MathDB

Problem 4

Part of 2000 VJIMC

Problems(2)

2-coloring 2n-gon, sequence of side lengths are equal

Source: VJIMC 2000 1.4

7/27/2021
Let us choose arbitrarily nn vertices of a regular 2n2n-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a non-decreasing sequence and do the same with the blue-blue distances. Prove that the sequences are equal.
geometry
existence of balls satisfying Lebesgue inequality

Source: VJIMC 2000 2.4

7/26/2021
Let B\mathcal B be a family of open balls in Rn\mathbb R^n and c<λ(⋃B)c<\lambda\left(\bigcup\mathcal B\right) where λ\lambda is the nn-dimensional Lebesgue measure. Show that there exists a finite family of pairwise disjoint balls {Ui}i=1k⊆B\{U_i\}^k_{i=1}\subseteq\mathcal B such that ∑j=1kλ(Uj)>c3n.\sum_{j=1}^k\lambda(U_j)>\frac c{3^n}.
real analysisinequalitiesMeasure theory