## Shares

From the 2004 Tournament of Towns:

Each day, the price of the shares of the corporation “Soap Bubble, Limited” either increases or decreases by n%, where n is an integer such that 0<n<100. The price is calculated with unlimited precision. Does there exist an n for which the price can take the same value twice?

## The Mathlon

A, B and C are the participants in a Mathlon – an athletics competition made up of several events. In each event x points are awarded for finishing 1st, y points for finishing 2nd, and z points for finishing 3rd, where x>y>z are positive integers. In the end, A scored 22 points, and B and C each scored 9 points. B won the 100 metres. How many events were there in total, and who finished second in the high jump?

## Queuing

From the 2011 South African Junior Olympiad:

Several people line up in single file. A solitary latecomer wishes to join the queue. Prove that it is always possible for them to join the line so that the number of men in front of them is equal to the number of women behind them.

## Binary strings

From the 2010 Australian Maths Challenge:

There are sixteen different ways of writing four-digit strings using 1s and 0s. Three of these strings are 1010, 0100 and 1001. These three can be found as substrings of 101001. There is a string of nineteen 1s and 0s which contains all sixteen strings of length 4 exactly once. If this string starts with 1111, what are the last four digits?

## 42 Points

From the 2007 Australian Maths Competition:

There are 42 points $P_1, P_2, ..., P_{42}$ , placed in order on a straight line so that each distance from $P_i$ to $P_{i+1}$ is  $\dfrac{1}{i}$  for $1\leq i\leq 41$. What is the sum of the distances between every pair of these points?

## Rugs

From the 1998 University of Waterloo Fermat Contest:

Three rugs have a combined area of 200 sq m. By overlapping rugs to cover a floor of area 140 sq m, the area which is covered by exactly two layers of rug is 24 sq m. What area of floor is covered by three layers of rug?

## The Vault: My favourite puzzle

Having spent nine days in Hamburg at a maths conference, I return to present what is to date my favourite ever maths problem. From the Tournament of the Towns:

Let n be a positive integer. Consider the largest odd factor of each of the numbers n+1, n+2, …, 2n. Prove that their sum is n2.