USAMO 1987

Find all solutions to $\left(m^{2}+n\right)\left(m+n^{2}\right)=(m-n)^{3}$, where $m$ and $n$ are non-zero integers.

The feet of the angle bisectors of the triangle $\mathrm{ABC}$ form a right-angled triangle. If the right-angle is at $\mathrm{X}$, where $\mathrm{AX}$ is the bisector of angle $\mathrm{A}$, find all possible values for angle $\mathrm{A}$.

$X$ is the smallest set of polynomials $p(x)$ such that: (1) $p(x)=x$ belongs to $X$; and (2) if $r(x)$ belongs to $X$, then $x r(x)$ and $(x+(1-x) r(x))$ both belong to $X$. Show that if $r(x)$ and $s(x)$ are distinct elements of $X$, then $\mathrm{r}(\mathrm{x}) \neq \mathrm{s}(\mathrm{x})$ for any $0<\mathrm{x}<1$.

$\mathrm{M}$ is the midpoint of $\mathrm{XY}$. The points $\mathrm{P}$ and $\mathrm{Q}$ lie on a line through $\mathrm{Y}$ on opposite sides of $\mathrm{Y}$, such that $|\mathrm{XQ}|=$ $2|\mathrm{MP}|$ and $|\mathrm{XY}| / 2<|\mathrm{MP}|<3|\mathrm{XY}| / 2$. For what value of $|\mathrm{PY}| /|\mathrm{QY}|$ is $|\mathrm{PQ}|$ a minimum?

$\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{\mathrm{n}}$ is a sequence of $0 \mathrm{~s}$ and $1 \mathrm{~s}$. $\mathrm{T}$ is the number of triples $\left(\mathrm{a}_{i}, \mathrm{a}_{j}, \mathrm{a}_{\mathrm{k}}\right)$ with $\mathrm{i}<\mathrm{j}<\mathrm{k}$ which are not equal to $(0,1,0)$ or $(1,0,1)$. For $1 \leq i \leq n, f(i)$ is the number of $j<i$ with $a_{j}=a_{i}$ plus the number of $j>i$ with $a_{j} \neq a_{i}$. Show that $\mathrm{T}=\mathrm{f}(1)(\mathrm{f}(1)-1) / 2+\mathrm{f}(2)(\mathrm{f}(2)-1) / 2+\ldots+\mathrm{f}(\mathrm{n})(\mathrm{f}(\mathrm{n})-1) / 2$. If $\mathrm{n}$ is odd, what is the smallest value of T?