For a prime number \(p\) denote by \(\lfloor\frac{p}{2}\rfloor\) the element of \(\mathbb{Z}/p\mathbb{Z}\), which corresponds to the largest integer which is smaller than \(\frac{p}{2}\). Prove that all the elements \(\{1, 2^2, 3^2, ..., \lfloor\frac{p}{2}\rfloor^2\}\) are different in \(\mathbb{Z}/p\mathbb{Z}\).