Find all polynomials \( P \) with real coefficients such that \( P(x) \in \mathbb{Q} \) implies \( x \in \mathbb{Q} \).

Suppose that * is an associative and commutative binary operation on the set of rational numbers such that 

1. 0*0=0
2. (a+c)*(b+c)=(a*b)+c for all rational numbers a,b,c.

Prove that either

1. a*b=max(a,b) for all rational numbers a,b, or
2. a*b=min(a,b) for all rational number a,b.

Let n>1 be an integer and let x>1 be a real number. Prove that if
\(\sqrt[n]{x+\sqrt{x^2-1}}+\sqrt[n]{x-\sqrt{x^2-1}}\)
is a rational number, then x is rational.

Let a, b, c, d be positive rational numbers. Prove that if \(\sqrt a+\sqrt b+\sqrt c+\sqrt d\) is rational, then each of \(\sqrt a\), \(\sqrt b\), \(\sqrt c\), and \(\sqrt d\) is rational.