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2022 Combo Div 1 P2 (Div 2 P6)

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February 28, 2022
combinatorics

Problem Statement

A sequence of pairwise distinct positive integers is called averaging if each term after the first two is the average of the previous two terms. Let MM be the maximum possible number of terms in an averaging sequence in which every term is less than or equal to 20222022 and let NN be the number of such distinct sequences (every term less than or equal to 20222022) with exactly MM terms. What is M+N?M+N? (Two sequences a1,a2,⋯ ,ana_1, a_2, \cdots, a_n and b1,b2,⋯ ,bnb_1, b_2, \cdots, b_n are said to be distinct if ai≠bia_i \neq b_i for some integer 1≤i≤n1 \leq i \leq n).
Proposed by Kyle Lee
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