Problems

Is there a bounded function $f:\mathbb{R}\to \mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies the inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$ 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 $f:\mathbb{R}\to \mathbb{R}$ 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 $a^2{b}^2(a^2{b}^2+4)=2(a^6+b^6)$. 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 $\cos a+\cos b+\cos c>\sin a+\sin b+\sin c$.