Find the smallest positive roots of the polynomials $x^{2011}+2011 x-1$ and $x^{2011}-2011 x+1$.

Compare the smallest positive roots of the polynomials $x^{2011}+2011 x-1$ and $x^{2011}-2011 x+1$.
Let $x_{1}>0$ be the root of the equation $x^{2011}+2011 x-1$, and $x_{2}>0$ be the root of the equation $x^{2011}-2011 x+1$. Then $x_{1}^{2001}+2011 x_{1}-1=0, x_{2}^{2001}-2001 x_{2}+1=0$. Adding these equalities term by term, we get $x_{1}^{2001}+x_{2}^{2001}+2001\left(x_{1}-x_{2}\right)=0$. Therefore, $x_{1}-x_{2}=-\frac{x_{1}^{2001}+x_{2}^{2001}}{2011}<0$. Thus, $x_{1}<x_{2}$.

## Answer

The smallest positive root of the first polynomial is smaller.

Authors: Folklore, Kanunnikov A.L., Sergeev I.N.

A curve on the plane in some coordinate system (Cartesian) is the graph of the function $y=\sin x$. Can the same curve be the graph of the function $y=\sin ^{2} x$ in another coordinate system: if yes, what are its origin and units of length on the axes (relative to the original coordinates and units of length)?

## Solution

Note that to solve the problem, it is sufficient to present at least one sought coordinate system. Let $y=\sin x$. Since

then for $y=2z-1$ (or, in other words, $y=2 z-1$) we get $z=\sin ^{2} t$. The origin $O^{\prime}$ of the new coordinate system $O^{\prime} t z$ has coordinates relative to the old system Oxy, and the unit of measurement on each axis of the new system is twice as large compared to the old coordinate system.

Note (prove this independently), that in the problem the following answers are possible: the new coordinate system can have a center with coordinates in the old system, where , and the unit of measurement on each axis of the new system is twice as large compared to the old one.

## Answer

Yes, it can. For example, a new coordinate system with its center having coordinates in the old system, and the unit of measurement on each axis of the new system being twice as large compared to the old one, fits.