## % Change

From the 2013 South Africa Olympiad:

A is a 2-digit number and B is a 3-digit number such that A increased by B% equals B reduced by A%. Find all possible pairs (A, B).

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## Two Taps

The scenario is my own, but the idea is a classic:

A bath takes 20 minutes to fill with just the cold tap, or 30 minutes to fill with just the hot tap. How long would it take to fill the bath with both taps together?

## The answer in the problem

A PuzzleCritic Original:

Let x be the answer to this problem, where x is a positive integer, and let y be the sum of its digits. Calculate 2x-2y.

## Numbers on a board

Here is a gentle thinker from the 2014 Grey Kangaroo:

Dean’s teacher asks him to write several different positive integers on the board. Exactly two of them are to be divisible by 2 and exactly 13 of them are to be divisible by 13. M is the greatest of these numbers.

What is the least possible value of M?

## A school night out

From the 2004 Turkey Junior Maths Olympiad:

One evening, more than $\frac{1}{3}$ of the studentsĀ at a school go to the cinema. On the same evening, more than $\frac{3}{10}$ go to the theatre and more than $\frac{4}{11}$ go to a concert. What is the smallest possible number of students at the school?

## William and his words

I had a very enjoyable bank holiday weekend, one highlight of which was finally cracking a problem that had caught my eye months ago but had been left unattempted until the aforementioned weekend. For now I will simply post the problem statement – it is one of the strangest puzzles I have ever come across and is definitely worth thinking about before looking at a solution.

From the 2007 Tournament of the Towns:

For each letter in the English alphabet, William assigns an English word which contains that letter. His first document consists only of the word assigned to the letter A. In each subsequent document, he replaces each letter of the preceding document by its assigned word. The fortieth document begins with “Till whatsoever star that guides my moving.” Prove that this sentence reappears later in this document.