Prove for each triangle $ABC$ the inequality
$$\frac14\lt \frac{IA\cdot IB\cdot IC}{l_A l_B l_C}\le\frac8{27},$$
where $I$ is the incenter and $l_A,l_B,l_C$ are the lengths of the angle bisectors of $ABC$.
Let $n>6$ and let $a_1\lt a_2\lt \cdots\lt a_k$ be all natural numbers that are less than $n$ and relatively prime to $n$. Show that if $a_1,a_2,\ldots,a_k$ is an arithmetic progression, then $n$ is a prime number or a natural power of two.
Let $S=\{1,2,3,\ldots,280\}$. Find the minimal natural number $n$ such that in any $n$-element subset of $S$ there are five numbers that are pairwise relatively prime.
Suppose $G$ is a connected graph with $n$ edges. Prove that it is possible to label the edges of $G$ from 1 to $n$ in such a way that in every vertex $v$ of $G$ with two or more incident edges, the set of numbers labeling those edges has no common divisor greater than 1.
Let $ABC$ be a triangle and $M$ an interior point in $ABC$. Show that at least one of the angles $\angle MAB$, $\angle MBC$, and $\angle MCA$ is less than or equal to $30^\circ$.
Given a real number $a>1$, construct an infinite and bounded sequence $x_0,x_1,x_2,\ldots$ such that for all natural numbers $i$ and $j$, $i\ne j$, the following inequality holds:
$$|x_i-x_j||i-j|^a\ge1.$$