2005 Canadian MO Problems

Problem 1

Consider an equilateral triangle of side length $n$ , which is divided into unit triangles, as shown. Let $f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in our path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example of one such path is illustrated below for $n=5$ . Determine the value of $f(2005)$ .

Solution

Problem 2

Let $(a,b,c)$ be a Pythagorean triple, i.e., a triplet of positive integers with $a^2+b^2=c^2$ .

Prove that $(c/a + c/b)^2 > 8$ .
Prove that there does not exist any integer $n$ for which we can find a Pythagorean triple $(a,b,c)$ satisfying $(c/a + c/b)^2 = n$ .

Solution

Problem 3

Let $S$ be a set of $n\ge 3$ points in the interior of a circle.

Show that there are three distinct points $a,b,c\in S$ and three distinct points $A,B,C$ on the circle such that $a$ is (strictly) closer to $A$ than any other point in $S$ , $b$ is closer to $B$ than any other point in $S$ and $c$ is closer to $C$ than any other point in $S$ .
Show that for no value of $n$ can four such points in $S$ (and corresponding points on the circle) be guaranteed.

Solution

Problem 4

Let $ABC$ be a triangle with circumradius $R$ , perimeter $P$ and area $K$ . Determine the maximum value of $KP/R^3$ . Solution

Problem 5

Let's say that an ordered triple of positive integers $(a,b,c)$ is $n$ -\emph{powerful} if $a \le b \le c$ , $\gcd(a,b,c) = 1$ , and $a^n + b^n + c^n$ is divisible by $a+b+c$ . For example, $(1,2,2)$ is 5-powerful.

Determine all ordered triples (if any) which are $n$ -powerful for all $n \ge 1$ .
Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.

Solution

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2005 Canadian MO

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2005 Canadian MO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Resources