21st CanMO 1989


1. How many permutations of 1, 2, 3, ... , n have each number larger than all the preceding numbers or smaller than all the preceding numbers?
2. Each vertex of a right angle triangle of area 1 is reflected in the opposite side. What is the area of the triangle formed by the three reflected points?
3. Tranform a number by taking the sum of its digits. Start with 1989¹⁹⁸⁹ and make four transformations. What is the result?
4. There are five ladders. There are also some ropes. Each rope attaches a rung of one ladder to a rung of another ladder. No ladder has two ropes attached to the same rung. A monkey starts at the bottom of each ladder and climbs. Each time it reaches a rope, it traverses the rope to the other ladder and continues climbing up the other ladder. Show that each monkey eventually reaches the top of a different ladder.
5. For every permutation a₁, a₂, ... , a_n of 1, 2, 4, 8, ... , 2^n-1 form the product of all n partial sums a₁ + a₂ + ... + a_k. Find the sum of the inverses of all these products.

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