In triangle $ABC$, find a point $M$ on the median $AD$ such that

$$
\overline{M A}^{2}+\overline{M B}^{2}+\overline{M C}^{2}
$$

is minimized.
Let $A D=k, M D=x, A M=k-x, B C=a$, then according to Carnot's theorem,

$$
\overline{M A}^{2}+\overline{M B}^{2}+\overline{M C}^{2}=3 x^{2}-2 k x+\frac{a^{2}}{2}
$$

this expression is minimal when $x=\frac{k}{3}$, that is, when the point $M$ coincides with the centroid.

(Ferencz Kertész, Szeged.)

Number of solutions: 32.