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Problem 812

AMC 12 late, AIME early

Number theory Difficulty 4.5 Prove it Irish Mathematical Olympiad · Ireland

Prove that
$\frac{2}{3} + \frac{4}{5} + \frac{6}{7} + \dots + \frac{2010}{2011}$
is not an integer.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let $S$ be the sum. Then
$1005 - S = \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \dots + \frac{1}{2011} = T.$
Then $S$ is an integer if and only if $T$ is an integer. Let $M = 3 \cdot 5 \cdot 7 \cdot 9 \dots 2009$ . If $T$ is an integer then $MT$ is an integer.
$MT = \frac{M}{3} + \frac{M}{5} + \frac{M}{7} + \frac{M}{9} + \dots + \frac{M}{2009} + \frac{M}{2011}.$
Each term is an integer except the last one which is not since $2011$ is prime. Hence $T$ is not an integer.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.