Putnam 1968

Given $n$ points on the sphere $\{(x, y, z): x^2 + y^2 + z^2 = 1\}$, find the upper bound of the sum of the squares of the distances between them in terms of $n$.

The temperatures in Chicago and Detroit are $x^\circ$ and $y^\circ$, respectively. These temperatures are not assumed to be independent; namely, we are given: \begin{itemize} \item[(i)] $P(x^\circ = 70^\circ)$, the probability that the temperature in Chicago is $70^\circ$, \item[(ii)] $P(y^\circ = 70^\circ)$, and \item[(iii)] $P(\max(x^\circ, y^\circ) = 70^\circ)$. \end{itemize} Determine $P(\min(x^\circ, y^\circ) = 70^\circ)$.

Let $p$ be a prime number. Let $J$ be the set of all $2 \times 2$ matrices $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ whose entries are chosen from $\{0, 1, 2, \dots, p-1\}$ and satisfy the conditions $a+d \equiv 1 \pmod{p}$ and $ad-bc \equiv 0 \pmod{p}$. Determine how many members $J$ has.