Problem 1. Determine the sum $a^{2}+b^{2}$, if $a+b=24$ and $a \cdot b=143$.
Solution. We have

$$
\begin{aligned}
a^{2}+b^{2} & =a^{2}+b^{2}+2 a b-2 a b \\
& =(a+b)^{2}-2 a b \\
& =24^{2}-2 \cdot 143=290
\end{aligned}
$$