[Bezout's Theorem. Factorization]

What remainder does $x+x^{3}+x^{9}+x^{27}+x^{81}+x^{243}$ give when divided by $x-1$?
By the theorem of Bezout, this remainder is equal to $P(1)=6$.

## Answer

6. 

What is greater: $\frac{2.00000000004}{(1.00000000004)^{2}+2.00000000004}$ or $\frac{2.00000000002}{(1.00000000002)^{2}+2.00000000002}$?

## Solution

Let $a=1.00000000004$ and $b=1.00000000002$. Then $a>b$, and the expressions under consideration are equal to $\frac{1+a}{1+a+a^{2}}$ and $\frac{1+b}{1+b+b^{2}}$. It remains to note that

$\frac{1+b}{1+b+b^{2}}-\frac{1+a}{1+a+a^{2}}=\frac{(a-b)(a+b+a b)}{\left(1+a+a^{2}\right)\left(1+b+b^{2}\right)}>0$

## Answer

The second expression is greater.