NZ-NZMO 2022

$ABCD$ is a rectangle with side lengths $AB = CD = 1$ and $BC = DA = 2$ . Let $M$ be the midpoint of $AD$ . Point $P$ lies on the opposite side of line $MB$ to $A$ , such that triangle $MBP$ is equilateral. Find the value of $\angle PCB$ .

Is it possible to pair up the numbers $0, 1, 2, 3, \ldots , 61$ in such a way that when we sum each pair, the product of the 31 numbers we get is a perfect fifth power?

Find all real numbers $x$ and $y$ such that

$$\frac{x^{2} + y^{2} = 2,}{2 - y} +\frac{y^{2}}{2 - x} = 2.$$

On a table, there is an empty bag and a chessboard containing exactly one token on each square. Next to the table is a large pile that contains an unlimited supply of tokens. Using only the following types of moves what is the maximum possible number of tokens that can be in the bag?

• Type 1: Choose a non-empty square on the chessboard that is not in the rightmost column. Take a token from this square and place it, along with one token from the pile, on the square immediately to its right.

• Type 2: Choose a non-empty square on the chessboard that is not in the bottommost row. Take a token from this square and place it, along with one token from the pile, on the square immediately below it.

• Type 3: Choose two adjacent non-empty squares. Remove a token from each and put them both into the bag.

The sequence $x_{1},x_{2},x_{3},\ldots$ is defined by $x_{1} = 2022$ and $x_{n + 1} = 7x_{n} + 5$ for all positive integers $n$ . Determine the maximum positive integer $m$ such that

$${\frac{x_{n}(x_{n}-1)(x_{n}-2)\ldots(x_{n}-m+1)}{m!}}$$

is never a multiple of 7 for any positive integer $n$ .

Let a positive integer $n$ be given. Determine, in terms of $n$ , the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$ .

Let $M$ be the midpoint of side $BC$ of acute triangle $ABC$ . The circle centered at $M$ passing through $A$ intersects the lines $AB$ and $AC$ again at $P$ and $Q$ , respectively. The tangents to this circle at $P$ and $Q$ meet at $D$ . Prove that the perpendicular bisector of $BC$ bisects segment $AD$ .

Find all real numbers $x$ such that $-1 < x \leq 2$ and

$$\sqrt{2 - x} + \sqrt{2 + 2x} = \sqrt{\frac{x^{4} + 1}{x^{2} + 1}} +\frac{x + 3}{x + 1}.$$