Let $ABCD$ be a quadrilateral with $AB=BC=CD=DA$. Let $MN$ and $PQ$ be segments perpendicular to diagonal $BD$, at distance $d\gt BD/2$ from each other, where $M\in AD$, $N\in DC$, $P\in AB$, $Q\in BC$. Show that the perimeter of hexagon $AMNCQP$ is independent of the positions of $MN,PQ$ provided their distance remains constant.
Let $P_1,P_2,P_3,P_4$ be four points on a circle. Let $I_1$ be the incentre of $P_2P_3P_4$, $I_2$ the incentre of $P_1P_3P_4$, $I_3$ the incentre of $P_1P_2P_4$, and $I_4$ the incentre of $P_1P_2P_3$. Prove that $I_1,I_2,I_3,I_4$ are the vertices of a rectangle.
The National Marriage Council wishes to invite $n$ couples to form $17$ discussion groups under these conditions:
1. Each group consists entirely of males or entirely of females.
2. The sizes of any two groups differ by $0$ or $1$.
3. Every group has at least one member.
4. Each person belongs to exactly one group.
Find and justify all values of $n\le1996$ for which this is possible.
Let $a,b,c$ be the side lengths of a triangle. Prove that
$$\sqrt{a+b-c}+\sqrt{b+c-a}+\sqrt{c+a-b}\le\sqrt a+\sqrt b+\sqrt c,$$
and determine when equality holds.
