From the 2013 South Africa Olympiad:
A is a 2-digit number and B is a 3-digit number such that A increased by B% equals B reduced by A%. Find all possible pairs (A, B).
The information presented yields the equation
This leads to
Since A is an integer, it follows that 50+B divides 50B. Well, 50+B also divides 50(50+B) = 2500+50B, so in fact 50+B must divide 2500. Remembering that B is a 3-digit number yields just three possible values of B: 200, 450 and 575. Substituting back into the equation above yields corresponding values for A of 40, 45 and 46 respectively. As a final check, we do indeed have
40 increased by 200% = 200 reduced by 40% = 120,
45 increased by 450% = 450 reduced by 45% = 247.5,
46 increased by 575% = 575 reduced by 46% = 310.5.
Therefore the set of possible pairs (A, B) is (40, 200), (45, 450), (46, 575).
A beautiful problem, borne out of a natural question that anyone fluent in percentages might ask themselves.