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 $PA = BC = P D = EF$ (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 $x_1, x_2, x_3, . . . x_n$ are real numbers between $0$ and $ 1$ with sum $s$. Prove that $$\prod_{i=1}^{n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{n} (1 - x_i) \le 1.$$

A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly?

Let $X$ be the intersection of the diagonals $AC$ and $BD$ of convex quadrilateral $ABCD$. Let $P$ be the intersection of lines $AB$ and $CD$, and let $Q$ be the intersection of lines $PX$ and $AD$. Suppose that $\angle ABX = \angle XCD = 90^o$. Prove that $QP$ is the angle bisector of $\angle BQC$.

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 3$. Prove that $$a^a + b^b + c^c \ge 3$$

Show that for all positive integers $k$, there exists a positive integer n such that $n2^k -7$ is a perfect square.

An equilateral triangle is partitioned into smaller equilateral triangular pieces. Prove that two of the pieces are the same size.