2005 Canadian MO Problems
Consider an equilateral triangle of side length , which is divided into unit triangles, as shown. Let 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 . Determine the value of .
Let be a Pythagorean triple, i.e., a triplet of positive integers with .
- Prove that .
- Prove that there are no integer and Pythagorean triple satisfying .
Let be a set of points in the interior of a circle.
- Show that there are three distinct points and three distinct points on the circle such that is (strictly) closer to than any other point in , is closer to than any other point in and is closer to than any other point in .
- Show that for no value of can four such points in (and corresponding points on the circle) be guaranteed.
Let be a triangle with circumradius , perimeter and area . Determine the maximum value of .
Let's say that an ordered triple of positive integers is -powerful if , , and is divisible by . For example, is 5-powerful.
- Determine all ordered triples (if any) which are -powerful for all .
- Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.