
Problem 4. Find all pairs $(x, y)$ of integer numbers such that $x^{3}=$ $y^{3}+2 y^{2}+1$. Nikolay Nikolov and Emil Kolev


Solution: It is obvious that $x>y$. On the other hand $xy^{3}+2 y^{2}+1 \Longleftrightarrow y(y+3)>0$. Therefore if $y>0$ or $y<-3$ the problem has no solution. Direct verification yields all pairs $(x, y)$ which satisfy the equality $x^{3}=y^{3}+2 y^{2}+1$, namely $(-2,-3),(1,-2)$ and $(1,0)$.
