Prove that the polynomial `x^9999+x^8888+x^7777+...+x^1111+1` is divisible by `x^9+x^8+x^7+....+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 function `f(x)` if `(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)`

 Find all function `f(x)` if `(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)`

Solution

If `x+1/x=2` Find the value of `x^5+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 Olympiad 2022: Find the value of `1/(a^2023)+1/(b^2023)+1/(c^2023)`

     If `a+b+c=2022` and `1/a+1/b+1/c=1/2022` Find the value of `1/(a^2023)+1/(b^2023)+1/(c^2023)`

Solution