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.

Click for solution Let's look at APQCNM when distance from B to PQ is h1, distance from D to MN is h2, AM = NC = a, AP = QC = b. AP = QC and AM = CN because ABCD is a rombus and PQ and MN are perpendicular to BD. When PQ is moved up by k units, MN is moved up by k units. As we move up PQ by t units, QC and AP increase by a rt, where r is constant. This ratio r is the sane for MN (except it's negative) because we have a rombus. Hence after the shift of PQ and MN, AP and QC increase by tr units, AM and CN decrease by rt units. So, AP + QC + AM + CN is const (*) Let angle DBC be w. Then angle BDC is w. Then PQ + MN = tgw * 2 * h1 + tgw * 2 * h2 = 2 * tgw * (BD - distance between PQ and MN) = const. Hence MN + PQ is const. Adding this to (*) gets the desired result.

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

If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of 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.