## 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.