Problems

The area of the coloured figure equals \(48\)cm\(^2\). Find the length of the side of the smallest square.

The director of a bank has forgotten the combination to open the safe! He only remembers the first \(8\) out of \(10\) digits, and that the whole number was divisible by \(45\). Help him out and find all possible pairs of digits which could complete the combination. \[20242025**\]

There are two hourglasses, a small one and a large one. The sand in the small one takes \(7\) minutes to fall all the way through, and the sand in the large one takes \(11\) minutes. How can you measure \(15\) minutes using those?

Multiply an odd number by the two numbers either side of it. Prove that the final product is divisible by \(24\).

Mattia is thinking of a big positive integer. He tells you what this number to the power of \(4\) is. Unfortunately it’s so large that you tune out, and only hear that the final digit is \(4\). How do you know that he’s lying?

You might want to know what day of the week your birthday is this year. Mathematician John Conway invented an algorithm called the ‘Doomsday Rule’ to determine which day of the week a particular date falls on. It works by finding the ‘anchor day’ for the year that you’re working in. For \(2025\), the anchor day is Friday. Certain days in the calendar always fall on the anchor day. Some memorable ones are the following:
‘\(0\)’ of March - which is \(29\)th February in a leap year, and \(28\)th February otherwise.

\(4\)th April, \(6\)th June, \(8\)th August, \(10\)th October and \(12\)th December. These are easier to remember as \(4/4\), \(6/6\), \(8/8\), \(10/10\) and \(12/12\).

\(9\)th May, \(11\)th July, \(5\)th September and \(7\)th November. These are easier to see as \(9/5\), \(11/7\), \(5/9\) and \(7/11\). A mnemonic for them is “9-5 at the 7-11".
Then find the nearest one of these dates to the date that you’re looking for and find remainders.

For example, \(\pi\) day, (\(14\)th March, which is written \(3/14\) in American date notation. It’s also Albert Einstein’s birthday) is exactly \(14\) days after ‘\(0\)’th March, so is the same day of the week - Friday in \(2025\).

What day of the week will \(25\)th December be in \(2025\)?

\(6\) friends get together for a game of three versus three basketball. In how many ways can they be split into two teams? The order of the two teams doesn’t matter, and the order within the teams doesn’t matter.

That is, we count A,B,C vs. D,E,F as the same splitting as F,D,E vs A,C,B.

Below is a regular octagon. Given that its side length is \(1\), what’s the difference between the area of the red rectangle and the rest of the octagon?

\(x\), \(y\) and \(z\) are all integers. We’re told that \[\begin{align} x^3yz&=6\\ xy^3z&=24\\ xyz^3&=54. \end{align}\] What’s \(xyz\)?

In the diagram, all the small squares are of the same size. What fraction of the large square is shaded?