Školjka

%V0 Let $P$ be a point inside a triangle $ABC$ such that $$\angle APB - \angle ACB = \angle APC - \angle ABC.$$ Let $D$, $E$ be the incenters of triangles $APB$, $APC$, respectively. Show that the lines $AP$, $BD$, $CE$ meet at a point.

%V0 A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.

%V0 Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

%V0 Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

%V0 Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies $$\angle PBA+\angle PCA = \angle PBC+\angle PCB.$$ Show that $AP \geq AI$, and that equality holds if and only if $P=I$.

%V0 Let $ABC$ be a triangle with $AB = AC$ . The angle bisectors of $\angle C AB$ and $\angle AB C$ meet the sides $B C$ and $C A$ at $D$ and $E$ , respectively. Let $K$ be the incentre of triangle $ADC$. Suppose that $\angle B E K = 45^\circ$ . Find all possible values of $\angle C AB$ . Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea