NZ-NZMO 2019

How many positive integers less than 2019 are divisible by either 18 or 21, but not both?

Find all real solutions to the equation

$$(x^{2} + 3x + 1)^{x^{2} - x - 6} = 1.$$

In triangle $ABC$ , points $D$ and $E$ lie on the interior of segments $AB$ and $AC$ , respectively, such that $AD = 1$ , $DB = 2$ , $BC = 4$ , $CE = 2$ and $EA = 3$ . Let $DE$ intersect $BC$ at $F$ . Determine the length of $CF$ .

Show that the number $122^{n} - 102^{n} - 21^{n}$ is always one less than a multiple of 2020, for any positive integer $n$ .

Find all positive integers $n$ such that $n^{4} - n^{3} + 3n^{2} + 5$ is a perfect square.

Let $V$ be the set of vertices of a regular 21-gon. Given a non-empty subset $U$ of $V$ , let $m(U)$ be the number of distinct lengths that occur between two distinct vertices in $U$ . What is the maximum value of $\frac{m(U)}{|U|}$ as $U$ varies over all non-empty subsets of $V$ ?

Let $ABCDEF$ be a convex hexagon containing a point $P$ in its interior such that $PABC$ and $PDEF$ are congruent rectangles with $PA = BC = PD = EF$ (and $AB = PC = DE = PF$ ). Let $\ell$ be the line through the midpoint of $AF$ and the circumcentre of $PCD$ . Prove that $\ell$ passes through $P$ .

Suppose that $x_{1},x_{2},x_{3},\ldots x_{n}$ are real numbers between 0 and 1 with sum $s$ . Prove that

$$\sum_{i = 1}^{n}{\frac{x_{i}}{s + 1 - x_{i}}} + \prod_{i = 1}^{n}(1 - x_{i})\leq 1.$$