Call triangle $ABC$ great if for every point $D$ on side $BC$, with $P,Q$ the perpendicular feet from $D$ to lines $AB,AC$, the reflection of $D$ in line $PQ$ lies on the circumcircle of $ABC$.
Prove that $ABC$ is great if and only if $\angle A=90^\circ$ and $AB=AC$.
A positive integer is fancy if it is a sum
$$2^{a_1}+2^{a_2}+\cdots+2^{a_{100}},$$
where the exponents are nonnegative integers, not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is fancy.
Let $AB,AC$ be distinct noncollinear rays. A circle $\omega$ with center $O$ is tangent to ray $AC$ at $E$ and ray $AB$ at $F$. Let $R$ lie on segment $EF$. The line through $O$ parallel to $EF$ meets line $AB$ at $P$. Let $N=PR\cap AC$, and let $M$ be the intersection of $AB$ with the line through $R$ parallel to $AC$. Prove that $MN$ is tangent to $\omega$.
Dreamland has 2016 cities. Starways establishes one-way flights between pairs of cities so that every city has exactly one outgoing flight. Find the smallest positive integer $k$ such that, for every such flight system, the cities can be partitioned into $k$ groups with no city able to reach a different city in its own group using at most 28 flights.
