# 2006 Canadian MO Problems

## Problem 1

Let be the number of ways distributing candies to children so that each child receives at most two candies. For example, , , and . Evaluate .

## Problem 2

Let be an acute angled triangle. Inscribe a rectangle in this triangle so that is on , on , and and on . Describe the locus of the intersections of the diagonals of all possible rectangles .

## Problem 3

In a rectangular array of nonnegative real numbers with rows and columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that .

## Problem 4

Consider a round robin tournament with teams, where two teams play exactly one match and there are no ties. We say that the teams , , and form a *cycle triplet* if beats , beats , and beats .

(a) Find the minimum number of cycle triplets possible.

(b) Find the maximum number of cycle triplets possible.

## Problem 5

The vertices of right triangle inscribed in a circle divide the three arcs, we draw a tangent intercepted by the lines and . If the tangency points are , , and , show that the triangle is equilateral.