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.

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© John Scholes
26 September 2003
Last corrected/updated 30 Dec 03