USAMO 1993

$\mathrm{n}>1$, and $\mathrm{a}$ and $\mathrm{b}$ are positive real numbers such that $\mathrm{a}^{\mathrm{n}}-\mathrm{a}-1=0$ and $\mathrm{b}^{2 \mathrm{n}}-\mathrm{b}-3 \mathrm{a}=0$. Which is larger?

The diagonals of a convex quadrilateral meet at right angles at $X$. Show that the four points obtained by reflecting $\mathrm{X}$ in each of the sides are cyclic.

Let $\mathrm{S}$ be the set of functions $\mathrm{f}$ defined on reals in the closed interval $[0,1]$ with non-negative real values such that $\mathrm{f}(1)=1$ and $\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \leq \mathrm{f}(\mathrm{x}+\mathrm{y})$ for all $\mathrm{x}, \mathrm{y}$ such that $\mathrm{x}+\mathrm{y} \leq 1$. What is the smallest $\mathrm{k}$ such that $\mathrm{f}(\mathrm{x}) \leq$ $\mathrm{kx}$ for all $\mathrm{f}$ in $\mathrm{S}$ and all $\mathrm{x}$ ?

The sequence $a_{n}$ of odd positive integers is defined as follows: $a_{1}=r, a_{2}=s$, and $a_{n}$ is the greatest odd divisor of $a_{n-1}+a_{n-2}$. Show that, for sufficiently large $n, a_{n}$ is constant and find this constant (in terms of $r$ and $s$ ).

A sequence $\mathrm{x}_{\mathrm{n}}$ of positive reals satisfies $\mathrm{x}_{n-1} \mathrm{x}_{\mathrm{n}+1} \leq \mathrm{x}_{\mathrm{n}}{ }^{2}$. Let $\mathrm{a}_{\mathrm{n}}$ be the average of the terms $\mathrm{x}_{0}, \mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}$ and $b_{n}$ be the average of the terms $x_{1}, x_{2}, \ldots, x_{n}$. Show that $a_{n} b_{n-1} \geq a_{n-1} b_{n}$.