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?