IMO 2019 Shortlist C9
For any two different real numbers x and y, we define Dpx,yq to be the unique integer d satisfying 2d ď |x ´ y| ă 2d1 . Gi...
Category: Combinatorics
Problem
For any two different real numbers x and y, we define Dpx,yq to be the unique integer d satisfying 2d ď |x ´ y| ă 2d`1 . Given a set of reals F, and an element x P F, we say that the scales of x in F are the values of Dpx,yq for y P F with x ‰ y. Let k be a given positive integer. Suppose that each member x of F has at most k different scales in F (note that these scales may depend on x). What is the maximum possible size of F? (Italy) 8 Bath — UK, 11th–22nd July 2019