A numeric set \(M\) containing 2003 distinct numbers is such that for every two distinct elements \(a, b\) in \(M\), the number \(a^2+ b\sqrt 2\) is rational. Prove that for any \(a\) in \(M\) the number \(q\sqrt 2\) is rational.
The number \(x\) is such a number that exactly one of the four numbers \(a = x - \sqrt{2}\), \(b = x-1/x\), \(c = x + 1/x\), \(d = x^2 + 2\sqrt{2}\) is not an integer. Find all such \(x\).
The numbers \(x\), \(y\) and \(z\) are such that all three numbers \(x + yz\), \(y + zx\) and \(z + xy\) are rational, and \(x^2 + y^2 = 1\). Prove that the number \(xyz^2\) is also rational.
Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).