[Solutions] British Team Selection Test 2022

1. For a given positive integer $k$, we call an integer $n$ a $k$-number if both of the following conditions are satisfied
• The integer $n$ is the product of two positive integers which differ by $k$.
• The integer $n$ is $k$ less than a square number.
Find all $k$ such that there are infinitely many $k$-numbers.
2. Find all functions $f$ from the positive integers to the positive integers such that for all $x, y$ we have $$2 y f\left(f\left(x^{2}\right)+x\right)=f(x+1) f(2 x y) .$$
3. The cards from $n$ identical decks of cards are put into boxes. Each deck contains $50$ cards, labelled from $1$ to $50$. Each box can contain at most $2022$ cards. A pile of boxes is said to be regular if that pile contains equal numbers of cards with each label. Show that there exists some $N$ such that, if $n \geq N$, then the boxes can be divided into two non-empty regular piles.
4. Let $A B C$ be an acute angled triangle with circumcircle $\Gamma$. Let $l_{B}$ and $l_{C}$ be the lines perpendicular to $B C$ which pass through $B$ and $C$ respectively. A point $T$ lies on the minor arc $B C$. The tangent to $\Gamma$ at $T$ meets $l_{B}$ and $l_{C}$ at $P_{B}$ and $P_{C}$ respectively. The line through $P_{B}$ perpendicular to $A C$ and the line through $P_{C}$ perpendicular to $A B$ meet at a point $Q$. Given that $Q$ lies on $B C$, prove that the line $A T$ passes through $Q$. (A minor arc of a circle is the shorter of the two arcs with given endpoints.)

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