Berkeley Math Circle #0hx4

Berkeley Math CircleHard

From module Telescoping Sums and Products

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Recurrence relationsTelescoping seriesChebyshev polynomials

Problem

Problem: Let $a > 2$ be given, and define a sequence $a_{0}, a_{1}, a_{2}, \ldots$ by

a_{0} = 1, \quad a_{1} = a, \quad a_{n+1} = \left(\frac{a_{n}^{2}}{a_{n-1}^{2}} - 2\right) \cdot a_{n} .

Show that for all integers $k \geq 0$ , we have

\sum_{i=0}^{k} \frac{1}{a_{i}} < \frac{a+2-\sqrt{a^{2}-4}}{2} .

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