Prove that the polynomial \[ x^{9999}+x^{8888}+x^{7777}+\cdots+x^{1111}+1 \] is divisible by \[ x^9+x^8+x^7+\cdots+x+1 \]

 Problem: 01

Prove that the polynomial `x^9999+x^8888+x^7777+...+x^1111+1` is divisible by `x^9+x^8+x^7+....+x+1`

Solution

Find all functions \(f(x)\) satisfying: \[ (x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2) \]

 

Find all functions \(f(x)\) satisfying:

\[ (x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2) \]

Solution

Problem If $$ x+\frac{1}{x}=2, $$ find the value of $$ x^5+\frac{1}{x^5}. $$

 If \x+1/x=2\ Find the value of \x^5+1/x^5\

As we had: `x+1/x=2` `rightarrow(x+1/x)^2=4` `leftrightarrowx^2+1/x^2=2`

We continue with `(x^2+1/x^2)(x+1/x)=4` `leftrightarrowx^3+x+1/x+1/x^3=4`

                                `leftrightarrowx^3+1/x^3=2`

Vietnamese Mathematical Olympiad 2022 Let \(a\), \(b\), and \(c\) be the roots of the equation $$ x^3-x^2-1=0. $$ Find the value of $$ \frac{1}{a^{2023}}+\frac{1}{b^{2023}}+\frac{1}{c^{2023}}. $$

Let \(a\), \(b\), and \(c\) be nonzero real numbers satisfying

$$ a+b+c=2022, $$ and $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2022}. $$

Find the value of

$$ \frac{1}{a^{2023}}+\frac{1}{b^{2023}}+\frac{1}{c^{2023}}. $$

Solution