CA,AB$ at $D,E,F$, respectively. $X$ is a point inside $\triangle ABC$ such that the incircle of $\triangle XBC$ touches $BC$ at $D$ also, touches $CX$ and $XB$ at $Y$ and $Z$, respectively. Prove that $EFZY$ is a cyclic quadrilateral.

Problem 38 (Israel, 1995)   Let $PQ$ be the diameter of semicircle $H$. Circle $k$ is internally tangent to $H$ and tangent to $PQ$ at $C$. Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB$ is perpendicular to $PQ$ and is also tangent to $k$. Prove that $AC$ bisects $\angle PAB$.

Problem 39 (Romania, 1997)   Let $ABC$ be a triangle, $D$ a point on side $BC$, and $\omega$ the circumcicle of $ABC$. Show that the circles tangent to $\omega$, $AD$, $BD$ and to $\omega$, $AD$, $DC$ are also tangent to each other if and only if $\angle BAD=\angle CAD$.

Problem 40 (Russia, 1995)   We are given a semicircle with diameter $AB$ and center $O$, and a line which intersects the semicircle at $C$ and $D$ and line $AB$ at $M$ ( $MB<MA,\,\,MD<MC$.) Let $K$ be the second point of intersection of the circumcircles of $\triangle AOC$ and $\triangle DOB$. Prove that $\angle MKO=90^{\circ}$.