MOD IMO

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2216   
Let $f(n)$ denote the sum of the digits of $n$. Find $f(f(f(4444^{4444})))$.

2699   

Let sequence $g(n)$ satisfy $g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$ where $n\ge 1$. Show that if $n$ is a prime greater than 5, then $n\mid g(n)[g(n)+1]$.


3627   
Show that every integer $k > 1$ has a multiple which is less than $k^4$ and can be written in base 10 using at most 4 different digits.

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