HMMT February 2019 #0k64

HMMT February 2019Hard

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.

Vieta's formulasRecurrence relationsFloors and ceilingsTelescoping seriesInfinite descent / root flipping

Problem

Problem:

The sequence of integers $\{a_{i}\}_{i=0}^{\infty}$ satisfies $a_{0}=3$ , $a_{1}=4$ , and

a_{n+2}=a_{n+1} a_{n}+\left\lceil\sqrt{a_{n+1}^{2}-1} \sqrt{a_{n}^{2}-1}\right\rceil

for $n \geq 0$ . Evaluate the sum

\sum_{n=0}^{\infty}\left(\frac{a_{n+3}}{a_{n+2}}-\frac{a_{n+2}}{a_{n}}+\frac{a_{n+1}}{a_{n+3}}-\frac{a_{n}}{a_{n+1}}\right) .

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