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Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational.

I tried using 3 of those numbers - a, b and c. And I checked each of the possible situations but I'm not sure if my maths teacher is going to accept it.

The problem is from the All-Russian Olympiad 2004/2005 9th Day 2.

I tried using 3 of those numbers - a, b and c. And I checked each of the possible situations but I'm not sure if my maths teacher is going to accept it.

The problem is from the All-Russian Olympiad 2004/2005 9th Day 2.

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