Proof
First of all, we choose a point, say A. Then, there are five lines joining A
to the other points. They are AB, AC, AD, AE and AF.
Since we have two different colours, by the Pigeonhole Principle, at least
three lines must be in the same colour, say AB, AC and AD.
We may assume these three lines are red.
Now, we consider the triangle BCD. If one side of triangle BCD is red,
then we have a red triangle, otherwise BCD will a blue triangle.
Therefore, either a red or a blue triangle must exist and it is
impossible to have a draw.
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