15th British Mathematical Olympiad 1979 Problems

15th British Mathematical Olympiad 1979 Problems

1.  Find all triangles ABC such that AB + AC = 2 and AD + BD = √5, where AD is the altitude.

2.  Three rays in space have endpoints at O. The angles between the pairs are α, β, γ, where 0 < α < β < γ. Show that there are unique points A, B, C, one on each ray, so that the triangles OAB, OBC, OCA all have perimeter 2s. Find their distances from O.

3.  Show that the sum of any n distinct positive odd integers whose pairs all have different differences is at least n(n² + 2)/3.

4.  f(x) is defined on the rationals and takes rational values. f(x + f(y) ) = f(x) f(y) for all x, y. Show that f must be constant.

5.  Let p(n) be the number of partitions of n. For example, p(4) = 5: 1 + 1 + 1 + 1, 1 + 1 + 2, 2 + 2, 1 + 3, 4. Show that p(n+1) ≥ 2p(n) - p(n-1).

6.  Show that the number 1 + 10⁴ + 10⁸ + ... + 10⁴ⁿ is not prime for n > 0.