Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows: $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$. Show that $a_n$ is always a strictly positive integer.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y) $$for all real numbers $x$ and $y$.

Let a sequence $(a_i)_{i=10}^{\infty}$ be defined as follows: $a_{10}$ is some positive integer, which can of course be written in base 10. For $i \geq 10$ if $a_i > 0$, let $b_i$ be the positive integer whose base-$(i + 1)$ representation is the same as $a_i$'s base-$i$ representation. Then let $a_{i + 1} = b_i - 1$. If $a_i = 0$, $a_{i + 1} = 0$. For example, if $a_{10} = 11$, then $b_{10} = 11_{11} (= 12_{10})$; $a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})$; $b_{11} = 10_{12} (= 12_{10})$; $a_{12} = 11$. Does there exist $a_{10}$ such that $a_i$ is strictly positive for all $i \geq 10$?

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

Let $n > 1$ be a positive integer. Prove that every term of the sequence $$ n - 1, n^n - 1, n^{n^2} - 1, n^{n^3} - 1, \dots $$has a prime divisor that does not divide any of the previous terms.

Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.

Let $ABCD$ be a cyclic quadrilateral with its diagonals intersecting at $E$. Let $M$ be the midpoint of $AB$. Suppose that $ME$ is perpendicular to $CD$. Show that either $AC$ is perpendicular to $BD$, or $AB$ is parallel to $CD$.

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

A pawn is a chess piece which attacks the two squares diagonally in front if it. What is the maximum number of pawns which can be placed on an $n \times n$ chessboard such that no two pawns attack each other?

On the board, we write the integers $1, 2, 3, \dots, 2019$. At each minute, we pick two numbers on the board $a$ and $b$, delete them, and write down the number $s(a + b)$ instead, where $s(n)$ denotes the sum of the digits of the integer $n$. Let $N$ be the last number on the board at the end. Is it possible to get $N = 19$? Is it possible to get $N = 15$?

A square is divided into $N^2$ equal smaller non-overlapping squares, where $N \geq 3$. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). Show that it is possible to find a broken line composed of $4$ segments for $N = 3$. Find the minimum number of segments in this broken line for arbitrary $N$.