30th Swedish 1990


1. Let d₁, d₂, ... , d_k be the positive divisors of n = 1990!. Show that ∑ d_i/√n = ∑ √n/d_i.
2. The points A₁, A₂, ... , A_2n are equally spaced in that order along a straight line with A₁A₂ = k. P is chosen to minimise ∑ PA_i. 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.