Let $\operatorname{rad}(k)$ denote the product of prime divisors of a natural number $k$ (define $\operatorname{rad}(1)=1$). A sequence $(a_n)$ is defined by setting $a_1$ arbitrarily, and $a_{n+1}=a_n+\operatorname{rad}(a_n)$ for $n\ge1$. Prove that the sequence $(a_n)$ contains arithmetic progressions of arbitrary length.

Circles $\omega_1,\omega_2,\omega_3$ with centers $O_1,O_2,O_3$, respectively, are externally tangent to each other. The circle $\omega_1$ touches $\omega_2$ at $P_1$ and $\omega_3$ at $P_2$. For any point $A$ on $\omega_1$, $A_1$ denotes the point symmetric to $A$ with respect to $O_1$. Show that the intersection points of $AP_2$ with $\omega_3$, $A_1P_3$ with $\omega_2$, and $AP_3$ with $A_1P_2$ lie on a line.

A cube of side $n$ is cut into $n^3$ unit cubes, and m of these cubes are marked so that the centers of any three marked cubes do not form a right-angled triangle with legs parallel to sides of the cube. Find the maximum possible value of $m$.

Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions: (i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$; (ii) $f(f(f(0)))=0$. Prove that $f(0)=0$.

Given a natural number $n$, find the number of quadruples $(x,y,u,v)$ of integers with $1\le x,y,y,v\le n$ satisfy the following inequalities: \begin{align*} &1\le v+x-y\le n,\\ &1\le x+y-u\le n,\\ &1\le u+v-y\le n,\\ &1\le v+x-u\le n. \end{align*}

In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then $$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$

Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.

Let $n\ge2$. For any two $n$-vectors $\vec x=(x_1,\ldots,x_n)$ and $\vec y=(y_1,\ldots,y_n)$, we define $$f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.$$Prove that if $f\left(\vec x,\vec x\right)\ge0$, and $f\left(\vec y,\vec y\right)\ge0$, then $\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right)$.

Two points $A$ and $B$ move around two different circles in the plane with the same angular velocity. Suppose that there is a point $C$ which is equidistant from $A$ and $B$ at every moment. Prove that, at some moment, $A$ and $B$ will coincide.

In a country with $n$ towns, the distance between the towns numbered $i$ and $j$ is denoted by $x_{ij}$. Suppose that the total length of every cyclic route which passes through every town exactly once is the same. Prove that there exist numbers $a_i,b_i$ ($i=1,\ldots,n$) such that $x_{ij}=a_i+b_j$ for all distinct $i,j$.

Let $m,n,k$ be positive integers with $m\ge2$ and $k\ge\log_2(m-1)$. Prove that $$\prod_{s=1}^n\frac{ms-1}{ms}<\sqrt[2^{k+1}]{\frac1{2n+1}}.$$

Given distinct prime numbers $p_1,\ldots,p_s$ and a positive integer $n$, find the number of positive integers not exceeding $n$ that are divisible by exactly one of the $p_i$.