14th British Mathematical Olympiad 1978 Problems1.  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).

13th British Mathematical Olympiad 1977 Problems1.  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.

12th British Mathematical Olympiad 1976 Problems1.  ABC is a triangle area k. Let d be the length of the shortest line segment which bisects the area of the triangle. Find d. Give an example of a curve which bisects the area and has length < d.

11th British Mathematical Olympiad 1975 Problems1.  Find all positive integer solutions to [11/3] + [21/3] + ... + [(n3 - 1)1/3] = 400. 2.  The first k primes are divided into two groups. n is the product of the first group and n is the product of the second group. M is any positive integer divisible only by primes in the first group and N is any positive

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10th British Mathematical Olympiad 1974 Problems1.  C is the curve y = 4x2/3 for x ≥ 0 and C' is the curve y = 3x2/8 for x ≥ 0. Find curve C" which lies between them such that for each point P on C" the area bounded by C, C" and a horizontal line through P equals the area bounded by C", C and a vertical line through P.

9th British Mathematical Olympiad 1973 Problems1.  A variable circle touches two fixed circles at P and Q. Show that the line PQ passes through one of two fixed points. State a generalisation to ellipses or conics. 2. 

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8th British Mathematical Olympiad 1972 Problems

8th British Mathematical Olympiad 1972 Problems 1.  The relation R is defined on the set X. It has the following two properties: if aRb and bRc then cRa for distinct elements a, b, c; for distinct elements a, b either aRb or bRa but not both. What is the largest possible number of elements in X?