Problem 3. Determine the natural numbers $a, b$ and $c$ that satisfy the relations:

$$
\begin{aligned}
& a^{2}+a b+a c=20 \\
& a b+b^{2}+b c=30 \text{ and } \\
& a c+b c+c^{2}=50
\end{aligned}
$$
Solution. The given relations are written in the form

$$
\begin{gathered}
a(a+b+c)=20 \\
b(a+b+c)=30 \\
c(a+b+c)=50
\end{gathered}
$$

By adding them, we get

$$
\begin{gathered}
a(a+b+c)+b(a+b+c)+c(a+b+c)=100 \\
(a+b+c)(a+b+c)=100, \text { i.e. } \\
(a+b+c)^{2}=100
\end{gathered}
$$

Given that $a, b$, and $c$ are natural numbers, it follows that $a+b+c=10$. From this equality and the given conditions in the problem, we obtain that the required numbers are $a=2, b=3$, and $c=5$.