Putnam 1964 A3

Let $P_{1}, P_{2}, \ldots$ be a sequence of distinct points which is dense in the interval $(0,1)$ . The points $P_{1}, P_{2}, \ldots, P_{n-1}$ decompose the interval into $n$ parts, and $P_{n}$ decomposes one of these into two parts. Let $a_{n}$ and $b_{n}$ be the lengths of these two intervals. Prove that $\sum_{n=1}^{\infty} a_{n} b_{n}\left(a_{n}+b_{n}\right)=1 / 3$ (A sequence of points in an interval is said to be dense when every subinterval contains at least one point of the sequence.)

Dataset text PutnamGAP. Quoted with source.

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