1970 IMO Problems/Problem 3

Problem

The real numbers $a_0, a_1, \ldots, a_n, \ldots$ satisfy the condition:

$1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots$ .

The numbers $b_{1}, b_{2}, \ldots, b_n, \ldots$ are defined by

$b_n = \sum_{k=1}^{n} \left( 1 - \frac{a_{k-1}}{a_{k}} \right)$

(a) Prove that $0 \leq b_n < 2$ for all $n$ .

(b) given $c$ with $0 \leq c < 2$ , prove that there exist numbers $a_0, a_1, \ldots$ with the above properties such that $b_n > c$ for large enough $n$ .

Solution

Contradiction to (a): Let $a_{k}=k$ . Thus $b_n = \sum_{k=1}^{n} \left( 1 - \frac{k-1}{k} \right)=\sum_{k=1}^{n} \frac{1}{k}$ and that sum tends to infinity as $k$ tends to infinity.

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1970 IMO (Problems) • Resources
Preceded by Problem 2	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 4
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1970 IMO Problems/Problem 3

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