Let $ABC$ be a triangle $A \neq B$ and let $P \in (AB)$ be a point for which denote $m(\widehat{ACP}) = x$ and $m(\widehat{BCP}) = y$. Prove that $\frac{\sin A \sin B}{\sin(A-B)} = \frac{\sin x \sin y}{\sin(x-y)}$ if and only if $PA = PB$.
A triangle has vertices $A$, $B$, $C$. A point $P$ is chosen on side $AB$, and lines through $P$ parallel to the other sides create smaller triangles $APQ$ and $BPR$ and a parallelogram $PQCR$. Find the minimum value of the maximum of the areas of triangles $APQ$ and $BPR$ as a fraction of the area of $ABC$.
Let $A, B, C, D, E$ be the vertices of a regular pentagon inscribed in a circle of radius $r$. Let $F$ be the midpoint of side $AB$. Find the circumradius $AO$ in terms of the side length $x = AB$.
Let $f(n)$ be the number of ways to write a positive integer as an ordered sum of three non-negative integers, where each integer is chosen from $\{0, 1, 2, \ldots, 2n-1\}$ (i.e., using $n$ colours with values $0$ to $2n-1$). Find $f(n)$.
Let $P_0 = (0,0)$, $P_1 = (0,4)$, $P_2 = (4,0)$, $P_3 = (-4,-4)$, $P_4 = (2,4)$, $P_5 = (4,6)$ (or similar points). Find the region of all points closer to $P_0$ than to any of $P_1, P_2, P_3, P_4, P_5$, and compute its perimeter.