Problems

A gang of three jewel thieves has stolen some gold coins and wants to divide them fairly. However, they each have one unusual rule: (i) The first thief wants the number of coins to be divisible by \(3\) so they can split it evenly. (ii) The second thief wants the number of coins to be divisible by \(5\) because she wants to split her share with her four siblings. (iii) The third thief wants the number of coins to be divisible by \(7\) since he wants to split his share amongst seven company stocks.

However, they’re stuck as the number of coins isn’t divisible by any of these numbers. In fact, the number of coins is \(1\) more than a multiple of \(3\), \(3\) more than a multiple of \(5\) and \(5\) more than a multiple of \(7\).
What’s the smallest number of coins they could have? (And if you’re feeling generous, how would you help them out?)

Munira wants to put \(6\) books on her shelf, \(4\) of which are red and \(2\) of which are blue. The four red ones are a small paperback, a small hardback, a large paperback and a large hardback. The two blue ones are both paperback, one small and one large. She doesn’t want the two blue ones next to each other. In how many ways can she do this?

Imagine a cube that’s turquoise on the front, pink on top, yellow on the right, white on left, dark blue on back and orange on the bottom. If Arne rotates this \(180^{\circ}\) about the line through the middles of the turquoise and dark blue sides, then does it again, he gets back to the original cube. If Arne rotates this \(90^{\circ}\) about that same line, then does that three more times, then he also gets back to the original cube.
Is there a rotation he could do, and then do twice more, to get back to the original cube?

Arne has a cube which is pink on top and orange on bottom, yellow on right and white on left, turquoise on front and dark blue at the back. He rotates this once so that it looks different. Could he perform the same rotation four more times and get back to the original colouring?

Can you tile the plane with regular octagons?

Sam the magician shuffles his hand of six cards: joker, ace (\(A\)), ten, jack (\(J\)), queen (\(Q\)) and king (\(K\)). After his shuffle, the relative order of joker, \(A\) and \(10\) is now \(A\), \(10\), joker. Also, the relative order of \(J\), \(Q\) and \(K\) is now \(Q\), \(K\) and \(J\).
For example, he could have \(A\), \(Q\), \(10\), joker, \(K\), \(J\) - but not \(A\), \(Q\), \(10\), joker, \(J\), \(K\).

How many choices does Sam has for his shuffle?

Draw how to tile the whole plane with figures, composed from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), and \(5\times 5\) where squares of all sizes are used the same amount of times in the design of the figure.

Today we explore inequalities related to mean values of a set of positive real numbers. Let \(\{a_1,a_2,...,a_n\}\) be a set of \(n\) positive real numbers. Define:
Quadratic mean (QM) as \[\sqrt{\frac{a_1^2 + a_2^2 + ...a_n^2}{n}}\] Arithmetic mean (AM) as \[\frac{a_1 + a_2 + ...+a_n}{n}\] Geometric mean (GM) as \[\sqrt[n]{a_1a_2...a_n}\] Harmonic mean (HM) as \[\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ... \frac{1}{a_n}}.\] Then the following inequality holds: \[\sqrt{\frac{a_1^2 + a_2^2 + ...a_n^2}{n}} \geq \frac{a_1 + a_2 + ...+a_n}{n} \geq \sqrt[n]{a_1a_2...a_n} \geq \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ... \frac{1}{a_n}}.\] We will prove \(QM\geq AM\) and infer the \(GM \geq HM\) part from \(AM \geq GM\) in the examples. However, the \(AM\geq GM\) part itself is more technical. The Mean Inequality is a well known theorem and you can use it in solutions today and refer to it on olympiads.

Today we will solve some problems using algebraic tricks, mostly related to turning a sum into a product or using an identity involving squares.
The ones particularly useful are: \((a+b)^2 = a^2 +b^2 +2ab\), \((a-b)^2 = a^2 +b^2 -2ab\) and \((a-b) \times (a+b) = a^2 -b^2\). While we are at squares, it is also worth noting that any square of a real number is never a negative number.

The evil warlock found a mathematics exercise book and replaced all the decimal numbers with the letters of the alphabet. The elves in his kingdom only know that different letters correspond to different digits \(\{0,1,2,3,4,5,6,7,8,9\}\) and the same letters correspond to the same digits. Help the elves to restore the exercise book to study.