IMO 2019 Shortlist C1
The infinite sequence a0, a1, a2, ... of (not necessarily different) integers has the following properties: 0 ď ai ď i for...
Category: Combinatorics
Problem
The infinite sequence a0, a1, a2, ... of (not necessarily different) integers has the
following properties: 0 ď ai ď i for all integers i ě 0, and
ˆ
k
a0
˙
ˆ k a1 ˙ ¨¨¨ `
ˆ
k
ak
˙
“ 2k
for all integers k ě 0.
Prove that all integers N ě 0 occur in the sequence (that is, for all N ě 0, there exists i ě 0
with ai “ N).
(Netherlands)