2014
Problem - 90
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.
Answer
937
We know there are only two three-digit number whose squares end with themselves: $625$ and $376$. Therefore, $N$ must end with either $625$ or $376$. There are some MOD tricks we can play to narrow down the choices, but it is not too hard to try all the possible thousands digit and find $N=9376$. Therefore the answer is $\boxed{937}$.