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problem related to largest k such that 2^k|n

Source: 2006 VMEO III Shortlist SL N1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__viet

October 28, 2021

number theorypower of 2

Problem Statement

f(n)

denotes the largest integer

k

such that that

2^k|n

2006

integers

a_i

are such that

a_1<a_2<...<a_{2016}

. Is it possible to find integers

k

where

1 \le k\le 2006

and

f(a_i-a_j)\ne k

for every

1 \le j \le i \le 2006