APMO 1996

Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$ . Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$ , with $M \in AD$ , $N \in DC$ , $P \in AB$ , and $Q \in BC$ . Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.

Let $m$ and $n$ be positive integers such that $n \leq m$ . Prove that $$2^n n! \leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n$$

Given four concyclic points. For each subset of three points take the incenter. Show that the four incenters from a rectangle.

The National Marriage Council wishes to invite $n$ couples to form 17 discussion groups under the following conditions:

(1) All members of a group must be of the same sex; i.e. they are either all male or all female.

(2) The difference in the size of any two groups is 0 or 1.

(3) All groups have at least 1 member.

(4) Each person must belong to one and only one group.

Find all values of $n$ , $n \leq 1996$ , for which this is possible. Justify your answer.

Let $a$ , $b$ , $c$ be the lengths of the sides of a triangle. Prove that $$\sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}$$ and determine when equality occurs.