Algebra Problem No.6

The inequality is:

$\displaystyle (2^n + 3^n)^{2020} < (2^{2020} + 3^{2020})^n$

$\displaystyle \iff 2020\ln{(2^n + 3^n)} < n\ln{(2^{2020} + 3^{2020})}$

$\displaystyle \iff \frac{\ln{(2^n + 3^n)}}{n} < \frac{\ln{(2^{2020} + 3^{2020})}}{2020}$

We have the function $\displaystyle f(x) = \frac{\ln{(2^x + 3^x)}}{x}$ is a monotonically decreasing function.

So $\displaystyle f(n) < f(2020) \iff n > 2020 \implies n \in [2021;9999] \implies$ there are $\blue{\boxed{7979}}$ numbers.