The Second British Mathematical Olympiad
1966年第二届英国奥林匹克数学竞赛 
 Find the greatest and least values of f(x) = (x^{4} + x^{2} + 5)/(x^{2} + 1)^{2} for real x .
 For which distinct, real a, b, c are all the roots of ±√(x  a) ±√(x  b) ±√(x  c) = 0 real ?
 Sketch y^{2} = x^{2}(x + 1)/(x  1). Find all stationary values and describe the behaviour for large x .
 A_{1}, A_{2}, A_{3}, A_{4} are consecutive vertices of a regular ngon. 1/A_{1}A_{2} = 1/A_{1}A_{3} + 1/A_{1}A_{4}. What are the possible values of n ?
 A spanner has an enclosed hole which is a regular hexagon side 1. For what values of s can it turn a square nut side s ?
 Find the largest interval over which f(x) = √(x  1) + √(x + 24  10√(x  1) ) is real and constant .
 Prove that √2, √3 and √5 cannot be terms in an arithmetic progression .
 Given 6 different colours, how many ways can we colour a cube so that each face has a different colour? Show that given 8 different colours, we can colour a regular octahedron in 1680 ways so that each face has a different colour .
 The angles of a triangle are A, B, C. Find the smallest possible value of tan A/2 + tan B/2 + tan C/2 and the largest possible value of tan A/2 tan B/2 tan C/2 .
 One hundred people of different heights are arranged in a 10 x 10 array. X, the shortest of the 10 people who are the tallest in their row, is a different height from Y, the tallest of the 10 people who are the shortest in their column. Is X taller or shorter than Y ?
 (a) Show that given any 52 integers we can always find two whose sum or difference is a multiple of 100.
(b) Show that given any set 100 integers, we can find a nonempty subset whose sum is a multiple of 100.

点击此处查看相关视频讲解 

