15th Swedish 1975


1. A is the point (1, 0), L is the line y = kx (where k > 0). For which points P (t, 0) can we find a point Q on L such that AQ and QP are perpendicular?
2. Is there a positive integer n such that the fractional part of (3 + √5)ⁿ > 0.99?
3. Show that aⁿ + bⁿ + cⁿ ≥ ab^n-1 + bc^n-1 + ca^n-1 for real a, b, c ≥ 0 and n a positive integer.
4. P₁, P₂, P₃, Q₁, Q₂, Q₃ are distinct points in the plane. The distances P₁Q₁, P₂Q₂, P₃Q₃ are equal. P₁P₂ and Q₂Q₁ are parallel (not antiparallel), similarly P₁P₃ and Q₃Q₁, and P₂P₃ and Q₃Q₂. Show that P₁Q₁, P₂Q₂ and P₃Q₃ intersect in a point.
5. Show that n divides 2ⁿ + 1 for infinitely many positive integers n.
6. f(x) is defined for 0 ≤ x ≤ 1 and has a continuous derivative satisfying \|f '(x)\| ≤ C \|f(x)\| for some positive constant C. Show that if f(0) = 0, then f(x) = 0 for the entire interval.

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