9th Swedish 1969

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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 ≥ ... ≥ an is a sequence of reals. b1, b2, b3, ... bn is any rearrangement of the sequence B1 ≥ B2 ≥ ... ≥ Bn. Show that ∑ aibi ≤ &sum aiBi.
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 ... an in binary. Show that if a1 - a2 + a3 - ... + (-1)n-1an = 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.

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© John Scholes
jscholes@kalva.demon.co.uk
23 September 2003
Last corrected/updated 23 Sep 03