Problems

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 $f(\alpha (x+y))=f(x)+f(y)$?

We are given a table of size $n\times n$. $n-1$ of the cells in the table contain the number $1$. The remainder contain the number $0$. We are allowed to carry out the following operation on the table:

1. Pick a cell.

2. Subtract 1 from the number in that cell.

3. Add 1 to every other cell in the same row or column as the chosen cell.

Is it possible, using only this operation, to create a table in which all the cells contain the same number?

Solve the equation $(x+1{)}^3=x^3$.

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.

Prove that in any set of 117 unique three-digit numbers it is possible to pick 4 non-overlapping subsets, so that the sum of the numbers in each subset is the same.

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 $x$.