# [Solutions] United Kingdom Mathematical Olympiad For Girls 2018

1. a) Write down the full factorisation of the expression $a^{2}-b^{2}$.
• Show that $359999$ is not prime.
• Show that $249919$ is not prime.
b) Write down the full factorisation of $a^{2}+2 a b+b^{2}$.
• Show that $9006001$ is not prime.
• Show that $11449$ is not prime.
2. Triangle $A B C$ is isosceles, with $A B=B C=1$ and angle $A B C$ equal to $120^{\circ}$. A circle is tangent to the line $A B$ at $A$ and to the line $B C$ at $C$. What is the radius of the circle? (You should state clearly any geometrical facts or theorems you use in each step of your calculation. For example, if one of your steps calculates the size of an angle in a triangle you might justify that particular step with "because angles in a triangle add up to $180$ degrees.")
3. a) Sheila the snail leaves a trail behind her as she moves along gridlines in Grid 1. She may only move in one direction along a gridline, indicated by arrows. Let $b$, $c$, $d$ be the number of different trails Sheila could make while moving from $A$ to $B$, $C$, $D$ respectively. Explain why $b=c+d$.
b) Ghastly the ghost lives in a haunted mansion with $27$ rooms arranged in a $3 \times 3 \times 3$ cube. He may pass unhindered between adjacent rooms, moving through the walls or ceilings. He wants to move from the room in the bottom left corner of the building to the room farthest away in the top right corner, passing through as few rooms as possible. Unfortunately, a trap has been placed in the room at the centre of the house and he must avoid it at all costs. How many distinct paths through the house can he take?
4. Each of $100$ houses in a row are to be painted white or yellow. The residents are quite particular and request that no three neighbouring houses are all the same colour.
a) Explain why no more than $67$ houses can be painted yellow.
b) In how many different ways may the houses be painted if exactly $67$ are painted yellow?
5. Sophie lays out $9$ coins in a $3 \times 3$ square grid, one in each cell, so that each coin is tail side up. A move consists of choosing a coin and turning over all coins which are adjacent to the chosen coin. For example if the centre coin is chosen then the four coins in cells above, below, left and right of it would be turned over.
a) Sophie records the number of times she has chosen each coin in a $3 \times 3$ table. Explain how she can use this table to determine which way up every coin in the grid is at the end of a sequence of moves.
b) Is it possible that after a sequence of moves all coins are tail side down?
c) If instead Sophie lays out 16 coins in the cells of a $4 \times 4$ grid, so that each coin is tail side up, is it possible that after a sequence of moves all coins are tail side down? (In parts b) and c), if you think that it is possible, you should specify a sequence of moves, after which all coins are tail side down. If you think it is not possible, you should give a proof to show that it can't be done, no matter which sequence of moves Sophie chooses to do.)
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