We say an integer $n$ is naoish if $n \geq 90$ and the second-to-last digit of $n$ (in decimal notation) is equal to $9$. For example, $10798$, $1999$ and $90$ are naoish, whereas $9900$, $2009$ and $9$ are not. Nino expresses 2020 as a sum: \[ 2020=n_{1}+n_{2}+\ldots+n_{k} \]where each of the $n_{j}$ is naoish. What is the smallest positive number $k$ for which Nino can do this?

A round table has $2 N$ chairs around it. Due to social distancing guidelines, no two people are allowed to sit next to each other. How many different ways are there to choose seats around the table on which $N-1$ guests can be seated?

Circles $\Omega_{1},$ centre $Q,$ and $\Omega_{2},$ centre $R,$ touch externally at $B .$ A third circle, $\Omega_{3},$ which contains $\Omega_{1}$ and $\Omega_{2},$ touches $\Omega_{1}$ and $\Omega_{2}$ at $A$ and $C,$ respectively. Point $C$ is joined to $B$ and the line $B C$ is extended to meet $\Omega_{3}$ at $D$. Prove that $Q R$ and $A D$ intersect on the circumference of $\Omega_{1}$.

Let $n$ be a positive integer. An $n$-level honeycomb is a plane region covered with regular hexagons of side-length 1 connected along edges, such that the centres of the boundary hexagons are lined up along a regular hexagon of side-length $n \sqrt{3}$. The diagram shows a 2-level honeycomb from which the central hexagon has been removed. A trex is a sequence of 3 hexagons with collinear centres such that the middle hexagon shares an edge with each of its neighbours in the trex. An $n$-level honeycomb from which the central size-1 hexagon has been removed is to be completely covered by trexes without any overlaps. Find all values of $n$ for which this is possible.

Let $a, b, c>0 .$ Prove that \[ \sqrt[7]{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt[7]{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}} \geq 3. \]

Pat has a pentagon, each of whose vertices is coloured either red or blue. Once an hour, Pat recolours the vertices as follows. Any vertex whose two neighbors were the same colour for the last hour, becomes blue for the next hour. Any vertex whose two neighbors were different colours for the last hour, becomes red for the next hour. Show that there is at least one vertex which is blue after the first recolouring and remains blue for ever.

A function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies the following for all $n \in \mathbb{N}$ : \begin{align*} f(1) &=1 \\ f(f(n)) &=n \\ f(2 n) &=2 f(n)+1 \end{align*}Find the value of $f(2020)$.

Determine the last (rightmost) three decimal digits of $n$ where: \[ n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019. \]

A trapezium $A B C D,$ in which $A B$ is parallel to $D C,$ is inscribed in a circle of radius $R$ and centre $O .$ The non-parallel sides $D A$ and $C B$ are extended to meet at $P$ while diagonals $A C$ and $B D$ intersect at $E .$ Prove that $|O E| \cdot|O P|=R^{2}$.

Show that there exists a hexagon $A B C D E F$ in the plane such that the distance between every pair of vertices is an integer.