Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

Let $M$ be the circumcenter of an acute-angled triangle $ABC$. The circumcircle of triangle $BMA$ intersects $BC$ at $P$ and $AC$ at $Q$. Show that $CM \perp PQ$.

Let ($b_n$) be a sequence such that $b_n \geq 0 $ and $b_{n+1}^2 \geq \frac{b_1^2}{1^3}+\cdots+\frac{b_n^2}{n^3}$ for all $n \geq 1$. Prove that there exists a natural number $K$ such that \[\sum_{n=1}^{K} \frac{b_{n+1}}{b_1+b_2+ \cdots + b_n} \geq \frac{1993}{1000}\]

Some towns are connected by roads, with at most one road between any two towns. Let $v$ be the number of towns and $e$ be the number of roads. Prove that $(a)$ if $e<v-1$, then there are two towns such that one cannot travel between them; $(b)$ if $2e>(v-1)(v-2)$, then one can travel between any two towns.

Points $E$ and $C$ are chosen on a semicircle with diameter $AB$ and center $O$ such that $OE \perp AB$ and the intersection point $D$ of $AC$ and $OE$ is inside the semicircle. Find all values of $\angle{CAB}$ for which the quadrilateral $OBCD$ is tangent.

Determine all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q^+}$ that satisfy: \[f\left(x+\frac{y}{x}\right) = f(x)+f\left(\frac{y}{x}\right)+2y \:\text{for all}\: x, y \in \mathbb{Q^+}\]