Problem 3

Part of 1999 VJIMC

Problems(2)

distance independent of n points on ellipsoid

Source: VJIMC 1999 1.3

A_1,\ldots,A_n

be points of an ellipsoid with center

O

\mathbb R^n

OA_i

i=1,\ldots,n

, are mutually orthogonal. Prove that the distance of the point

O

from the hyperplane

A_1A_2\ldots A_n

does not depend on the choice of the points

A_1,\ldots,A_n

countable set of balls, finite subsets have disjoint interiors

Source: VJIMC 1999 2.3

Suppose that we have a countable set

A

of balls and a unit cube in

\mathbb R^3

. Assume that for every finite subset

B

A

it is possible to put all balls of

B

into the cube in such a way that they have disjoint interiors. Show that it is possible to arrange all the balls in the cube so that all of them have pairwise disjoint interiors.

geometryinfinity