Problems

1. (Sergey Yurin, 9th grade) In an isosceles triangle $ABC$, $AB=AC$, and $\angle A=20^\circ$. Point $P$ is taken on the side $AC$ such that $AP=BC$. Find $\angle PBC$.
   

   Use the idea of excenters and incenters to solve the following problems.

2. (Carleton University Mathematics Competition for High School Students, 1976)
   $ABC$ is an isosceles triangle with $\angle ABC = \angle ACB= 80^\circ$. $P$ is the point on $AB$ such that $\angle PCB = 70^\circ$. $Q$ is the point on $AC$ such that $\angle QBC = 60^\circ$. Find $\angle PQA$.
3. (Pythagoras Olympiad in The Netherlands, 1980) In triangle $ABC$, point $D$ is such that $\angle DCA=\angle DCB=\angle DBC= 10^\circ$ and $\angle DBA =20^\circ$. Find the measure of $\angle CAD$.
4. (Alberta High School Mathematics Competition, 1989-90) In quadrilateral $ABCD$ with diagonals $BD$ and $AC$, $\angle ABD=40^\circ$, $\angle CBD=70^\circ$, $\angle CDB=50^\circ$, $\angle ADB=80^\circ$. Find the measure of $\angle CAD$.
5. (Junior Problem A-6, Tournament of Towns, Spring 1997) Let $P$ be a point inside triangle $ABC$ with $AB=BC$, $\angle ABC=80^\circ$, $\angle PAC =40^\circ$ and $\angle ACP=30^\circ$. Find the measure of $\angle BPC$.
6. Senior Problem A-2, Tournament of Towns, Spring 1997) $D$ is the point on $BC$ and $E$ is the point on $CA$ such that $AD$ and $BE$ are the bisectors of $\angle A$ and $\angle B$ of triangle $ABC$. If $DE$ is the bisector of $\angle ADC$, find the measure of $\angle A$.