International Mathematics Tournament of Towns

For general information about the Tournament click here.

Some past Problems

• On the island of Camelot live 13 grey, 15 brown and 17 crimson chameleons. If two chameleons of different colours meet they both simultaneously change colour to the third colour (e.g. if a grey and a brown chameleon meet each other they both change to crimson). Is it possible that they will eventually all be the same colour? (1984)

• Prove that among 18 consecutive three digit numbers there must be at least one which is divisible by the sum of its digits. (1984)

• A 7 by 7 square is made up of 16 (1 by 3) tiles and 1 (1 by 1) tile. Prove that the 1 by 1 tile lies either at the centre of the square or adjoins one of its boundaries. (1984)

• A machine gives out five pennies for each nickel inserted into it. The machine also gives out five nickels for each penny. Can Peter, who starts out with one penny, use the machine in such a way as to end up with an equal number of nickels and pennies? (1987)

• We are given 101 rectangles with sides of integer lengths not exceeding 100. Prove that among these 101 rectangles there are 3 rectangles, say A, B and C, such that A will fit inside B and B inside C. (1989)

• A regular hexagon is divided internally into parallelograms of equal area. Prove that the number of these parallelograms is divisible by three. (1989)

• Points M and N are taken on the hypotenuse of a right triangle ABC so that BC=BM and AC=AN. Prove that the angle MCN is equal to 45 degrees. (1993)

• Consider an arbitrary "figure" F (non convex polygon). A chord of F is defined to be a segment which lies entirely within F and whose ends are on the boundary.
  1. Does there exist a chord of F that divides its area in half?
  2. Prove that for any F there exists a chord such that the area of each of the two parts of F is not less than 1/3 of the area of F.
  3. Can the number 1/3 in (b) be changed to a greater one?
  (1994)

• Prove that the number 40..09 (with at least one zero) is not a perfect square. (1995)

• Can one paint four points in the plane red and another four points black so that any three points of the same colour are vertices of a parallelogram whose fourth vertex is a point of the other colour? (1996)

• Several strips and a circle of radius 1 are drawn on the plane. The sum of the widths of the strips is 100. Prove that one can translate each strip parallel to itself so that together they cover the circle. (1997)

Books containing past problems and solutions may be obtained from the Australian Mathematics Trust and going to its bookshop.