Let's determine the smallest positive value of $a$ for which the equation

$$
\frac{\frac{x-a}{2}+\frac{x-2 a}{3}}{\frac{x+4 a}{5}-\frac{x+3 a}{4}}=\frac{\frac{x-3 a}{4}+\frac{x-4 a}{5}}{\frac{x+2 a}{3}-\frac{x+a}{2}}
$$

has an integer solution for $x$.
Let's multiply both sides of the equation by $3 \cdot 4 \cdot 5=60$. After simplification, we get

$$
\frac{50 x-70 a}{3(a-x)}=\frac{27 x-93 a}{10(a-x)}
$$

Excluding the value $x=a$, we can multiply both sides by $30(a-x)$. After rearrangement, we get

$$
419 x=421 a
$$

from which

$$
x=\frac{421}{419} a
$$

According to the problem, $a>0$, so $x>0$. Since $x$ is directly proportional to $a$, the smallest positive integer value of $x$ (when $x=1$) will also give the smallest value of $a$.

Thus,

$$
x=1=\frac{421}{419} a
$$

from which

$$
a=\frac{419}{421}
$$

Zsuzsanna Pataki (Bp., XIV., Chemical Industry t. I. o. t.)