The maximum of $\binom{n}{x+1}-\binom{n}{x}$ The following question comes from an American Olympiad problem. The reason why I am posting it here is that, although it seems really easy, it allows for some different and really interesting solutions. Do you want to give a try?
Let $n$ be one million. Find the maximum value attained by $\binom{n}{x+1}-\binom{n}{x}$, where $0<x<n$ is an integer. 

Edit: I saw the answers below, and I can say that I posted this one because there is at least one really nice solution, which is not so "mechanical" :) The value of $n$ has no particular meaning, it stands just for a "sufficiently large integer"..

$$t_x=\binom n{x+1}-\binom nx=\frac{(n-2x-1)}{(x+1)}\binom nx$$
$$\frac{t_x}{t_{x-1}}=\frac{n-x+1}{(x+1)(n-2x+1)(n-2x)}=z$$
If $z\ge1$:
$$\frac{n-x+1}{(x+1)(n-2x+1)(n-2x)}\ge1\implies x\le z_0(\text{say})$$
Then maximum occurs at :
$$x=\lfloor z_0\rfloor$$
