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Bounded Sequence and Cubes

Source: Benelux Mathematical Olympiad 2011, Problem 3

May 7, 2011

geometry3D geometrymodular arithmeticinequalitiesalgebra proposedalgebra

Problem Statement

k

is an integer, let

\mathrm{c}(k)

denote the largest cube that is less than or equal to

k

. Find all positive integers

p

for which the following sequence is bounded:

a_0 = p

and

a_{n+1} = 3a_n-2\mathrm{c}(a_n)

for

n \geqslant 0