35th Canadian Mathematical Olympiad Problems 2003

35th Canadian Mathematical Olympiad Problems 20031.  The angle between the hour and minute hands of a standard 12-hour clock is exactly 1o. The time is an integral number n of minutes after noon (where 0 < n < 720). Find the possible values of n. 2.  What are the last three digits of 2003N, where N = 20022001. 3.  Show that if n > 1 is an integer then (1 + 1/3 + 1/5

29th Canadian Mathematical Olympiad Problems 1997

29th Canadian Mathematical Olympiad Problems 19971.  How many pairs of positive integers have greatest common divisor 5! and least common multiple 50! ? 2.  A finite number of closed intervals of length 1 cover the interval [0, 50]. Show that we can find a subset of at least 25 intervals with every pair disjoint. 3. 

26th Canadian Mathematical Olympiad Problems 1994

26th Canadian Mathematical Olympiad Problems 19941.  Find -3/1! + 7/2! - 13/3! + 21/4! - 31/5! + ... + (19942 + 1994 + 1)/1994! 2.  Show that every power of (√2 - 1) can be written in the form √(k+1) - √k. 3.

25th Canadian Mathematical Olympiad Problems 1993

25th Canadian Mathematical Olympiad Problems 19931.  Show that there is a unique triangle such that (1) the sides and an altitude have lengths with are 4 consecutive integers, and (2) the foot of the altitude is an integral distance from each vertex. 2.  Show that the real number k is rational iff the sequence k, k + 1, k + 2, k + 3, ... contains three (distinct)

24th Canadian Mathematical Olympiad Problems 1992

24th Canadian Mathematical Olympiad Problems 19921.  Show that n! is divisible by (1 + 2 + ... + n) iff n+1 is not an odd prime. 2.  Show that x(x - z)2 + y(y - z)2 ≥ (x - z)(y - z)(x + y - z) for all non-negative reals x, y, z. When does equality hold? 2.  Three

17th Canadian Mathematical Olympiad Problems 1985

17th Canadian Mathematical Olympiad Problems 19851.  A triangle has sides 6, 8, 10. Show that there is a unique line which bisects the area and the perimeter. 2.  Is there an integer which is doubled by moving its first digit to the end? [For example, 241 does not work because 412 is not 2 x 241.]