USAMO 1988 Problem 3

Let $X$ be the set $\{1,2, \ldots, 20\}$ and let $\mathrm{P}$ be the set of all 9-element subsets of $\mathrm{X}$. Show that for any map f: $\mathrm{P}$ $\rightarrow X$ we can find a 10 -element subset $Y$ of $X$, such that $f(Y-\{k\}) \neq k$ for any $k$ in $Y$.