29th CanMO 1997


1. 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. Show that 1/44 > (1/2)(3/4)(5/6) ... (1997/1998) > 1/1999.
4. Two opposite sides of a parallelogram subtend supplementary angles at a point inside the parallelogram. Show that the line joining the point to a vertex subtends equal angles at the two adjacent vertices.
5. Find ∑_0≤k≤n (-1)^k nCk /(k³ + 9k² + 26k + 24), where nCk is the binomial coefficient n!/( k! (n-k)! ).

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