Problems

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any $k=1,2,3,\dots$ the sum of the first $k$ terms of the sequence is divisible by $k$?

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Prove that for any positive integer $n$ the inequality

is true.

Find all the functions $f:\mathbb{R}\to \mathbb{R}$ which satisfy the inequality $f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)$ for all $x,y,z$.

Find the sum $1/3+2/3+2^2/3+2^3/3+\cdots +2^{1000}/3$.

A number set $M$ contains $2003$ distinct positive numbers, such that for any three distinct elements $a,b,c$ in $M$, the number $a^2+bc$ is rational. Prove that we can choose a natural number $n$ such that for any $a$ in $M$ the number $a\sqrt{n}$ is rational.

A numeric set $M$ containing 2003 distinct numbers is such that for every two distinct elements $a,b$ in $M$, the number $a^2+b\sqrt{2}$ is rational. Prove that for any $a$ in $M$ the number $q\sqrt{2}$ is rational.

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?