BMO Round 1, 2022

Combinatorics Difficulty 5.0 AIME, harder Find the answer United Kingdom

Marvin has been tasked with writing down every list of integers with the following properties:

(i) The list contains $71$ terms.

(ii) The first term is $1$ .

(iii) Every term after the first is equal to either the previous term, or the sum of all previous terms.

When Marvin is finished, how many of the lists will have a sum equal to $999{,}999$ ?

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