OMCC 2002

For what integers $n\ge 3$ is it possible to accommodate, in some order, the numbers $1,2,\cdots, n$ in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?

Let $ABC$ be an acute triangle, and let $D$ and $E$ be the feet of the altitudes drawn from vertexes $A$ and $B$ , respectively. Show that if, $$Area[BDE]\le Area[DEA]\le Area[EAB]\le Area[ABD]$$ then, the triangle is isosceles.

For every integer $a>1$ an infinite list of integers is constructed $L(a)$ , as follows:

[list] $a$ is the first number in the list $L(a)$ .[/list] [list] Given a number $b$ in $L(a)$ , the next number in the list is $b+c$ , where $c$ is the largest integer that divides $b$ and is smaller than $b$ .[/list] Find all the integers $a>1$ such that $2002$ is in the list $L(a)$ .

Let $ABC$ be a triangle, $D$ be the midpoint of $BC$ , $E$ be a point on segment $AC$ such that $BE=2AD$ and $F$ is the intersection point of $AD$ with $BE$ . If $\angle DAC=60^{\circ}$ , find the measure of the angle $FEA$ .

Find a set of infinite positive integers $S$ such that for every $n\ge 1$ and whichever $n$ distinct elements $x_1,x_2,\cdots, x_n$ of S, the number $x_1+x_2+\cdots +x_n$ is not a perfect square.

A path from $(0,0)$ to $(n,n)$ on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its $2n+1$ vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices $(0,0)$ , $(n,0)$ , $(n,n)$ , $(0,n)$ into two parts with equal area.