2006 Canadian MO Problems

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Problem 1

Let $f(n,k)$ be the number of ways distributing $k$ candies to $n$ children so that each child receives at most two candies. For example, $f(3,7)=0$, $f(3,6)=1$, and $f(3,4)=6$. Evaluate $f(2006,1)+f(2006,4)+f(2006,7)+\dots+f(2006,1003)$.


Problem 2

Let $ABC$ be an acute angled triangle. Inscribe a rectangle $DEFG$ in this triangle so that $D$ is on $AB$, $E$ on $AC$, and $F$ and $G$ on $BC$. Describe the locus of the intersections of the diagonals of all possible rectangles $DEFG$.


Problem 3

In a rectangular array of nonnegative real numbers with $m$ rows and $n$ 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 $m = n$.


Problem 4

Consider a round robin tournament with $2n+1$ teams, where two teams play exactly one match and there are no ties. We say that the teams $X$, $Y$, and $Z$ form a cycle triplet if $X$ beats $Y$, $Y$ beats $Z$, and $Z$ beats $X$.

(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 $ABC$ inscribed in a circle divide the three arcs, we draw a tangent intercepted by the lines $AB$ and $AC$. If the tangency points are $D$, $E$, and $F$, show that the triangle $DEF$ is equilateral.