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Maximum Value of Function in Range

Source: 1989 IrMO Paper 1 Problem 3

September 29, 2017

functionalgebra

Problem Statement

A function

f

is defined on the natural numbers

\mathbb{N}

and satisfies the following rules:
(a)

f(1)=1

;
(b)

f(2n)=f(n)

and

f(2n+1)=f(2n)+1

for all

n\in \mathbb{N}

.
Calculate the maximum value

m

of the set

\{f(n):n\in \mathbb{N}, 1\le n\le 1989\}

, and determine the number of natural numbers

n

, with

1\le n\le 1989

, that satisfy the equation

f(n)=m

.

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