MathDB

A4

Part of 2023 Balkan MO Shortlist

Problems(1)

Infinitely many close pairs of members of a sequence

Source: BMO SL 2023 A4

5/3/2024
Prove that there exists a real c<34c<\frac{3}{4}, such that for each sequence x1,x2,…x_1, x_2, \ldots satisfying 0≤xi≤10 \leq x_i \leq 1 for all ii, there exist infinitely many (m,n)(m, n) with m>nm>n, such that ∣xm−xn∣≤cm.|x_m-x_n|\leq \frac{c} {m}.
algebra