BALKAN-SL 2016 C1

Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_{1}, b_{2}, \ldots, b_{K}$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_{1}, b_{2}, \ldots, b_{K}$.