# [Solutions] British Team Selection Test 2021

1. A positive integer $n$ is called good if there is a set of divisors of $n$ whose members sum to $n$ and include $1$. Prove that every positive integer has a multiple which is good.
2. Eliza has a large collection of $a \times a$ and $b \times b$ tiles where $a$ and $b$ are positive integers. She arranges some of these tiles, without overlaps, to form a square of side length $n$. Prove that she can cover another square of side length $n$ using only one of her two types of tile.
3. Let $A B C$ be a triangle with $A B>A C$. Its circumcircle is $\Gamma$ and its incentre is $I$. Let $D$ be the contact point of the incircle of $A B C$ with $B C$. Let $K$ be the point on $\Gamma$ such that $\angle A K I$ is a right angle. Prove that $A I$ and $K D$ meet on $\Gamma$.
4. Matthew writes down a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of positive integers. Each $a_{n}$ is the smallest positive integer, different from all previous terms in the sequence, such that the mean of the terms $a_{1}, a_{2}, \ldots, a_{n}$ is an integer. Prove that the sequence defined by $a_{i}-i$ for $i=1,2,3, \ldots$ contains every integer exactly once.
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