# 2013-2014 Team Mathematics Challenge

You can browse the problems and solutions below or view the download/print the PDF copy of problems.Problem 1

Let \(a_1 = 1\) and \(a_{n+1} = (n + 1)(a_n + 1)\). Find \[\prod_{n=1}^\infty\left(1+\frac{1}{a_n}\right).\]

There were excellent entries to this problem from Billy Smells, Chums, the Mathemagicians, and One Man Army.

However, the judges were particularly struck by the solution by Hobbit & Hercules, which goes just that little bit further and thinks hard about various methods of computing \(e\).

The following picture shows a (not to scale) triangle which is divided into four regions. The areas of three of the regions are shown. Find the area of the shaded region.

The judges knew of several methods to solve this problem. However, the submitted solutions used methods that the judges hadn't thought of!

An honourable mention goes to a valiant effort by One Man Army, but the solution of choice is that by Chums.

Given an acute angled triangle \(T\), show that there is a unique tetrahedron whose faces are all congruent to \(T\) and find the volume of this tetrahedron.

This problem was found very difficult, and no team got very far on it. Several teams had a couple of ideas (the Mathemagicians, and Chums in particular). However, we reproduce here a comment by Billy Smells.

You may be interested to read the solution by Fionntan Roukema, who suggested the problem in the first place.

For which \(n\in\mathbb{N}\) can a square be divided into \(n\) (not necessarily congruent) squares?

There were several glorious attempts at this, including those of Chums and Billy Smells. However the solution the judges liked the most was that of πη, which, once a minor slip in the algebra at the end was corrected, was pretty as pie.

Define an \(n\)â€“colouring of the plane to be a function \[f : \mathbb{R}^2 \rightarrow \left\{c_1, \ldots, c_n\right\}.\] In other words, each point in the plane is coloured one of n colours. For which \(n \in \mathbb{N}\) does there exist an \(n\)â€“colouring of the plane with the property that no two points in \(\mathbb{R}^2\) at distance 1 apart have the same colour?

This was a hard problem. The team that got the furthest was Chums again. They leave a gap in their solution, not saying whether it's possible with four, five or six colours. Impressively, they rediscovered all that is known about this very hard open problem!

**Judges:** James Cranch,
Jayanta Manoharmayum, and
Fionntan Roukema.

**O.B.S. prize:** (overall best submission) Chums (Sam Baker, James Hesltine).

**M.V.S. prize:** (most valuable solution) Hobbit and Hercules (Lawrence Beesely-Peck, Robert Nicolaides).

**Honourable mentions:** Billy Smells (William Thurley, Rhys Munden, Sam Watson, Jordan Williamson),
the Mathemagicians (Dan Graves, Ben Patey), One Man Army (Okigbo Aghaji), \(\pi\eta\) (Sam Mutter).

**Best team name:** \(\pi\eta\) (Sam Mutter).

You can browse past years problems and solutions here.