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u_{2023} wanted, u_{k+2} >= 2 + u_K, u_{\ell+5} <=5 + u_\ell

Source: P1 Francophone Math Olympiad Junior 2023

May 2, 2023

inequalitiesnumber theorySequence

Problem Statement

Let

u_0, u_1, u_2, \ldots

be integers such that

u_0 = 100

;

u_{k+2} \geqslant 2 + u_k

for all

k \geqslant 0

; and

u_{\ell+5} \leqslant 5 + u_\ell

for all

\ell \geqslant 0

. Find all possible values for the integer

u_{2023}