Problems

Find all functions $f(x)$ such that $f(2x+1)=4x^2+14x+7$.

Prove that for any natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_1,a_2,a_3,\dots$, for which $a_1^2+a_2^2+\cdots +a_k^2$ is divisible by $a_1+a_2+\cdots +a_k$ for all $k\ge 1$.

The quadratic trinomials $f(x)$ and $g(x)$ are such that $f^{\prime }(x)g^{\prime }(x)\ge |f(x)|+|g(x)|$ for all real $x$. Prove that the product $f(x)g(x)$ is equal to the square of some trinomial.

Two different numbers $x$ and $y$ (not necessarily integers) are such that $x^2-2000x=y^2-2000y$. Find the sum of $x$ and $y$.

A cubic polynomial $f(x)$ is given. Let’s find a group of three different numbers $(a,b,c)$ such that $f(a)=b$, $f(b)=c$ and $f(c)=a$. It is known that there were eight such groups $[a_i,b_i,c_i]$, $i=1,2,\dots ,8$, which contains 24 different numbers. Prove that among eight numbers of the form $a_i+b_i+c_i$ at least three are different.

The expression $a{x}^2+bx+c$ is an exact fourth power for all integers $x$. Prove that $a=b=0$.