1. Problem: $A B C D$ is a rectangle with side lengths $A B=C D=1$ and $B C=D A=2$. Let $M$ be the midpoint of $A D$. Point $P$ lies on the opposite side of line $M B$ to $A$, such that triangle $M B P$ is equilateral. Find the value of $\angle P C B$.
Solution: (Kevin Shen)
$M$ is the midpoint of $A D$, by symmetry $M B=M C$. The side lengths of an equilateral triangle are all equal, so $M B=M P$.
As $M B=M C=M P, M$ is the circumcenter of triangle $B C P$. For any chord of any circle, the angle subtended at the center is always double the angle subtended at the circumference. Since $\angle P C B$ and $\angle P M B$ are both subtended by arc $B P$, we get
$$
\angle P C B=\frac{1}{2} \angle P M B=\frac{1}{2} 60^{\circ}=30^{\circ}
$$
as required.