BG-MO 2001

Find all real values $t$ for which there exist real numbers $x$ , $y$ , $z$ satisfying : $3x^2 + 3xz + z^2 = 1$ , $3y^2 + 3yz + z^2 = 4$ , $x^2 - xy + y^2 = t$ .

Let $p$ be a prime number congruent to $3$ modulo $4$ , and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$ . Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$ .