Friday, December 23, 2022

Prove that the polynomial x9999+x8888+x7777+...+x1111+1x9999+x8888+x7777+...+x1111+1 is divisible by x9+x8+x7+....+x+1x9+x8+x7+....+x+1

 Problem: 01

Prove that the polynomial x9999+x8888+x7777+...+x1111+1x9999+x8888+x7777+...+x1111+1 is divisible by x9+x8+x7+....+x+1x9+x8+x7+....+x+1

Solution


Thursday, December 22, 2022

Find all function f(x)f(x) if (x-y)f(x+y)-(x+y)f(x-y)=4xy(x2-y2)(xy)f(x+y)(x+y)f(xy)=4xy(x2y2)

 Find all function f(x)f(x) if (x-y)f(x+y)-(x+y)f(x-y)=4xy(x2-y2)(xy)f(x+y)(x+y)f(xy)=4xy(x2y2)

Solution

Sunday, September 11, 2022

If x+1x=2x+1x=2 Find the value of x5+1x5x5+1x5

 If x+1x=2x+1x=2 Find the value of x5+1x5x5+1x5

As we had: x+1x=2x+1x=2 (x+1x)2=4(x+1x)2=4 x2+1x2=2x2+1x2=2

We continue with (x2+1x2)(x+1x)=4(x2+1x2)(x+1x)=4 x3+x+1x+1x3=4x3+x+1x+1x3=4

                                x3+1x3=2x3+1x3=2

Friday, September 9, 2022

Vietnamese Olympiad 2022: Find the value of 1a2023+1b2023+1c20231a2023+1b2023+1c2023

     If a+b+c=2022a+b+c=2022 and 1a+1b+1c=120221a+1b+1c=12022 Find the value of 1a2023+1b2023+1c20231a2023+1b2023+1c2023

Solution