Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Water Pistols

Stage: 5 Challenge Level:

A group of children are in a field and no two pairs of children are at the same distance apart. Each child has a water pistol and shoots the child nearest to them.

For different numbers of children, can you construct configurations in which $0, 1, 2, 3, \cdots$ children get wet? Can you construct configurations in which $0, 1, 2, 3, \cdots$ children stay dry? Is there a largest or smallest number who could get wet in each case? Do any patterns emerge?

It is a hot day and all of the children want to get wet? Can you place them so that this is possible. Prove your results carefully.