Let $a_0 \lt a_1 \lt a_2 \lt \dotsb$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that $$
a_n \lt \frac{a_0+a_1+a_2+\dotsb+a_n}{n} \le a_{n+1}.
$$
Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is *peaceful* if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^\circ$. Point $H$ is the foot of the perpendicular from $A$ to $\overline{BD}$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and $$
\angle CHS - \angle CSB = 90^{\circ},
\quad \angle THC - \angle DTC = 90^{\circ}.
$$ Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be points on $\overline{AP}$ and $\overline{AQ}$, respectively, such that $P$ is the midpoint of $\overline{AM}$ and $Q$ is the midpoint of $\overline{AN}$. Prove that $\overline{BM}$ and $\overline{CN}$ meet on the circumcircle of $\triangle ABC$.
For every positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac 1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most $99 + \frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.
A set of lines in the plane is in *general position* if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its *finite regions*. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. Note: Results with $\sqrt n$ replaced by $c\sqrt n$ will be awarded points depending on the value of the constant $c$.