31st Swedish 1991


1. Find all positive integers m, n such that 1/m + 1/n - 1/(mn) = 2/5.
2. x, y are positive reals such that x - √x ≤ y - 1/4 ≤ x + √x. Show that y - √y ≤ x - 1/4 ≤ y + √y.
3. The sequence x₀, x₁, x₂, ... is defined by x₀ = 0, x_k+1 = [(n - ∑₀^k x_i)/2]. Show that x_k = 0 for all sufficiently large k and that the sum of the non-zero terms x_k is n-1.
4. x₁, x₂, ... , x₈ is a permutation of 1, 2, ... , 8. A move is to take x₃ or x₈ and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at 1, 2, ... , 8.
5. Show that there are infinitely many odd positive integers n such that in binary n has more 1s than n².
6. Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

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