Problems

Is there a bounded function \(f\colon \mathbb{R} \rightarrow \mathbb{R}\) such that \(f (1)> 0\) and \(f (x)\) satisfies the inequality \(f^2 (x + y) \geq f^2 (x) + 2f (xy) + f^2 (y)\) for all \(x, y \in \mathbb{R}\)?

Ten pairwise distinct non-zero numbers are such that for each two of them either the sum of these numbers or their product is a rational number.

Prove that the squares of all numbers are rational.

The polynomial \(P (x)\) of degree \(n\) has \(n\) distinct real roots.

What is the largest number of its coefficients that can be equal to zero?

For which \(\alpha\) does there exist a function \(f\colon \mathbb{R} \rightarrow \mathbb{R}\) that is not a constant, such that \(f (\alpha (x + y)) = f (x) + f (y)\)?

On a function \(f (x)\) defined on the whole line of real numbers, it is known that for any \(a > 1\) the function \(f (x)\) + \(f (ax)\) is continuous on the whole line. Prove that \(f (x)\) is also continuous on the whole line.

Does there exist a function \(f (x)\) defined for all \(x \in \mathbb{R}\) and for all \(x, y \in \mathbb{R}\) satisfying the inequality \(| f (x + y) + \sin x + \sin y | < 2\)?

We call a number \(x\) rational if it can be represented as \(x=\frac{p}{q}\) for coprime integers \(p\) and \(q\). Otherwise we call the number irrational.
Non-zero numbers \(a\) and \(b\) satisfy the equality \(a^2b^2 (a^2b^2 + 4) = 2(a^6 + b^6)\). Prove that at least one of them is irrational.

The real numbers \(x\) and \(y\) are such that for any distinct prime odd \(p\) and \(q\) the number \(x^p + y^q\) is rational. Prove that \(x\) and \(y\) are rational numbers.

The functions \(f (x) - x\) and \(f (x^2) - x^6\) are defined for all positive \(x\) and increase. Prove that the function

also increases for all positive \(x\).

The sum of the positive numbers \(a, b, c\) is \(\pi / 2\). Prove that \(\cos a + \cos b + \cos c > \sin a + \sin b + \sin c\).