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Prove for natural number $n$ there are natural numbers $k_1,k_2,...k_n$ such that all of them are greater than $1$, each pair of them are co-prime, and we can write $k_1 k_2 ... k_n - 1$ as product of two consecutive numbers.

Any hints how to prove the statement?

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This is problem 1 from USAMO 2008, and many solutions can be found here

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