USAMO 1988

The repeating decimal $0 . \mathrm{ab} \ldots \mathrm{kpq} \ldots \mathrm{u}=\mathrm{m} / \mathrm{n}$, where $\mathrm{m}$ and $\mathrm{n}$ are relatively prime integers, and there is at least one decimal before the repeating part. Show that $\mathrm{n}$ is divisible by 2 or 5 (or both). [For example, $0.011 \underline{36}=0.01136363636 \ldots=1 / 88$ and 88 is divisible by 2.]

The cubic $x^{3}+a x^{2}+b x+c$ has real coefficients and three real roots $r \geq s \geq t$. Show that $k=a^{2}-3 b \geq 0$ and that $\sqrt{ } \mathrm{k} \leq \mathrm{r}-\mathrm{t}$.

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 $Y$.

$A B C$ is a triangle with incenter I. Show that the circumcenters of IAB, IBC, ICA lie on a circle whose center is the circumcenter of $\mathrm{ABC}$.

Let $p(x)$ be the polynomial $(1-x)^{a}\left(1-x^{2}\right)^{b}\left(1-x^{3}\right)^{c} \ldots\left(1-x^{32}\right)^{k}$, where $a, b, \ldots, k$ are integers. When expanded in powers of $x$, the coefficient of $x^{1}$ is -2 and the coefficients of $x^{2}, x^{3}, \ldots, x^{32}$ are all zero. Find $\mathrm{k}$.