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Sets of chain divisors

Source: Romanian IMO TST 2006, day 5, problem 2

May 23, 2006

inductioncombinatorics proposedcombinatorics

Problem Statement

Let

m

and

n

be positive integers and

S

be a subset with

(2^m-1)n+1

elements of the set

\{1,2,3,\ldots, 2^mn\}

. Prove that

S

contains

m+1

distinct numbers

a_0,a_1,\ldots, a_m

such that

a_{k-1} \mid a_{k}

for all

k=1,2,\ldots, m

.

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