# [Solutions] British Mathematical Olympiad 2020

1. Alice and Bob take it in turns to write numbers on a blackboard. Alice starts by writing an integer $a$ between $-100$ and 100 inclusive on the board. On each of Bob's turns he writes twice the number Alice wrote last. On each of Alice's subsequent turns she writes the number $45$ less than the number Bob wrote last. At some point, the number $a$ is written on the board for a second time. Find the possible values of $a$.
2. A triangle has side lengths $a, a$ and $b$. It has perimeter $P$ and area $A$. Given that $b$ and $P$ are integers, and that $P$ is numerically equal to $A^{2}$, find all possible pairs $(a, b)$.
3. A square piece of paper is folded in half along a line of symmetry. The resulting shape is then folded in half along a line of symmetry of the new shape. This process is repeated until $n$ folds have been made, giving a sequence of $n+1$ shapes. If we do not distinguish between congruent shapes, find the number of possible sequences when (When $n=1$ there are two possible sequences.)
• $n=3$;
• $n=6$;
• $n=9$.
4. In the equation $$A^{A A}+A A=B, B B C, D E D, B E E, B B B, B B E$$ the letters $A$, $B$, $C$, $D$ and $E$ represent different base $10$ digits (so the right hand side is a sixteen digit number and $A A$ is a two digit number). Given that $C=9$, find $A$, $B$, $D$ and $E$.
5. Let points $A, B$ and $C$ lie on a circle $\Gamma$. Circle $\Delta$ is tangent to $A C$ at $A$. It meets $\Gamma$ again at $D$ and the line $A B$ again at $P$. The point $A$ lies between points $B$ and $P$. Prove that if $A D=D P$, then $B P=A C$.
6. Given that an integer $n$ is the sum of two different powers of $2$ and also the sum of two different Mersenne primes, prove that $n$ is the sum of two different square numbers. (A Mersenne prime is a prime number which is one less than a power of two.)
7. Evie and Odette are playing a game. Three pebbles are placed on the number line; one at $-2020$, one at $2020$, and one at $n$, where $n$ is an integer between $-2020$ and $2020$. They take it in turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their sum is odd; Evie wins if their sum is even. For how many values of $n$ can Evie guarantee victory if
• Odette goes first;
• Evie goes first?
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