9th Swedish 1969


1. Find all integers m, n such that m³ = n³ + n.
2. Show that tan π/3n is irrational for all positive integers n.
3. a₁ ≥ a₂ ≥ ... ≥ a_n is a sequence of reals. b₁, b₂, b₃, ... b_n is any rearrangement of the sequence B₁ ≥ B₂ ≥ ... ≥ B_n. Show that ∑ a_ib_i ≤ &sum a_iB_i.
4. Define g(x) as the largest value of \|y² - xy\| for y in [0, 1]. Find the minimum value of g (for real x).
5. Let N = a₁a₂ ... a_n in binary. Show that if a₁ - a₂ + a₃ - ... + (-1)^n-1a_n = 0 mod 3, then N = 0 mod 3.
6. Given 3n points in the plane, no three collinear, is it always possible to form n triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?

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