

1. Let d_{1}, d_{2}, ... , d_{k} be the positive divisors of n = 1990!. Show that ∑ d_{i}/√n = ∑ √n/d_{i}.


2. The points A_{1}, A_{2}, ... , A_{2n} are equally spaced in that order along a straight line with A_{1}A_{2} = 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.

