2022-2023 SoMaS Team Mathematics Challenge

The Secretary of State for Education is hiding in an American style motel with five rooms facing into the car park. The rooms are connected by secret doors which allow the Secretary to move from Room \(n\) to Rooms \(n\pm1\) if they exist. You know that:

• if you enter a room and the Secretary is present then they will not run away (it would be undignified to do so);
• if you exit an empty room, the Secretary will scurry though a secret door into an adjacent room;
• the secret doors are so secret that you will never be able to find or use them.

Can you devise a strategy which will allow you to find the Secretary and ask them why your lecturers are on strike?

A fat cat, a middle manager, and a lowly academic would like to share a delcious square cake. Can the fat cat cut the cake into pieces that can be reassembled to form 3 square cakes into different sizes so they get the largest cake and the academic gets the smallest? If so, what is the minimal number of pieces that you can find?

Let \(a_0, a_1, a_2, \dots\) and \(b_0, b_1, b_2,\dots\) be two sequences of real numbers defined by the recurrence relations \[a_{n+1}=\frac{1}{\sqrt{2}}\sqrt{1-\sqrt{1-a_n^2}}, \qquad b_{n+1}=\frac{\sqrt{1+b_n^2}-1}{b_n}\] for \(n\geq 0\) and initial terms \(\displaystyle a_0:=\frac{1}{\sqrt{2}}\), \(b_0:=1\). Show that \[2^{n+2}a_n~<~\pi ~<~2^{n+2}b_n.\]

Let \(g: \mathbb{R} \rightarrow \mathbb{R}\) be a function that is twice differentiable and satisfies \[g(0)=2, \quad g'(0)=-2, \quad \text{and} \hspace{0.3em} g(1)=1.\] Prove that the function \[f(x):=g(x) g'(x)+g''(x)\] has at least one root.

What's the largest prime you can find whose first four digits are ``2022"? You should explain how you verified that it was prime.

For every football world cup since 1970, Panini have sold collectible sticker albums. A completed album will have a sticker for every squad member at the world cup. This year, there are 670 stickers to collect. Stickers are sold in packets of five. One packet costs 90p, and the five stickers are chosen at random; you won't know which five stickers you have got until you've bought the packet. The album costs £12.99.

1. What is the expected cost of completing the album?
2. Suppose you and a friend buy one album each. You buy sticker packets at the same time, so you can swap duplicates before buying your next packet. What is the expected cost for the person who completes the album first?

SoMaS is going to award an a honorary doctorate to the all-time greatest overall runner. A table of world records is given below. Using data for either women or men:

1. Plot the data as log (time) versus log (distance);
2. Fit a low degree polynomial to the plot in (i);
3. Draw some conclusions from the residues and propose a winner for the SoMaS running award.