Putnam 1964

Find all continuous positive functions $f(x)$ , for $0\leq x \leq 1$ , such that $$\int_{0}^{1} f(x) dx =1,$$ $$\int_{0}^{1} xf(x) dx =\alpha,$$ $$\int_{0}^{1} x^2 f(x) dx =\alpha^2,$$ where $\alpha$ is a given real number.

Let $p_n$ be a bounded sequence of integers which satisfies the recursion $$p_n =\frac{p_{n-1} +p_{n-2} + p_{n-3}p _{n-4}}{p_{n-1} p_{n-2}+ p_{n-3} +p_{n-4}}.$$ Show that the sequence eventually becomes periodic.

Prove that there exists a constant $K$ such that the following inequality holds for any sequence of positive numbers $a_1 , a_2 , a_3 , \ldots:$ $$\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 +\ldots + a_n } \leq K \sum_{n=1}^{\infty} \frac{1}{a_{n}}.$$

Let $S$ be a finite subset of a straight line. Say that $S$ has the repeated distance property if every value of the distance between two points of $S$ (except the longest) occurs at least twice. Show that if $S$ has the repeated distance property then the ratio of any two distances between two points of $S$ is rational.

Let $u_k$ be a sequence of integers, and let $V_n$ be the number of those which are less than or equal to $n$ . Show that if $$\sum_{k=1}^{\infty} \frac{1}{u_k } < \infty,$$ then $$\lim_{n \to \infty} \frac{ V_{n}}{n}=0.$$

Let $S$ be a set of $n>0$ elements, and let $A_1 , A_2 , \ldots A_k$ be a family of distinct subsets such that any two have a non-empty intersection. Assume that no other subset of $S$ intersects all of the $A_i.$ Prove that $k=2^{n-1}.$

Into how many regions do $n$ great circles, no three of which meet at a point, divide a sphere?

Let $u_n$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers. Prove that the series $$\sum_{n=1}^{\infty} \frac{1}{ u_n }$$ is convergent

This is rather simple, but I liked it :).

Show that a disk cannot be partitioned into two congruent subsets.