Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.
Let be a quadrilateral . Let and be two segments perpendicular to the diagonal and such that the distance between them is , with , and . Show that the perimeter of hexagon does not depend on the position of and so long as the distance between them remains constant.
Let and be positive integers such that . Prove that
Let be four points on a circle, and let be the incentre of the triangle ; be the incentre of the triangle be the incentre of the triangle be the incentre of the triangle . Prove that are the vertices of a rectangle.
The National Marriage Council wishes to invite couples to form discussion groups under the following conditions:
1. All members of a group must be of the same sex; i.e. they are either all male or all female.
2. The difference in the size of any two groups is 0 or .
3. All groups have at least member.
4. Each person must belong to one and only one group.
Find all values of , for which this is possible. Justify your answer.
Let be the lengths of the sides of a triangle. Prove that
and determine when equality occurs.