12th CanMO 1980


1. If the 5-digit decimal number a679b is a multiple of 72 find a and b.
2. The numbers 1 to 50 are arranged in an arbitrary manner into 5 rows of 10 numbers each. Then each row is rearranged so that it is in increasing order. Then each column is arranged so that it is in increasing order. Are the rows necessarily still in increasing order?
3. Find the triangle with given angle A and given inradius r with the smallest perimeter.
4. A fair coin is tossed repeatedly. At each toss 1 is scored for a head and 2 for a tail. Show that the probability that at some point the score is n is (2 + (-1/2)ⁿ)/3.
5. Do any polyhedra other than parallelepipeds have the property that all cross sections parallel to any given face have the same perimeter?

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