2015 UNSW School Mathematics Competition - Problems and Solutions

Junior Division - Problems and Solutions
Solutions by Denis Potapov, UNSW Australia.

Problem 1

Every point on a line is painted using two different colours: black and white.
Prove that there are always points $A_1$, $A_2$ and $A_3$ of the same colour such that
  A_1A_2 = A_2A_3\,.