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a nice problem

Source: TST Romania 2003, Fourth Round - Created by Valentin Vornicu

February 17, 2004
modular arithmeticpigeonhole principleinequalitiesnumber theoryrelatively primenumber theory solved

Problem Statement

Let P\mathcal{P} be the set of all primes, and let MM be a subset of P\mathcal{P}, having at least three elements, and such that for any proper subset AA of MM all of the prime factors of the number −1+∏p∈Ap -1+\prod_{p\in A}p are found in MM. Prove that M=PM= \mathcal{P}.
Valentin Vornicu
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