### 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:

There are n people in a field and no two pairs of people are at the same distance apart. Everyone has a water pistol and shoots at and hits the nearest person to them. Show that if n is even everyone may get wet but if n is odd we can be sure that there will be someone who does not get wet.

Here are the solutions to the different parts of the problem from three different people.

Roy M. proved that for an even number of people there must be some circumstances where they all get wet. "For the first problem it is possible that everyone is placed in pairs so that around the field each person shoots, and gets shot by, their pairs. Since there is an even number of people this will work and is therefore a possibility."

Gary from Winchester College proved that when there are three people one stays dry: "Let them be X, Y and Z with XY the shortest distance, thus X and Y each get shot by the other, we are left with 1 shot and 1 person Z, as he cannot shoot himself he has to shoot the one who is nearest to him...whoever that is, and Z is left unshot!"

Now suppose n is an odd number greater than 3. Chen of The Chinese High School uses an argument that reduces the number under consideration from n to n-2 which is the basis for a proof of this result by mathematical induction. "Since the distance between any two people is unique, we can consider the shortest length between 2 people. Then, these two people necessarily spray water to each other, i.e. they are both wet. Now, consider the remaining (n-2) people. If anyone of the (n-2) people spray water at the first 2 people, then there would be at least one person who is not wet. Hence, we consider the second shortest length among these n people (and this line does not connect to the 2 people who are previously chosen) and the two people which the length connects also spray water at each other. By continuing this sequence of actions till there are 3 people remaining, we see that it is impossible for everyone to be wet when n is odd, as the case for n=3 has already been proved to be impossible."