EGMO 2020

The positive integers $a_0, a_1, a_2, \ldots, a_{3030}$ satisfy $$2a_{n + 2} = a_{n + 1} + 4a_n \text{ for } n = 0, 1, 2, \ldots, 3028.$$ Prove that at least one of the numbers $a_0, a_1, a_2, \ldots, a_{3030}$ is divisible by $2^{2020}$ .

Find all lists $(x_1, x_2, \ldots, x_{2020})$ of non-negative real numbers such that the following three conditions are all satisfied:

• $x_1 \le x_2 \le \ldots \le x_{2020}$ ;
• $x_{2020} \le x_1 + 1$ ;
• there is a permutation $(y_1, y_2, \ldots, y_{2020})$ of $(x_1, x_2, \ldots, x_{2020})$ such that $$\sum_{i = 1}^{2020} ((x_i + 1)(y_i + 1))^2 = 8 \sum_{i = 1}^{2020} x_i^3.$$

A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example, $(2, 1, 2)$ is a permutation of $(1, 2, 2)$ , and they are both permutations of $(2, 2, 1)$ . Note that any list is a permutation of itself.

Let $ABCDEF$ be a convex hexagon such that $\angle A = \angle C = \angle E$ and $\angle B = \angle D = \angle F$ and the (interior) angle bisectors of $\angle A, ~\angle C,$ and $\angle E$ are concurrent.

Prove that the (interior) angle bisectors of $\angle B, ~\angle D,$ and $\angle F$ must also be concurrent.

Note that $\angle A = \angle FAB$ . The other interior angles of the hexagon are similarly described.

A permutation of the integers $1, 2, \ldots, m$ is called fresh if there exists no positive integer $k < m$ such that the first $k$ numbers in the permutation are $1, 2, \ldots, k$ in some order. Let $f_m$ be the number of fresh permutations of the integers $1, 2, \ldots, m$ .

Prove that $f_n \ge n \cdot f_{n - 1}$ for all $n \ge 3$ .

For example, if $m = 4$ , then the permutation $(3, 1, 4, 2)$ is fresh, whereas the permutation $(2, 3, 1, 4)$ is not.

Consider the triangle $ABC$ with $\angle BCA > 90^{\circ}$ . The circumcircle $\Gamma$ of $ABC$ has radius $R$ . There is a point $P$ in the interior of the line segment $AB$ such that $PB = PC$ and the length of $PA$ is $R$ . The perpendicular bisector of $PB$ intersects $\Gamma$ at the points $D$ and $E$ .

Prove $P$ is the incentre of triangle $CDE$ .

Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$ , $a_3 = 4$ , and for all $n \ge 4$ , $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$ Determine all integers $m$ such that every term of the sequence is a square.