2. Find all positive solutions of the equation

$$
x^{101} + 100^{99} = x^{99} + 100^{101}
$$

Do not forget to justify your answer.
Solution. The equation is reduced to the form

$$
x^{99}\left(x^{2}-1\right)=100^{99}\left(100^{2}-1\right)
$$

Obviously, the solution is $x=100$. For $0<x<1$, the left side is less than the right side. For $x>1$, the left side is an increasing function, so the required value is taken no more than once. Therefore, there are no other roots. Answer. 100.