For a certain triangle, we are given the radius of the circumscribed circle and the sum of the squares of the sides. What is the geometric locus of the centroid of the triangle?
According to problem 1234,

$$
p^{2}=\frac{1}{9}\left(a^{2}+b^{2}+c^{2}\right)=\text { const }
$$

Connecting $G$ with $O$, the center of the circumscribed circle with radius $r$, we get

$$
p^{2}=(r+G O)(r-G O)=r^{2}-\overline{G O}^{2}
$$

and thus

$$
\overline{G O}^{2}=r^{2}-p^{2}=\text { const. }
$$

The geometric locus of the centroid of the triangle is therefore a circle that is concentric with the original circle and whose radius is $\sqrt{r^{2}-p^{2}}$.

(Sándor Pichler, Budapest.)

The problem was also solved by: N. Ehrenfeld, I. Erdélyi, H. Fodor, P. Hajdú, E. Kirchknopf, E. Kiss, J. Kiss, E. Mellinger, C. Neubauer, P. Sárközy, Gy. Schuster, A. Vilcsek, Term. kör. Budapest, VII. ker.