Competition problem — Number theory

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Number theory Difficulty 5.2 AIME, harder Prove it Ukraine

Can the number $1$ be represented as a sum of $2005$ different terms, each one of the form $\frac{1}{3n-1}$ with a positive integer $n$ ?

Solution

Якщо доданки даної суми звести до спільного знаменника, то за mod $3$ чисельник матиме остачу $1$ , а знаменник — остачу $2$ . Таким чином, така сума не може дорівнювати $1$ .

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