IMO SL N4(2008)

[Sazid Akhter Turzo]

This problem maybe quite easy but I've not solved it yet.
Let $n$ be a positive integer. Show that the numbers
\[\binom{2^n-1}{0},\binom{2^n-1}{1},\binom{2^n-1}{2},\cdot \cdot \cdot ,\binom{2^n-1}{2^{n-1}-1}\]
are congruent modulo $2^{n}$ to $1,3,5,\cdot \cdot \cdot ,2^{n}-1$ in some order.
Turzo

[Phlembac Adib Hasan]

Not realizing the last line.Better if you give an example.

[Sazid Akhter Turzo]

Oh-ho!!! I made serious mistake . It's actually $2^{n}$, not $2$. I've just edited it.
Turzo