International Mathematical Olympiad · 1993

Problems

Problem 1Let $f(x)=x^n+5x^{n-1}+3$ , where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two non-constant polynomials with integer coefficients.Solutions: 2
Problem 2Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB=\angle ACB+\frac{\pi}{2}$ and $AC\cdot BD=AD\cdot BC$ . (a) Calculate the ratio $\frac{AB\cdot CD}{AC\cdot BD}$ . (b) Prove that the tangents at $C$ to the circumcircles of $\triangle ACD$ and $\triangle BCD$ are perpendicular.Solutions: 3
Problem 3On an infinite chessboard, start with $n^2$ pieces in an $n$ by $n$ block, one per square. A move jumps horizontally or vertically over an adjacent occupied square to the unoccupied square immediately beyond, removing the jumped piece. Find those values of $n$ for which the game can end with only one piece remaining.Solutions: 1
Problem 4For three points $A,B,C$ in the plane, let $m(ABC)$ be the smallest length of the three heights of triangle $ABC$ , and set $m(ABC)=0$ when the points are collinear. Given $A,B,C$ , prove that for every point $X$ in the plane, $m(ABC)\le m(ABX)+m(AXC)+m(XBC)$ .Solutions: 1
Problem 5Let $\mathbb{N}=\{1,2,3,\ldots\}$ . Determine whether there exists a strictly increasing function $f:\mathbb{N}\mapsto\mathbb{N}$ such that (i) $f(1)=2$ ; (ii) $f(f(n))=f(n)+n$ , $(n\in\mathbb{N})$ .Solutions: 1
Problem 6There are $n$ lamps $L_0,\ldots,L_{n-1}$ in a circle, where $n>1$ and $L_{n+k}=L_k$ . At step $s_i$ , if $L_{i-1}$ is lit, switch $L_i$ , otherwise do nothing. Initially all lamps are on. Show that (a) there is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are on again; (b) if $n=2^k$ , take $M(n)=n^2-1$ ; (c) if $n=2^k+1$ , take $M(n)=n^2-n+1$ .Solutions: 1