MathDB

Hello Everyone,
i'm trying to make a strong marathon for number theory .. which will be in Pre-Olympiad level
Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons.
it will be preferred to write the source of the problem. please , show detailed solutions , and please post some challenging Pre-Olympiad problems.. remember ,, different approaches are most welcome :)
now .. let's get our hands dirty :lol: :
let

f(n)

denote the sum of the digits of

n

. Now let N \equal{} {4444^{4444}}. Find

f\left( {f\left( {f\left( N \right)} \right)} \right)

Problem Statement