The Vault: Robbers

Here I present one of my favourites; a problem of rare invention. From the 1998 Tournament of the Towns:

(Two-person case) Two robbers stole a bag of coins from a merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the two robbers (that is, they both get coins with the same total value in pennies).

Prove that after one coin is removed, the number of coins remaining is even.

(General case) A gang of robbers stole a bag of coins from a merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (that is, they all get coins with the same total value in pennies).

Prove that after one coin is removed, the number of coins remaining is divisible by the number of robbers.

Solution

The Mathlon

From the 1986 Canadian Maths Olympiad:

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?

Solution