Solve the following equation:

$$
\frac{x}{x+1}+\frac{x+1}{x+2}+\frac{x+2}{x+3}-\frac{x+3}{x+4}-\frac{x+4}{x+5}-\frac{x+5}{x+6}=0
$$
If we combine the first and sixth, second and fifth, third and fourth terms on the left side, we get:

$$
\frac{5}{(x+1)(x+6)}+\frac{3}{(x+2)(x+5)}+\frac{1}{(x+3)(x+4)}=0
$$

Let

$$
(x+1)(x+6)=y
$$

then our given equation transforms into:

$$
\frac{5}{y}+\frac{3}{y+4}+\frac{1}{y+6}=0
$$

or

$$
3 y^{2}+24 y+40=0
$$

from which

$$
y=\frac{-12 \pm 2 \sqrt{6}}{3}
$$

Substituting this into (1), we get:

$$
3 x^{2}+21 x+30 \mp 2 \sqrt{6}=0
$$

From this,

$$
x=\frac{-21 \pm \sqrt{81 \pm 24 \sqrt{6}}}{6}
$$

(Imre Deutsch, Budapest.)

The problem was also solved by: I. Bartók, E. Deutsch, B. Enyedi, A. Haar, A. Liebner, I. Pivnyik, M. Popoviciu, G. Preisich, K. Riesz.