USAMO 1985

Do there exist 1985 distinct positive integers such that the sum of their squares is a cube and the sum of their cubes is a square?

Find all real roots of the quartic $\mathrm{x}^{4}-(2 \mathrm{~N}+1) \mathrm{x}^{2}-x+\mathrm{N}^{2}+N-1=0$ correct to 4 decimal places, where $\mathrm{N}=$ $10^{10}$.

A tetrahedron has at most one edge longer than 1 . What is the maximum total length of its edges?

A graph has $n>2$ points. Show that we can find two points $A$ and $B$ such that at least $[n / 2]-1$ of the remaining points are joined to either both or neither of $\mathrm{A}$ and $\mathrm{B}$.

$0<a_{1} \leq a_{2} \leq a_{3} \leq \ldots$ is an unbounded sequence of integers. Let $b_{n}=m$ if $a_{m}$ is the first member of the sequence to equal or exceed $n$. Given that $a_{19}=85$, what is the maximum possible value of $a_{1}+a_{2}+\ldots+$ $\mathrm{a}_{19}+\mathrm{b}_{1}+\mathrm{b}_{2}+\ldots+\mathrm{b}_{8} ?$