Asian Pacific Mathematics Olympiad · 1996

Problems

Problem 1Let $ABCD$ be a quadrilateral with $AB=BC=CD=DA$ . Let $MN$ and $PQ$ be segments perpendicular to diagonal $BD$ , with $M\in AD$ , $N\in DC$ , $P\in AB$ , and $Q\in BC$ , and with distance $d>BD/2$ between them. Show that the perimeter of hexagon $AMNCQP$ does not depend on the positions of $MN$ and $PQ$ while their distance remains $d$ .Solutions: 1
Problem 2Let $m,n$ be positive integers with $n\le m$ . Prove that $2^n n!\le\dfrac{(m+n)!}{(m-n)!}\le(m^2+m)^n$ .Solutions: 1
Problem 3Let $P_1,P_2,P_3,P_4$ be four points on a circle. For each $i$ , let $I_i$ be the incenter of the triangle formed by the other three points. Prove that $I_1,I_2,I_3,I_4$ are the vertices of a rectangle.Solutions: 1
Problem 4The National Marriage Council wishes to invite $n$ couples to form $17$ discussion groups. Each group contains members of only one sex; the sizes of any two groups differ by $0$ or $1$ ; every group is nonempty; and every person belongs to exactly one group. Find all $n\le1996$ for which this is possible.Solutions: 1
Problem 5Let $a,b,c$ 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 occurs.Solutions: 1