OIM 2004

It is given a 10011001 board divided in 11 squares. We want to amrk m squares in such a way that: 1: if 2 squares are adjacent then one of them is marked. 2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked.

Find the minimun number of squares we most mark.

In the plane are given a circle with center $O$ and radius $r$ and a point $A$ outside the circle. For any point $M$ on the circle, let $N$ be the diametrically opposite point. Find the locus of the circumcenter of triangle $AMN$ when $M$ describes the circle.

Let $n$ and $k$ be positive integers such as either $n$ is odd or both $n$ and $k$ are even. Prove that exists integers $a$ and $b$ such as $GCD(a,n) = GCD(b,n) = 1$ and $k = a + b$

Determine all pairs $(a,b)$ of positive integers, each integer having two decimal digits, such that $100a+b$ and $201a+b$ are both perfect squares.

Given a scalene triangle $ABC$ . Let $A'$ , $B'$ , $C'$ be the points where the internal bisectors of the angles $CAB$ , $ABC$ , $BCA$ meet the sides $BC$ , $CA$ , $AB$ , respectively. Let the line $BC$ meet the perpendicular bisector of $AA'$ at $A''$ . Let the line $CA$ meet the perpendicular bisector of $BB'$ at $B'$ . Let the line $AB$ meet the perpendicular bisector of $CC'$ at $C''$ . Prove that $A''$ , $B''$ and $C''$ are collinear.

Given a set $\mathcal{H}$ of points in the plane, $P$ is called an "intersection point of $\mathcal{H}$ " if distinct points $A,B,C,D$ exist in $\mathcal{H}$ such that lines $AB$ and $CD$ are distinct and intersect in $P$ . Given a finite set $\mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $j\geq0$ , $\mathcal{A}_{j+1}$ is the union of $\mathcal{A}_{j}$ and the intersection points of $\mathcal{A}_{j}$ . Prove that, if the union of all the sets in the sequence is finite, then $\mathcal{A}_{i}=\mathcal{A}_{1}$ for any $i\geq1$ .