I worked out the answer easily but the method is slow.
I'll write for the first one because I don't have much time now:
a+b(-Q
b+c(-Q
a+c(-Q
ab+bc+ac+b^2(-Q
ab+bc+ac+c^2(-Q
(b-c)(b+c)(-Q
b-c(- Q Analogically we can do for each pair of numbers.
b^2-2bc+c^2-B^2-2bc-c^2(-Q
=> bc (- Q Analogically...
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...