The axes $\mathrm{Az} O x, O y$ are perpendicular to each other. Point $C$ lies on $O y$, and points $A$ and $B$ lie on $O x$. Determine the position of point $M$ on $O x$ such that $\frac{M A \cdot M B}{\overline{M C}^{2}}$ takes a significant value. (Special case: $\overline{O C}^{2}=O A \cdot O B$).
Let $O A=a, O B=b, O C=c, O M=x$, then the function to be examined is:

$$
m=\frac{(x-a)(x-b)}{x^{2}+c^{2}}
$$

from which

$$
x^{2}(m-1)+x(a+b)+m c^{2}-a b=0
$$

and thus

$$
x=\frac{-(a+b) \pm \sqrt{(a+b)^{2}-4(m-1)\left(m c^{2}-a b\right)}}{2(m-1)}
$$

For $x$ to be real, the expression under the square root must be positive; rearranging this expression, we get:

$$
D=-4 c^{2} m^{2}+4(a b+c)^{2} m+(a-b)^{2}
$$

This expression is positive for those values of $m$ that lie between the roots of the equation $D=0$. The smaller root gives the minimum, and the larger root gives the maximum. If $c^{2}=a b$, then

$$
D=4 a b m^{2}-8 a b m-(a-b)^{2}=0
$$

from which

$$
m=1 \pm \frac{a+b}{2 \sqrt{a b}}
$$

(Szmodics Hildegárd, Kaposvár.)

The problem was also solved by: Aczél F., Bartók I., Bayer B., Haar A., Hirschfeld Gy., Pilczer P., Pivnyik I., Póka Gy., Sasvári J., Schlesinger A., Tóbiás L.