10th CanMO 1978


1. A square has tens digit 7. What is the units digit?
2. Find all positive integers m, n such that 2m² = 3n³.
3. Find the real solution x, y, z to x + y + z = 5, xy + yz + zx = 3 with the largest z.
4. ABCD is a convex quadrilateral with area 1. The lines AD, BC meet at X. The midpoints of the diagonals AC and BD are Y and Z. Find the area of the triangle XYZ.
5. Two players play a game on an initially empty 3 x 3 board. Each player in turn places a black or white piece on an unoccupied square of the board. Each player may play either color. When the board is full player A gets one point for every row, column or main diagonal with 0 or 2 black pieces on it. Player B gets one point for every row, column or main diagonal with 1 or 3 black pieces on it. Can the game end in a draw? Which player has a winning strategy if player A plays first? If player B plays first?
6. Sketch the graph of x³ + xy + y³ = 3.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.