IMC 2024 · Problem 2

For $n = 1, 2, \ldots$ let

$$S_n = \log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \ldots \cdot n^n}\right) - \log(\sqrt{n}),$$

where $\log$ denotes the natural logarithm. Find $\lim\limits_{n \to \infty} S_n$.