Determine all integers $m\ge2$ for which there exists an integer $n\ge1$ with $\gcd(m,n)=d$ and $\gcd(m,4n+1)=1$. Proposed by Gerhard Woeginger, Austria

Let $S$ be a set of $n\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\in S$. Proposed by Gerhard Woeginger, Austria

Prove that all postive real numbers $a,b,c$ with $a+b+c=4$ satisfy the inequality $$\frac{ab}{\sqrt[4]{3c^2+16}}+ \frac{bc}{\sqrt[4]{3a^2+16}}+ \frac{ca}{\sqrt[4]{3b^2+16}} \le\frac43 \sqrt[4]{12}$$

Let $P,Q,R$ be three points on a circle $k_1$ with $|PQ|=|PR|$ and $|PQ|>|QR|$. Let $k_2$ be the circle with center in $P$ that goes through $Q$ and $R$. The circle with center $Q$ through $R$ intersects $k_1$ in another point $X\ne R$ and intersects $k_2$ in another point $Y\ne R$. The two points $X$ and $R$ lie on different sides of the line through $PQ$. Show that the three points $P$, $X$, $Y$ lie on a common line.