8.4. Petya wrote down a line of three positive numbers, below it - a line of their pairwise sums, and below that - a line of the pairwise products of the numbers in the second line. The numbers in the third line coincided (in some order) with the numbers in the first line. Find these numbers.
Solution. Let Pete write down the numbers $a \leq b \leq c$, then the numbers in the third row are $(a+b)(a+c) \leq(a+b)(b+c) \leq(a+c)(b+c)$. Therefore, $a=(a+b)(a+c), b=(a+b)(b+c)$. Adding these equations, we get $a+b=(a+b)(a+b+2 c)$, from which $a+b+2 c=1$.

Similarly, $a+2 b+c=1, 2 a+b+c=1$, from which $a=b=c=\frac{1}{4}$.

Answer: $\frac{1}{4} ; \frac{1}{4} ; \frac{1}{4}$.

## Criteria:

7 points - complete solution;

3 points - equations $a+2 b+c=1, 2 a+b+c=1, a+b+2 c=1$ are obtained, further progress is absent;

2 points - the answer is written, the condition is checked, but it is not proven that there are no other numbers;

1 point - only the answer is written.