Asian Pacific Mathematics Olympiad · 2006

Problems

Problem 1Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ such that whenever real numbers $a_1,a_2,\ldots,a_n$ have an integer sum, there is an index $i$ for which $\left|a_i-\frac12\right|\ge f(n)$ .Solutions: 1
Problem 2Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden mean $\phi=\frac{1+\sqrt5}{2}$ . Here an integral power is $\phi^i$ with $i$ an integer, not necessarily positive.Solutions: 1
Problem 3Let $p\ge5$ be a prime. Let $r$ be the number of ways of placing $p$ identical checkers on a $p\times p$ checkerboard so that not all checkers are in the same row (they may all be in the same column). Show that $r$ is divisible by $p^5$ .Solutions: 1
Problem 4Let $A,B$ be distinct points on a given circle $O$ , and let $P$ be the midpoint of $AB$ . Let $O_1$ be the circle tangent to the line $AB$ at $P$ and tangent to $O$ . Let $\ell$ , different from $AB$ , be the tangent to $O_1$ through $A$ , and let $C$ be the intersection point other than $A$ of $\ell$ and $O$ . Let $Q$ be the midpoint of $BC$ , and let $O_2$ be the circle tangent to the line $BC$ at $Q$ and tangent to the line segment $AC$ . Prove that $O_2$ is tangent to $O$ .Solutions: 1
Problem 5In a circus, $n$ clowns dress and paint themselves using a selection of 12 distinct colours. Each clown must use at least five different colours. The ringmaster requires that no two clowns have exactly the same set of colours and that no more than 20 clowns use any one particular colour. Find the largest possible $n$ .Solutions: 1