Question 30
Natural numbers $a_{1}, a_{2}, \ldots, a_{2^{2016}}$ are given such that for all $n$, $1 \leqslant n \leqslant 2^{2016}$, we have
$$
a_{n} \leqslant 2016 \quad \text{and} \quad a_{1} a_{2} \cdots a_{n}+1 \text{ is a perfect square.}
$$
Prove that some of the numbers $a_{1}, a_{2}, \ldots, a_{2^{2016}}$ is equal to 1. (Dusan Djukic)
$$
a_{n} \leqslant 2016 \quad \text{and} \quad a_{1} a_{2} \cdots a_{n}+1 \text{ is a perfect square.}
$$
Prove that some of the numbers $a_{1}, a_{2}, \ldots, a_{2^{2016}}$ is equal to 1. (Dusan Djukic)