Let be a natural number $ n. $ Prove that there is a polynomial $ P\in\mathbb{Z} [X,Y] $ such that $ a+b+c=0 $ implies $$ a^{2n+1}+b^{2n+1}+c^{2n+1}=abc\left( P(a,b)+P(b,c)+P(c,a)\right) $$ Dan Brânzei

Consider, on a plane, the triangle $ ABC, $ vectors $ \vec x,\vec y,\vec z, $ real variable $ \lambda >0 $ and $ M,N,P $ such that $$ \left\{\begin{matrix} \overrightarrow{AM}=\lambda\cdot\vec x\\\overrightarrow{AN}=\lambda\cdot\vec y \\\overrightarrow{AP}=\lambda\cdot\vec z \end{matrix}\right. . $$Find the locus of the center of mass of $ MNP. $ Dan Brânzei and Gheorghe Iurea

Let be a subset of the interval $ (0,1) $ that contains $ 1/2 $ and has the property that if a number is in this subset, then, both its half and its successor's inverse are in the same subset. Prove that this subset contains all the rational numbers of the interval $ (0,1). $

Let be three discs $ D_1,D_2,D_3. $ For each $ i,j\in\{1,2,3\} , $ denote $ a_{ij} $ as being the area of $ D_i\cap D_j. $ If $ x_1,x_2,x_3\in\mathbb{R} $ such that $ x_1x_2x_3\neq 0, $ then $$ a_{11} x_1^2+a_{22} x_2^2+a_{33} x_3^2+2a_{12} x_1x_2+2a_{23 }x_2x_3+2a_{31} x_3x_1>0. $$ Vasile Pop

Find a relation between the angles of a triangle such that this could be separated in two isosceles triangles by a line. Dan Brânzei

Find the number of perfect squares of five digits whose last two digits are equal. Gheorghe Iurea

Consider the set $ \mathcal{M}=\left\{ \gcd(2n+3m+13,3n+5m+1,6n+6m-1) | m,n\in\mathbb{N} \right\} . $ Show that there is a natural $ k $ such that the set of its positive divisors is $ \mathcal{M} . $ Dan Brânzei

Let be a convex quadrilateral $ ABCD. $ On the semi-straight line extension of $ AB $ in the direction of $ B, $ put $ A_1 $ such that $ AB=BA_1. $ Similarly, define $ B_1,C_1,D_1, $ for the other three sides. a) If $ E,E_1,F,F_1 $ are the midpoints of $ BC,A_1B_1,AD $ respectively, $ C_1,D_1, $ show that $ EE_1=FF_1. $ b) Delete everything, but $ A_1,B_1,C_1,D_1. $ Now, find a way to construct the initial quadrilateral. Vasile Pop