Let $n$ be a positive integer and let $p, q>n$ be odd primes. Prove that the positive integers $1, 2, \ldots, n$ can be colored in $2$ colors, such that for any $x \neq y$ of the same color, $xy-1$ is not divisible by $p$ and $q$.

Let $ABCD$ be a non-isosceles trapezoid with $AB \parallel CD$. A circle through $A$ and $B$ meets $AD$, $BC$ at $E, F$. The segments $AF, BE$ meet at $G$. The circumcircles of $\triangle ADG$ and $\triangle BCG$ meet at $H$. Show that if $GD=GC$, $H$ is the orthocenter of $\triangle ABG$.

Let $a_1, a_2, \ldots$ be a strictly increasing sequence of positive integers, such that for any positive integer $n$, $a_n$ is not representable in the for $\sum_{i=1}^{n-1}c_ia_i$ for $c_i \in \{0, 1\}$. For every positive integer $m$, let $f(m)$ denote the number of $a_i$ that are at most $m$. Show that for any positive integers $m, k$, we have that $$f(m) \leq a_k+\frac{m} {k+1}.$$