14th British Mathematical Olympiad 1978 Problems

14th British Mathematical Olympiad 1978 Problems

1.  Find the point inside a triangle which has the largest product of the distances to the three sides.

2.  Show that there is no rational number m/n with 0 < m < n < 101 whose decimal expansion has the consecutive digits 1, 6, 7 (in that order).

3.  Show that there is a unique sequence a₁, a₂, a₃, ... such that a₁ = 1, a₂ > 1, a_n+1a_n-1 = a_n³ + 1, and all terms are integral.

4.  An altitude of a tetrahedron is a perpendicular from a vertex to the opposite face. Show that the four altitudes are concurrent iff each pair of opposite edges is perpendicular.

5.  There are 11000 points inside a cube side 15. Show that there is a sphere radius 1 which contains at least 6 of the points.

6.  Show that 2 cos nx is a polynomial of degree n in (2 cos x). Hence or otherwise show that if k is rational then cos kπ is 0, ±1/2, ±1 or irrational.