If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0
what can you say about the triangle?
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
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
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 Madar from Allerton High School 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 Wong 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
Now suppose n is an odd number greater than 3. Chen Jinquan 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."