2008 IMO 第 5 题

Let $n$ and $k$ be positive integers with $k \geq n$ and $k - n$ an even number. There are $2n$ lamps labelled 1, 2, ..., $2n$ each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $N$ be the number of such sequences consisting of $k$ steps and resulting in the state where lamps 1 through $n$ are all on, and lamps $n + 1$ through $2n$ are all off. Let $M$ be number of such sequences consisting of $k$ steps, resulting in the state where lamps 1 through $n$ are all on, and lamps $n + 1$ through $2n$ are all off, but where none of the lamps $n + 1$ through $2n$ is ever switched on. Determine $\frac{N}{M}$ .

令 $n$ 和 $k$ 为正整数，其中 $k \geq n$ 和 $k - n$ 为偶数。有 $2n$ 灯，标记为 1、2、...、$2n$，每个灯都可以打开或关闭。最初所有灯都关闭。我们考虑步骤顺序：每一步都会切换一盏灯(从开到关或从关到开)。令 $N$ 为由 $k$ 个步骤组成的此类序列的数量，并导致灯 1 到 $n$ 全部打开，而灯 $n + 1$ 到 $2n$ 全部关闭。令 $M$ 为由 $k$ 个步骤组成的此类序列的数量，导致灯 1 到 $n$ 全部打开，灯 $n + 1$ 到 $2n$ 全部关闭，但 $n + 1$ 到 $2n$ 中没有一个灯打开。确定 $\frac{N}{M}$ 。