Putnam 1970

Given a sequence $\{x_n\}$, $n=1, 2, \dots$, such that $\lim_{n \to \infty} \{x_n - x_{n-2}\} = 0$, find the value of \lim_{n \to \infty} \frac{x_n - x_{n-1}}{n}.

Determine the radius of the largest circle which can lie on the ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$ $(a > b > c)$.

Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i]$ $(i=1, 2, 3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen.