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
Find all function f(x)f(x) if (x-y)f(x+y)-(x+y)f(x-y)=4xy(x2-y2)(x−y)f(x+y)−(x+y)f(x−y)=4xy(x2−y2)
Solution
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=2↔x2+1x2=2
We continue with (x2+1x2)(x+1x)=4(x2+1x2)(x+1x)=4 ↔x3+x+1x+1x3=4↔x3+x+1x+1x3=4
↔x3+1x3=2↔x3+1x3=2
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