International Mathematical Olympiad · 1999

Problems

Problem 1Determine all finite sets $S$ of at least three points in the plane which satisfy the following condition: For any two distinct points $A$ and $B$ in $S$ , the perpendicular bisector of the line segment $AB$ is an axis of symmetry of $S$ .Solutions: 1
Problem 2Let $n\ge2$ be a fixed integer. (a) Find the least constant $C$ such that for all nonnegative real numbers $x_1,\dots,x_n$ , $\sum_{1\le i<j\le n}x_ix_j(x_i^2+x_j^2)\le C\left(\sum_{i=1}^n x_i\right)^4$ . (b) Determine when equality occurs for this value of $C$ .Solutions: 1
Problem 3Consider an $n\times n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. We say that two different squares on the board are adjacent if they have a common side. $N$ unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square. Determine the smallest possible value of $N$ .Solutions: 1
Problem 4Determine all pairs $(n,p)$ of positive integers such that $p$ is a prime, $n$ not exceeded $2p$ , and $(p-1)^n+1$ is divisible by $n^{p-1}$ .Solutions: 1
Problem 5Two circles $G_1$ and $G_2$ are contained inside the circle $G$ , and are tangent to $G$ at the distinct points $M$ and $N$ , respectively. $G_1$ passes through the center of $G_2$ . The line passing through the two points of intersection of $G_1$ and $G_2$ meets $G$ at $A$ and $B$ . The lines $MA$ and $MB$ meet $G_1$ at $C$ and $D$ , respectively. Prove that $CD$ is tangent to $G_2$ .Solutions: 1
Problem 6Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x-f(y))=f(f(y))+xf(y)+f(x)-1$ for all real numbers $x,y$ .Solutions: 1