Let $n$ be a positive integer. Show that there exist positive integers $a$ and $b$ such that \[ \frac{a^2 + a + 1}{b^2 + b + 1} = n^2 + n + 1. \]

Let $a, b, \alpha, \beta$ be real numbers such that $0 \leq a, b \leq 1$, and $0 \leq \alpha, \beta \leq \frac{\pi}{2}$. Show that if \[ ab\cos(\alpha - \beta) \leq \sqrt{(1-a^2)(1-b^2)}, \]then \[ a\cos\alpha + b\sin\beta \leq 1 + ab\sin(\beta - \alpha). \]

Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$, respectively, of an acute triangle $ABC$, $AB \neq AC$. Let $\omega_B$ be the circle centered at $M$ passing through $B$, and let $\omega_C$ be the circle centered at $N$ passing through $C$. Let the point $D$ be such that $ABCD$ is an isosceles trapezoid with $AD$ parallel to $BC$. Assume that $\omega_B$ and $\omega_C$ intersect in two distinct points $P$ and $Q$. Show that $D$ lies on the line $PQ$.

Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon.