NZ-NZMO 2020

Let $P(x) = x^{3} - 2x + 1$ and let $Q(x) = x^{3} - 4x^{2} + 4x - 1$ . Show that

if $P(r) = 0$ then $Q(r^{2}) = 0$

Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$ . Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$ .

You have an unlimited supply of square tiles with side length 1 and equilateral triangle tiles with side length 1. For which $n$ can you use these tiles to create a convex $n$ -sided polygon? The tiles must fit together without gaps and may not overlap.

Determine all prime numbers $p$ such that $p^{2} - 6$ and $p^{2} + 6$ are both prime numbers.

A sequence of $A s$ and $B s$ is called antipalindromic if writing it backwards, then turning all the $A s$ into $B s$ and vice versa, produces the original sequence. For example $A B B A A B$ is antipalindromic. For any sequence of $A s$ and $B s$ we define the cost of the sequence to be the product of the positions of the $A s$ . For example, the string $A B B A A B$ has cost $1\cdot 4\cdot 5 = 20$ . Find the sum of the costs of all antipalindromic sequences of length 2020.

Let $\triangle ABC$ be an acute triangle with $AB > AC$ . Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$ . Let $X$ be the intersection of $PQ$ and $BC$ . Let the intersection of the circumcircles of triangle $\triangle AXC$ and triangle $\triangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$ .

Josie and Ross are playing a game on a $20 \times 20$ chessboard. Initially the chessboard is empty. The two players alternately take turns, with Josie going first. On Josie's turn, she selects any two different empty cells, and places one white stone in each of them. On Ross' turn, he chooses any one white stone currently on the board, and replaces it with a black stone. If at any time there are 8 consecutive cells in a line (horizontally or vertically) all of which contain a white stone, Josie wins. Is it possible that Ross can stop Josie winning - regardless of how Josie plays?

For a positive integer $x$ , define a sequence $a_{0}, a_{1}, a_{2}, \ldots$ according to the following rules: $a_{0} = 1$ , $a_{1} = x + 1$ and

$$a_{n + 2} = x a_{n + 1} - a_{n} \mathrm{for~all~}n\geq 0.$$

Prove that there exist infinitely many positive integers $x$ such that this sequence does not contain a prime number.