2. Given three not necessarily integer numbers. If each of the given numbers is increased by 1, then the product of the numbers also increases by 1. If each of the numbers is increased by 2, then their product also increases by 2. Find these numbers.
Solution. Let the required numbers be $a, b, c$. According to the condition, $(a+1)(b+1)(c+1)=a b c+1$ and $(a+2)(b+$ $2)(c+2)=a b c+2$. Expanding the brackets, we get $a b+b c+c a+a+b+c=0$ and $2(a b+b c+c a)+4(a+b+c)=-6$. From this, $a+b+c=-3$ and $a b+b c+c a=3$. Since $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(a b+b c+c a)=9$, we have $a^{2}+b^{2}+c^{2}=3$. But then $2 \cdot 3-2 \cdot 3=2\left(a^{2}+\right.$ $\left.b^{2}+c^{2}\right)-2(a b+b c+c a)=\left(a^{2}-2 a b+b^{2}\right)+\left(b^{2}-2 b c+c^{2}\right)+\left(c^{2}-2 c a+a^{2}\right)=(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0$, which means $a=b=c$ and considering $a=b=c=-3$ we get $a=b=c=-1$.

Criteria. Only the answer - 1 point. Found $a+b+c$ and $a b+b c+c a-2$ points, if the answer is guessed after this - another 1 point. Found $a+b+c$ and $a^{2}+b^{2}+c^{2}-3$ points, if the answer is guessed after this - another 1 point. Written $a^{2}+a b+b^{2}+3 a+3 b+3=0-4$ points, if the answer is guessed after this - another 1 point. Any correct solution - 7 points.