1. On the board, the numbers $a, b$, and $c$ are written. They were erased, and in their place, the numbers $a^{2}+2 b c, b^{2}+2 c a$, $c^{2}+2 a b$ were written. After this, it turned out that the numbers on the board are the same as those at the beginning (possibly in a different order). Find all possible values of the sum $a+b+c$.

$(25$ points. $)$
Answer: 0 or 1.

From the condition, it follows that the sum of the "new" numbers coincides with the sum $s=a+b+c$ of the original numbers, that is,

$$
a^{2}+2 b c+b^{2}+2 c a+c^{2}+2 a b=s
$$

The expression on the left side coincides with the number $(a+b+c)^{2}=s^{2}$, so this equality is equivalent to: $s^{2}=s$, from which $s=0$ or $s=1$.

It remains to provide examples of numbers $a, b$, and $c$, for which the specified values of $s$ are achieved. For $s=1$, for example, the set $a=b=c=\frac{1}{3}$ will do. For $s=0$, you can take the set $a=b=-c=\frac{1}{3}$.

Criteria. 7 points for each correct value of the sum. 11 more points if examples of sets for which these values are achieved are provided.