From the 2003 Tournament of the Towns:

*Each term in a sequence of positive integers is obtained from the previous term by adding it to its largest digit. What is the greatest possible number of successive odd terms in such a sequence?*

From the 2003 Tournament of the Towns:

*Each term in a sequence of positive integers is obtained from the previous term by adding it to its largest digit. What is the greatest possible number of successive odd terms in such a sequence?*

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.*

It’s here! My new book of puzzles Elastic Numbers is now available on Amazon.

https://www.amazon.co.uk/Elastic-Numbers-Puzzles-Serious-Problem/dp/0995782601

https://www.amazon.com/gp/aw/d/0995782601

I hope you enjoy solving the problems as much as I enjoyed creating them.

Happy New Year! From the 2004 Georg Mohr Contest:

*Find all positive integers n such that a 2n x 2n chessboard can be covered by non-overlapping L-pieces, each covering 4 squares. Rotations and reflections are allowed.*

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?*

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?*

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.*