International Mathematical Olympiad · 2004

Problems

Problem 1Let $ABC$ be an acute-angled triangle with $AB\ne AC$ . The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of side $BC$ . The bisectors of angles $\angle BAC$ and $\angle MON$ intersect at $R$ . Prove that the circumcircles of triangles $BMR$ and $CNR$ have a common point lying on side $BC$ .Solutions: 1
Problem 2Find all polynomials $P$ with real coefficients such that for all real numbers $a,b,c$ with $ab+bc+ca=0$ , we have $P(a-b)+P(b-c)+P(c-a)=2P(a+b+c).$ Solutions: 1
Problem 3Define a hook to be a figure made up of six unit squares consisting of a row of three squares and a column of four squares sharing a corner square, together with one further square attached at the far end of the row (or any figure obtained from this one by rotations and reflections). Determine all $m\times n$ rectangles that can be covered without gaps and without overlaps by such hooks, with no part of a hook covering area outside the rectangle.Solutions: 1
Problem 4Let $n\ge 3$ be an integer and $t_1,t_2,\dots,t_n$ positive real numbers such that $n^2+1 > (t_1+t_2+\cdots+t_n)\left(\frac{1}{t_1}+\frac{1}{t_2}+\cdots+\frac{1}{t_n}\right).$ Show that $t_i,t_j,t_k$ are the lengths of the sides of a triangle for all $i,j,k$ with $1\le i<j<k\le n$ .Solutions: 1
Problem 5In a convex quadrilateral $ABCD$ , the diagonal $BD$ bisects neither the angle $\angle ABC$ nor the angle $\angle CDA$ . The point $P$ lies inside $ABCD$ and satisfies $\angle PBC=\angle DBA\quad\text{and}\quad\angle PDC=\angle BDA.$ Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$ .Solutions: 1
Problem 6We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ which have an alternating multiple.Solutions: 1