10th Swedish 1970


1. Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers.
2. 6 open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all 6 disks.
3. A polynomial with integer coefficients takes the value 5 at five distinct integers. Show that it does not take the value 9 at any integer.
4. Let p(x) = (x - x₁)(x - x₂)(x - x₃), where x₁, x₂ and x₃ are real. Show that p(x) p''(x) ≤ p'(x)² for all x.
5. A 3 x 1 paper rectangle is folded twice to give a square side 1. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making 6 holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?
6. Show that (n - m)!/m! ≤ (n/2 + 1/2)^n-2m for positive integers m, n with 2m ≤ n.

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