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Three sequences

Source: Cono Sur Olympiad 2017, problem 5

August 21, 2017
cono suralgebra

Problem Statement

Let aa, bb and cc positive integers. Three sequences are defined as follows:
[*] a1=aa_1=a, b1=bb_1=b, c1=cc_1=c[/*] [*] an+1=⌊anbn⌋a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor,  bn+1=⌊bncn⌋\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor,  cn+1=⌊cnan⌋\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor for n≥1n \ge 1[/*]
[list = a] [*]Prove that for any aa, bb, cc, there exists a positive integer NN such that aN=bN=cNa_N=b_N=c_N.[/*] [*]Find the smallest NN such that aN=bN=cNa_N=b_N=c_N for some choice of aa, bb, cc such that a≥2a \ge 2 y b+c=2a−1b+c=2a-1.[/*]
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