Asian Pacific Mathematics Olympiad · 1997

Problems

Problem 1Let $T_n=1+2+\cdots+n=n(n+1)/2$ and $S_n=1/T_1+1/T_2+\cdots+1/T_n$ . Prove that $1/S_1+1/S_2+\cdots+1/S_{1996}>1001$ .Solutions: 1
Problem 2Find an integer $n$ in the range $100\le n\le1997$ such that $n$ divides $2^n+2$ .Solutions: 1
Problem 3Let $ABC$ be a triangle. The bisector of angle $A$ meets segment $BC$ at $X$ and the circumcircle at $Y$ . Let $r_A=AX/AY$ , and define $r_B,r_C$ similarly. Prove that $r_A/\sin^2A+r_B/\sin^2B+r_C/\sin^2C\ge3$ , with equality if and only if the triangle is equilateral.Solutions: 1
Problem 4Let $P_1$ and $P_3$ be fixed points. Let $P_2$ lie on the line through $P_3$ perpendicular to $P_1P_3$ . Define $P_{n+1}$ as the foot of the perpendicular from $P_n$ to $P_{n-1}P_{n-2}$ . Show that the sequence converges to a point $P$ , and determine the locus of $P$ as $P_2$ varies.Solutions: 1
Problem 5 $n$ people are seated in a circle. A total of $nk$ coins are distributed among them, not necessarily equally. A move transfers one coin between two adjacent people. Find an algorithm using the minimum number of moves that leaves everyone with the same number of coins.Solutions: 1