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Divisors with special properties - Switzerland 2011

Source:

March 22, 2011

ceiling functionfloor functioninductionstrong inductionnumber theory proposednumber theory

Problem Statement

For any positive integer

n

let

f(n)

be the number of divisors of

n

ending with

1

9

in base

10

and let

g(n)

be the number of divisors of

n

ending with digit

3

7

in base

10

. Prove that

f(n)\geqslant g(n)

for all nonnegative integers

n

.
(Swiss Mathematical Olympiad 2011, Final round, problem 9)