Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

Fix an integer $n \geq 3$. Let $\mathcal{S}$ be a set of $n$ points in the plane, no three of which are collinear. Given different points $A,B,C$ in $\mathcal{S}$, the triangle $ABC$ is nice for $AB$ if $[ABC] \leq [ABX]$ for all $X$ in $\mathcal{S}$ different from $A$ and $B$. (Note that for a segment $AB$ there could be several nice triangles). A triangle is beautiful if its vertices are all in $\mathcal{S}$ and is nice for at least two of its sides. Prove that there are at least $\frac{1}{2}(n-1)$ beautiful triangles.

Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal. Determine the smallest possible degree of $f$. (Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$is $7$ because $7=3+4$.) Ankan Bhattacharya

An acute triangle $ABC$ is given and $H$ and $O$ be its orthocenter and circumcenter respectively. Let $K$ be the midpoint of $AH$ and $\ell$ be a line through $O. $ Let $P$ and $Q$ be the projections of $B$ and $C$ on $\ell. $ Prove that$$KP+KQ\ge BC$$

Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that $$\displaystyle P(x)=R(T(x))$$

Let $r,g,b$ be non negative integers and $\Gamma$ be a connected graph with $r+g+b+1$ vertices. Its edges are colored in red green and blue. It turned out that $\Gamma $ contains A spanning tree with exactly $r$ red edges. A spanning tree with exactly $g$ green edges. A spanning tree with exactly $b$ blue edges. Prove that $\Gamma$ contains a spanning tree with exactly $r$ red edges, $g$ green edges and $b$ blue edges.