MathDB
2022 DMM Tiebreaker Round - Duke Math Meet

Source:

October 2, 2023
DMMcombinatoricsalgebra

Problem Statement

p1. The sequence {xn}\{x_n\} is defined by xn+1={2xn−1,  if  12≤xn<12xn,  if  0≤xn<12x_{n+1} = \begin{cases} 2x_n - 1, \,\, if \,\, \frac12 \le x_n < 1 \\ 2x_n, \,\, if \,\, 0 \le x_n < \frac12 \end{cases} where 0≤x0<10 \le x_0 < 1 and x7=x0x_7 = x_0. Find the number of sequences satisfying these conditions.
p2. Let M={1,...,2022}M = \{1, . . . , 2022\}. For any nonempty set X⊆MX \subseteq M, let aXa_X be the sum of the maximum and the minimum number of XX. Find the average value of aXa_X across all nonempty subsets XX of MM.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Back to ProblemsView on AoPS