Determine whether or not there are any positive integral solutions of the simultaneous equations \begin{align*}x_1^2+x_2^2+\cdots+x_{1985}^2&=y^3,\\ x_1^3+x_2^3+\cdots+x_{1985}^3&=z^2\end{align*}with distinct integers $x_1$, $x_2$, $\ldots$, $x_{1985}$.

Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\]correct to four decimal places.

Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$. Determine the maximum value of the sum of the six distances.

There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of \[a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}.\]

These problems are copyright $\copyright$ Mathematical Association of America.