30th Swedish 1990

1.  Let d1, d2, ... , dk be the positive divisors of n = 1990!. Show that ∑ di/√n = ∑ √n/di.
2.  The points A1, A2, ... , A2n are equally spaced in that order along a straight line with A1A2 = k. P is chosen to minimise ∑ PAi. Find the minimum.
3.  Find all a, b such that sin x + sin a ≥ b cos x for all x.
4.  ABCD is a quadrilateral. The bisectors of ∠A and ∠B meet at E. The line through E parallel to CD meets AD at L and BC at M. Show that LM = AL + BM.
5.  Find all monotonic positive functions f(x) defined on the positive reals such that f(xy) f( f(y)/x) = 1 for all x, y.
6.  Find all positive integers m, n such that 117/158 > m/n > 97/131 and n ≤ 500.

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

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© John Scholes
9 Oct 2003
Last corrected/updated 9 Oct 03