|The 36th Canadian Mathematical Olympiads
- Find all ordered triples (x, y, z) of real numbers which satisfy the following system of equations:
xy = z - x - y
xz = y - x - z
yz = x - y - z
- How many ways can 8 mutually non-attacking rooks be placed on a 9 × 9 chessboard so that all 8 rooks are on squares of the same colour ?
[Two squares are said to be attacking each other if they are placed in the same row or column of the board.]
- Let A,B,C,D be four points on a circle (occurring in clockwise order), with AB < AD and BC > CD.
Let the bisector of angle BAD meet the circle at X and the bisector of angle BCD meet the circle at Y . Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that BD must be a diameter of the circle.
- Let p be an odd prime. Prove that
∑k from 1 to (p - 1) ( k^(2p+1)) ≡ p(p+1)/2 (mod p^2)
[Note that a ≡ b (mod m) means that a - b is divisible by m.]
- Let T be the set of all positive integer divisors of 2004^100. What is the largest possible number of elements that a subset S of T can have if no element of S is an integer multiple of any other element of S ?