2. Find all integer solutions $x$ and $y$ that satisfy the equation $x^{2} + xy + y^{2} = 1$.
I/2. We multiply the equation by 2 and rearrange it to $x^{2}+y^{2}+(x+y)^{2}=2$. Since the numbers $x$ and $y$ are integers, two of the terms must be equal to 1, and one must be 0, which gives us all the solutions $(x, y) \in$ $\{(1,0),(-1,0),(1,-1),(0,1),(0,-1),(-1,1)\}$.

Rearranged equation $x^{2}+y^{2}+(x+y)^{2}=2$: 3 points. Conclusion “two terms are equal to 1, the third is 0”: 1 point. Each correct pair of solutions based on one of the three possibilities: 1 point. If the contestant lists all correct solutions without proving there are no others: 2 points. For at least 3 correct solutions: 1 point.