MathDB

The set of all positive real numbers is partitioned into three mutually disjoint non-empty subsets:

\mathbb R^+ = A \cup B\cup C

and

A \cap B = B \cap C = C \cap A = \emptyset

whereas none of

A, B, C

is empty. [*] Show that one can choose

a \in A, b \in B

and

c \in C

such that

a,b, c

are the sides of a triangle. [*] Is it always possible to choose three numbers from three different sets

A,B,C

such that these three numbers are the sides of a right-angled triangle?

Problem Statement