3. Given the expression $\sqrt{x^{2}+y^{2}-z^{2}-2 x y}$. If $x=361979$ and $z=561990$, determine all values of $y$ for which the given expression takes its minimum possible value.
Solution. The smallest value of the given root can be 0. Therefore, it must be

$$
x^{2}+y^{2}-z^{2}-2 x y=0, \text { i.e. }(x+y-z)(x+y+z)=0
$$

From this, we get $x+y-z=0$ or $x+y+z=0$, which means $y=z-x$ or $y=-z-x$. The required values for $y$ are 200011 or -923969.