USAMO 1991

An obtuse angled triangle has integral sides and one acute angle is twice the other. Find the smallest possible perimeter.

For each non-empty subset of $\{1,2, \ldots, n\}$ take the sum of the elements divided by the product. Show that the sum of the resulting quantities is $n^{2}+2 n-(n+1) s_{n}$, where $s_{n}=1+1 / 2+1 / 3+\ldots+1 / n$.

Define the function $\mathrm{f}$ on the natural numbers by $\mathrm{f}(1)=2, \mathrm{f}(\mathrm{n})=2^{\mathrm{f}(\mathrm{n}-1)}$. Show that $\mathrm{f}(\mathrm{n})$ has the same residue mod $m$ for all sufficiently large $n$.

$\mathrm{a}$ and $\mathrm{b}$ are positive integers and $\mathrm{c}=\left(\mathrm{a}^{\mathrm{a}+1}+\mathrm{b}^{\mathrm{b}+1}\right) /\left(\mathrm{a}^{\mathrm{a}}+\mathrm{b}^{\mathrm{b}}\right)$. By considering $\left(\mathrm{x}^{\mathrm{n}}-\mathrm{n}^{\mathrm{n}}\right) /(\mathrm{x}-\mathrm{n})$ or otherwise, show that $c^{a}+c^{b} \geq a^{a}+b^{b}$.

$\mathrm{X}$ is a point on the side $\mathrm{BC}$ of the triangle $\mathrm{ABC}$. Take the other common tangent (apart from $\mathrm{BC}$ ) to the incircles of ABX and ACX which intersects the segments AB and AC. Let it meet AX at Y. Show that the locus of $\mathrm{Y}$, as $\mathrm{X}$ varies, is the arc of a circle.