RMO Practice Problems - Number Theory #3

Number Theory Level 5

Let \((p_{i}, q_{i}, r_{i})\) be the ordered triple of primes that satisfy the below condition

\[p_{i}|q_{i} ^{r_{i}} +1 \quad \quad \quad q_{i}|r_{i} ^{p_{i}} +1 \quad \quad \quad r_{i}|p_{i} ^{q_{i}}+ 1\]

Find the value of \(\sum_{i=1}^{n} p_{i} + q_{i} + r_{i}\) where \(n\) denotes the number of solutions.

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