Question 12
G6 (USA). Let $A_1A_2A_3A_4$ be a tetrahedron, $G$ its centroid, and $A'_1$, $A'_2$, $A'_3$, and $A'_4$ the points where the circumsphere of $A_1A_2A_3A_4$ intersects $GA_1$, $GA_2$, $GA_3$, and $GA_4$, respectively. Prove that
$$GA_1\cdot GA_2\cdot GA_3\cdot GA_4\le GA'_1\cdot GA'_2\cdot GA'_3\cdot GA'_4$$
and
$$\frac1{GA'_1}+\frac1{GA'_2}+\frac1{GA'_3}+\frac1{GA'_4}\le\frac1{GA_1}+\frac1{GA_2}+\frac1{GA_3}+\frac1{GA_4}.$$
$$GA_1\cdot GA_2\cdot GA_3\cdot GA_4\le GA'_1\cdot GA'_2\cdot GA'_3\cdot GA'_4$$
and
$$\frac1{GA'_1}+\frac1{GA'_2}+\frac1{GA'_3}+\frac1{GA'_4}\le\frac1{GA_1}+\frac1{GA_2}+\frac1{GA_3}+\frac1{GA_4}.$$