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China 2010 quiz4 problem 3

Source:

September 12, 2010

number theory unsolvednumber theory

Problem Statement

For integers

n>1

, define

f(n)

to be the sum of all postive divisors of

n

that are less than

n

. Prove that for any positive integer

k

, there exists a positive integer

n>1

such that

n<f(n)<f^2(n)<\cdots<f^k(n)

, where

f^i(n)=f(f^{i-1}(n))

for

i>1

and

f^1(n)=f(n)

.

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