16th British Mathematical Olympiad 1980 Problems

1.  Show that there are no solutions to aⁿ + bⁿ = cⁿ, with n > 1 is an integer, and a, b, c are positive integers with a and b not exceeding n.

2.  Find a set of seven consecutive positive integers and a polynomial p(x) of degree 5 with integer coefficients such that p(n) = n for five numbers n in the set including the smallest and largest, and p(n) = 0 for another number in the set.

3.  AB is a diameter of a circle. P, Q are points on the diameter and R, S are points on the same arc AB such that PQRS is a square. C is a point on the same arc such that the triangle ABC has the same area as the square. Show that the incenter I of the triangle ABC lies on one of the sides of the square and on the line joining A or B to R or S.

4.  Find all real a₀ such that the sequence a₀, a₁, a₂, ... defined by a_n+1 = 2ⁿ - 3a_n has a_n+1 > a_n for all n ≥ 0.

5.  A graph has 10 points and no triangles. Show that there are 4 points with no edges between them.