In the triangle $ABC$, the midpoint of side $BC$ is one vertex of another triangle, the other two vertices of which are cut by a line parallel to side $BC$ on the sides $AB$ and $AC$ of the given triangle. What will be the position of the cutting line if the area of the newly defined triangle takes on a significant value?
If the length of a cutting line is $y$, its distance from the base is $x$, the base of the triangle is $a$, and its height is $m$, then

$$
(m-x): m=y: a, \text{ hence } y=a-\frac{a}{m} x
$$

Multiply both sides of the equation by $\frac{x}{2}$:

$$
\frac{x y}{2}=-\frac{a x^{2}}{2 m}+\frac{a}{2} x
$$

The eminent value thus reaches its maximum when $x=\frac{m}{2}$.

(Géza Messik, Budapest.)

Number of solutions: 35.