3. Solve in the set of real numbers the equation

$$
x+y+z=2 \sqrt{x-2}+2 \sqrt{y-8}+10 \sqrt{z-4}-13 .
$$

Ioana Ciocirlan
## Solution.

The equation is defined for $x \in[2, \infty), y \in[8, \infty), z \in[4, \infty)$. (1p)

The equation is equivalent to $(\sqrt{x-2}-1)^{2}+(\sqrt{y-8}-1)^{2}+(\sqrt{z-4}-5)^{2}=0$. (3p) A sum of squares of real numbers is zero only if each number is zero. (1p) The unique solution is obtained: $x=3, y=9, z=29$. (2p)