Problems

Prove that for any positive integer $n$ the inequality

is true.

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 $x$.

Prove that if the numbers $x,y,z$ satisfy the following system of equations for some values of $p$ and $q$: $$\begin{array}{rl}y& =x^2+px+q,\\ z& =y^2+py+q,\\ x& =z^2+pz+q,\end{array}$$ then the inequality $x^{2}y+y^{2}z+z^{2}x\ge x^{2}z+y^{2}x+z^{2}y$ is satisfied.

Let $p$ and $q$ be positive numbers where $1/p+1/q=1$. Prove that $$a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n\le (a_1^p+\dots a_n^p{)}^{1/p}(b_1^q+\cdots +b_n^q{)}^{1/q}$$ The values of the variables are considered positive.

Solve the inequality: $\lfloor x\rfloor \times \{x\}<x-1$.