2003 JBMO Problems

Problem 1

Let $n$ be a positive integer. A number $A$ consists of $2n$ digits, each of which is 4; and a number $B$ consists of $n$ digits, each of which is 8. Prove that $A+2B+4$ is a perfect square.

Solution

Problem 2

Suppose there are $n$ points in a plane no three of which are collinear with the property that if we label these points as $A_1,A_2,\ldots,A_n$ in any way whatsoever, the broken line $A_1A_2\ldots A_n$ does not intersect itself. Find the maximum value of $n$ .

Solution

Problem 3

Let $D$ , $E$ , $F$ be the midpoints of the arcs $BC$ , $CA$ , $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$ , $B$ , $C$ , respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$ , and let $M$ be the midpoint of the segment $GH$ . Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$ , and let $N$ be the midpoint of the segment $KJ$ .

a) Find the angles of triangle $DMN$ ;

b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$ , then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$ .