26. For which natural numbers $n$ can we find integers $x, y, z$ such that

$$
x+y+z=0, \quad \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{n} ?
$$
$\square$

Answer: for even $n$.

For even $n=2k$, take $x=y=3k, z=-6k$. Let $x=d x_{1}, y=d y_{1}, z=d y_{1}$, where $x_{1}, y_{1}, z_{1}$ have no common divisor greater than 1. Then $x_{1}+y_{1}+z_{1}=0$ and $1 / x_{1}+1 / y_{1}+1 / z_{1}=d / n$. Among the numbers $x_{1}, y_{1}, z_{1}$, exactly one is even (otherwise, either their sum is odd, or they are all even, which is impossible). Therefore, the denominator of the fraction $1 / x_{1}+1 / y_{1}+1 / z_{1}$ (after simplification) is an even number, which means $n$ cannot be odd.

(Iberoamerican 2011)