USAMO 1977

For which positive integers $\mathrm{a}, \mathrm{b}$ does $\left(\mathrm{x}^{\mathrm{a}}+\ldots+\mathrm{x}+1\right)$ divide $\left(\mathrm{x}^{\mathrm{ab}}+\mathrm{x}^{\mathrm{ab}-\mathrm{b}}+\ldots+\mathrm{x}^{2 b}+\mathrm{x}^{\mathrm{b}}+1\right)$ ?

The triangles ABC and DEF have AD, BE and CF parallel. Show that $[\mathrm{AEF}]+[\mathrm{DBF}]+[\mathrm{DEC}]+[\mathrm{DBC}]+$ $[\mathrm{AEC}]+[\mathrm{ABF}]=3[\mathrm{ABC}]+3[\mathrm{DEF}]$, where $[\mathrm{XYZ}]$ denotes the signed area of the triangle XYZ. Thus $[\mathrm{XYZ}]$ is + area XYZ if the order $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ is anti-clockwise and - area $\mathrm{XYZ}$ if the order $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ is clockwise. So, in particular, $[\mathrm{XYZ}]=[\mathrm{YZX}]=-[\mathrm{YXZ}]$.

Prove that the product of the two real roots of $x^{4}+x^{3}-1=0$ is a root of $x^{6}+x^{4}+x^{3}-x^{2}-1=0$.

$\mathrm{ABCD}$ is a tetrahedron. The midpoint of $\mathrm{AB}$ is $\mathrm{M}$ and the midpoint of $\mathrm{CD}$ is $\mathrm{N}$. Show that $\mathrm{MN}$ is perpendicular to $\mathrm{AB}$ and $\mathrm{CD}$ iff $\mathrm{AC}=\mathrm{BD}$ and $\mathrm{AD}=\mathrm{BC}$.

The positive reals $\mathrm{v}, \mathrm{w}, \mathrm{x}, \mathrm{y}, \mathrm{z}$ satisfy $0<\mathrm{h} \leq \mathrm{v}, \mathrm{w}, \mathrm{x}, \mathrm{y}, \mathrm{z} \leq \mathrm{k}$. Show that $(\mathrm{v}+\mathrm{w}+\mathrm{x}+\mathrm{y}+\mathrm{z})(1 / \mathrm{v}+1 / \mathrm{w}+1 / \mathrm{x}$ $+1 / \mathrm{y}+1 / \mathrm{z}) \leq 25+6(\sqrt{ }(\mathrm{h} / \mathrm{k})-\sqrt{ }(\mathrm{k} / \mathrm{h}))^{2}$. When do we have equality?