Asian Pacific Mathematics Olympiad · 1994

Problems

Problem 1Let $f:\mathbb R\to\mathbb R$ satisfy $f(x)+f(y)+1\ge f(x+y)\ge f(x)+f(y)$ for all real $x,y$ ; $f(0)\ge f(x)$ for $x\in[0,1)$ ; and $-f(-1)=f(1)=1$ . Find all such functions.Solutions: 1
Problem 2Given a nondegenerate triangle $ABC$ , with circumcentre $O$ , orthocentre $H$ , and circumradius $R$ , prove that $|OH|<3R$ .Solutions: 1
Problem 3Find all positive integers $n$ that can be written as $n=a^2+b^2$ , where $a,b$ are relatively prime positive integers and every prime $p <= sqrt(n)$ divides $ab$ .Solutions: 1
Problem 4Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?Solutions: 1
Problem 5List $A$ contains the decimal numbers $10^k$ for integers $k\ge1$ . Lists $B$ and $C$ contain these same numbers written in bases $2$ and $5$ , respectively. Prove that for every integer $n>1$ , exactly one number in exactly one of $B$ or $C$ has exactly $n$ digits.Solutions: 1