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product with prime divisors of n

Source: Vietnam TST 1993 for the 34th IMO, problem 5

June 25, 2005

algebra unsolvedalgebra

Problem Statement

Let an integer

k > 1

be given. For each integer

n > 1

, we put

f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)

where

p_1, p_2, \ldots, p_r

are all distinct prime divisors of

n

. Find all values

k

for which the sequence

\{x_m\}

defined by

x_0 = a

and

x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots

is bounded for all integers

a > 1

.

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