Question 1
Let $ABCD$ be a tetrahedron with $BC=a$, $CA=b$, $AB=c$, $DA=a_1$, $DB=b_1$, $DC=c_1$. Prove that there is a unique point $P$ satisfying
$$PA^2+a_1^2+b^2+c^2=PB^2+b_1^2+c^2+a^2=PC^2+c_1^2+a^2+b^2=PD^2+a_1^2+b_1^2+c_1^2.$$
For this point prove $PA^2+PB^2+PC^2+PD^2\ge4R^2$, where $R$ is the circumradius of the tetrahedron. Find a necessary and sufficient condition on its edge lengths for equality.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
$$PA^2+a_1^2+b^2+c^2=PB^2+b_1^2+c^2+a^2=PC^2+c_1^2+a^2+b^2=PD^2+a_1^2+b_1^2+c_1^2.$$
For this point prove $PA^2+PB^2+PC^2+PD^2\ge4R^2$, where $R$ is the circumradius of the tetrahedron. Find a necessary and sufficient condition on its edge lengths for equality.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.