For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$? Russian Competition

Let $n$ be a positive integer. Show that there exists a polynomial $f$ of degree $n$ with integral coefficients such that $f^2=(x^2-1)g^2+1$, where $g$ is a polynomial with integral coefficients. * * *

Let $ABC$ be a triangle, let its $A$-symmedian cross the circle $ABC$ again at $D$, and let $Q$ and $R$ be the feet of the perpendiculars from $D$ on the lines $AC$ and $AB$, respectively. Consider a variable point $X$ on the line $QR$, different from both $Q$ and $R$. The line through $X$ and perpendicular to $DX$ crosses the lines $AC$ and $AB$ at $V$ and $W$, respectively. Determine the geometric locus of the midpoint of the segment $VW$. Adapted from American Mathematical Monthly

Fix an integer $n\geq4$. Let $C_n$ be the collection of all $n$–point configurations in the plane, every three points of which span a triangle of area strictly greater than $1.$ For each configuration $C\in C_n$ let $f(n,C)$ be the maximal size of a subconfiguration of $C$ subject to the condition that every pair of distinct points has distance strictly greater than $2.$ Determine the minimum value $f(n)$ which $f(n,C)$ achieves as $C$ runs through $C_n.$ Radu Bumbăcea and Călin Popescu

Fix integers $m \geq 3$ and $n \geq 3$. Each cell of an array with $m$ rows and $n$ columns is coloured one of two colours such that: (1) Both colours occur on every column; and (2) On every two rows the cells on the same column share colour on exactly $k$ columns. Show that, if $m$ is odd, then \[\frac{n(m-1)}{2m}\leq k\leq \frac{n(m-2)}{m}\] The Problem Selection Committee

Let $k$ be a positive integer, and let $a,b$ and $c$ be positive real numbers. Show that \[a(1-a^k)+b(1-(a+b)^k)+c(1-(a+b+c)^k)<\frac{k}{k+1}.\] * * *