TSTST 2026 Problem 5

Let $\Omega(k)$ denote the number of prime factors of $k$, counted with multiplicity. For example, $\Omega(20) = 3$, because $20 = 2 \cdot 2 \cdot 5$. Prove that for all positive integers $n$, there exists a positive integer $k$ strictly between $n^2$ and $(n+1)^2$ such that $\Omega(k)$ is odd.