RO-NMO 2004 Problem 103

Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$ . Prove that: $$2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} .$$

Radu Gologan