# [Solutions] United Kingdom Mathematical Olympiad For Girls 2015

1. a) Expand and simplify $$(a-b)\left(a^{2}+a b+b^{2}\right).$$ b) Find the value of $$\frac{2016^{3}+2015^{3}}{2016^{2}-2015^{2}}.$$
2. The diagram shows five polygons placed together edge-toedge: two triangles, a regular hexagon and two regular nonagons. Prove that each of the triangles is isosceles.
3. A ladybird is going for a wander around a $10 \times 10$ board, subject to the following three rules (see the diagram).
• She starts in the top left cell, labelled $S$.
• She only moves left, right or down, as indicated.
• She never goes back to a cell that she has already visited.
In how many different ways can she reach the bottom row of cells, shaded grey?
4. a) A tournament has $n$ contestants. Each contestant plays exactly one game against every other contestant. Explain why the total number of games is $\dfrac{1}{2} n(n-1)$.
b) In a particular chess tournament, every contestant is supposed to play exactly one game against every other contestant. However, contestant $A$ withdrew from the tournament after playing only ten games, and contestant $B$ withdrew after just one game. A total of $55$ games were played. Did $A$ and $B$ play each other?
5. a) The integer $N$ is a square. Find, with proof, all possible remainders when $N$ is divided by $16$.
b) Find all positive integers $m$ and $n$ such that $m !+76=n^{2}$. (The notation $m$ ! stands for the factorial of $m$, that is, $m !=m \times(m-1) \times \cdots \times 2 \times 1$. For example, $4 !=4 \times 3 \times 2 \times 1$.)
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