31st Swedish 1991

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1.  Find all positive integers m, n such that 1/m + 1/n - 1/(mn) = 2/5.
2.  x, y are positive reals such that x - √x ≤ y - 1/4 ≤ x + √x. Show that y - √y ≤ x - 1/4 ≤ y + √y.
3.  The sequence x0, x1, x2, ... is defined by x0 = 0, xk+1 = [(n - ∑0k xi)/2]. Show that xk = 0 for all sufficiently large k and that the sum of the non-zero terms xk is n-1.
4.  x1, x2, ... , x8 is a permutation of 1, 2, ... , 8. A move is to take x3 or x8 and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at 1, 2, ... , 8.
5.  Show that there are infinitely many odd positive integers n such that in binary n has more 1s than n2.
6.  Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original 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
9 Oct 2003
Last corrected/updated 9 Oct 03