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Estonia — Open Contests

Number theory Difficulty 6.6 National Olympiad Prove it Estonia

Jüri writes on blackboard some consecutive integers. It is known that the total number of these integers is greater than one and the least of them is greater than $2$ . Mari writes on blackboard consecutive integers, too, with the same total number of them as Jüri, but the least of them equals $1$ . Is it possible that the product of the integers written by Jüri divided by the product of the integers written by Mari is equal to the square of some integer?

Solution

For example, the product $8 \cdot 9$ of two consecutive integers divided by the product $1 \cdot 2$ of the first two positive integers equals $6^2$ .

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