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

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

Determine all triangles ABC such that all of \( \frac{AB}{BC}, \frac{BC}{CA}, \frac{CA}{AB}, \frac{\angle A}{\angle B}, \frac{\angle B}{\angle C}, \frac{\angle C}{\angle A}\) are rational.

Let \(r_1,r_2,r_3,\ldots\) be a sequence of all rational numbers in \( (0,1) \) except finitely many numbers. Let \(r_j=0.a_{j,1}a_{j,2}a_{j,3}\cdots\) be a decimal representation of \(r_j\). (For instance, if \(r_1=\frac{1}{3}=0.333333\cdots\), then \(a_{1,k}=3\) for any \(k\).)

Prove that the number \(0.a_{1,1}a_{2,2}a_{3,3}a_{4,4}\cdots\) given by the main diagonal cannot be a rational number.

Determine all Platonic solids that can be drawn with the property that all of its vertices are rational points.

Find the set of all rational numbers x such that 1, cos πx, and sin πx are linearly dependent over rational field Q.

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

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

Prove that either

- a*b=max(a,b) for all rational numbers a,b, or
- 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.