## Task $3 / 87$

The smallest positive real number $r$ is sought for which the equations

$$
r=p+q \quad ; \quad(r+1)^{2}=(p+1)^{2}+(q+1)^{2}
$$

hold with positive real numbers $p$ and $q$. Give $p$ and $q$!
By substitution, we have

$$
(p+q+1)^{2}=(p+1)^{2}+(q+1)^{2} \quad ; \quad pq=0.5
$$

Since $p, q > 0$, the inequality between the arithmetic mean and the geometric mean holds:

$$
0.5(p+q)=0.5 r \geq \sqrt{pq}=0.5 \sqrt{2}
$$

Thus, the smallest positive real number $r=\sqrt{2}$. From the equality of the two means, it follows that $p=q=0.5 r=0.5 \sqrt{2}$.