OMCPLP 2013

If Xiluva puts two oranges in each basket, four oranges are in excess. If she puts five oranges in each basket, one basket is in excess. How many oranges and baskets has Xiluva?

Let $ABC$ be an acute triangle. The circumference with diameter $AB$ intersects sides $AC$ and $BC$ at $E$ and $F$ respectively. The tangent lines to the circumference at the points $E$ and $F$ meet at $P$ . Show that $P$ belongs to the altitude from $C$ of triangle $ABC$ .

An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$ , $1986$ , $1996$ , $2008$ and it didn't occur in $1976$ , $1993$ , $2006$ , $2013$ . What is the first year in that the event will occur again?

Find all the pairs $(x,y)$ of positive integers that satisfy the equation $x^2-xy+2x-3y=2013$ .

Find all the numbers of $5$ non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.

Consider a triangle $ABC$ . Let $S$ be a circumference in the interior of the triangle that is tangent to the sides $BC$ , $CA$ , $AB$ at the points $D$ , $E$ , $F$ respectively. In the exterior of the triangle we draw three circumferences $S_A$ , $S_B$ , $S_C$ . The circumference $S_A$ is tangent to $BC$ at $L$ and to the prolongation of the lines $AB$ , $AC$ at the points $M$ , $N$ respectively. The circumference $S_B$ is tangent to $AC$ at $E$ and to the prolongation of the line $BC$ at $P$ . The circumference $S_C$ is tangent to $AB$ at $F$ and to the prolongation of the line $BC$ at $Q$ . Show that the lines $EP$ , $FQ$ and $AL$ meet at a point of the circumference $S$ .