## Task 1 - V00901

The sum of two numbers is 20, and the sum of their squares is 202. Solve the problem mathematically.
Let $x$ and $y$ be the two numbers we are looking for. Then we get the system of equations

$$
x+y=20 \quad(1) \quad ; \quad x^{2}+y^{2}=202
$$

Rearranging (1) for $y$ and substituting into (2) yields

$$
x^{2}+(20-x)^{2}=202 \Rightarrow 2 x^{2}-40 x+198=0 \Rightarrow x^{2}-20 x+99=0
$$

This quadratic equation has the solutions $x_{1}=9$ and $x_{2}=11$ with $y_{1}=11$ and $y_{2}=9$.

The numbers we are looking for are 9 and 11. They satisfy the conditions of the problem, as confirmed by the check.