3. Petya thought of five numbers. On the board, he wrote down their pairwise sums: $7, 9, 12, 16, 17, 19, 20, 21, 22$, 29. What numbers did Petya think of?
Solution. Since all the sums are different, all the numbers are also different. Let's arrange these numbers in ascending order and denote them as follows: $x_{1}<x_{2}<x_{3}<x_{4}<x_{5}$.

Then $x_{1}+x_{2}=7, x_{1}+x_{3}=9, x_{3}+x_{5}=22, x_{4}+x_{5}=29$. In addition, $4\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}\right)=7+9+12+16+17+19+20+21+22+29, x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=34$.

Solving the system, we sequentially find

$\left(x_{1}+x_{2}\right)+x_{3}+\left(x_{4}+x_{5}\right)=43, x_{3}=7, x_{1}=2, x_{2}=5, x_{5}=15, x_{4}=14$.

Answer: $2,5,7,14,15$.