13th British Mathematical Olympiad 1977 Problems

13th British Mathematical Olympiad 1977 Problems

1.  f(n) is a function on the positive integers with non-negative integer values such that: (1) f(mn) = f(m) + f(n) for all m, n; (2) f(n) = 0 if the last digit of n is 3; (3) f(10) = 0. Show that f(n) = 0 for all n.

2.  S is either the incircle or one of the excircles of the triangle ABC. It touches the line BC at X. M is the midpoint of BC and N is the midpoint of AX. Show that the center of S lies on the line MN.

3.  (1) Show that x(x - y)(x - z) + y(y - z)(y - x) + z(z - x)(z - y) ≥ 0 for any non-negative reals x, y, z.

(2) Hence or otherwise show that x⁶ + y⁶ + z⁶ + 3x²y²z² ≥ 2(y³z³ + z³x³ + x³y³) for all real x, y, z.

4.  x³ + qx + r = 0, where r is non-zero, has roots u, v, w. Find the roots of r²x³ + q³x + q³ = 0 (*) in terms of u, v, w. Show that if u, v, w are all real, then (*) has no real root x satisfying -1 < x < 3.

5.  Five spheres radius a all touch externally two spheres S and S' of radius a. We can find five points, one on each of the first five spheres, which form the vertices of a regular pentagon side 2a. Do the spheres S and S' intersect?

6.  Find all n > 1 for which we can write 26(x + x² + x³ + ... + xⁿ) as a sum of polynomials of degree n, each of which has coefficients which are a permutation of 1, 2, 3, ... , n.