Time allowed: 4 hours
No calculators are to be used
Each question is worth 7 points
Let be real numbers such that the polynomial
factorises into eight linear factors , with for . Determine all possible values of .
Suppose is a square piece of cardboard with side length . On a plane are two parallel lines and , which are also a units apart. The square is placed on the plane so that sides and intersect at and respectively. Also, sides and intersect at and respectively. Let the perimeters of and be and respectively. Prove that no matter how the square was placed, remains constant.
Let be an integer, and let be the largest prime number which is strictly less than . You may assume that . Let be a composite integer. Prove:
(a) if , then does not divide ;
(b) if , then divides .
Let be the sides of a triangle, with , and let be an integer. Show that
Given two positive integers and , find the smallest positive integer such that among any people, either there are of them who form pairs of mutually acquainted people or there are of them forming pairs of mutually unacquainted people.