2008 IMO Shortlist N4

Let $n$ be a positive integer. Show that the numbers $$\left(\begin{array}{c} 2^{n}-1 \\ 0 \end{array}\right), \quad\left(\begin{array}{c} 2^{n}-1 \\ 1 \end{array}\right), \quad\left(\begin{array}{c} 2^{n}-1 \\ 2 \end{array}\right), \quad \ldots, \quad\left(\begin{array}{c} 2^{n}-1 \\ 2^{n-1}-1 \end{array}\right)$$ are congruent modulo $2^{n}$ to $1,3,5, \ldots, 2^{n}-1$ in some order.

令 $n$ 为正整数。显示数字 $$\left(\begin{array}{c} 2^{n}-1 \\ 0 \end{array}\right), \quad\left(\begin{array}{c} 2^{n}-1 \\ 1 \end{array}\right), \quad\left(\begin{array}{c} 2^{n}-1 \\ 2 \end{array}\right), \quad \ldots, \quad\left(\begin{array}{c} 2^{n}-1 \\ 2^{n-1}-1 \end{array}\right)$$ 与 $2^{n}$ 到 $1,3,5, \ldots, 2^{n}-1$ 以某种顺序全等。