MathDB

Translations [Combinatorial Geometry]

Source: Iranian National Math Olympiad (Final exam) 2006

September 14, 2006

geometrygeometric transformationvectortrigonometrydilationratioreal analysis

Problem Statement

Assume that

C

is a convex subset of

\mathbb R^{d}

. Suppose that

C_{1},C_{2},\dots,C_{n}

are translations of

C

that

C_{i}\cap C\neq\emptyset

but

C_{i}\cap C_{j}=\emptyset

. Prove that

n\leq 3^{d}-1

Prove that

3^{d}-1

is the best bound. P.S. In the exam problem was given for

n=3