IV OBM

The key is to consider how many squares have their vertices on the perimeter of a given $(n+1)\times (n+1)$ array whose sides are parallel to the sides of the array.



The diagram shows that there are $n$ such squares. There are $(k+1-n{)}^2$ such arrays. So the total number of squares is $k\cdot 1^2+(k-1{)}^2\cdot 2+\cdots +1\cdot k^2={\sum }_1^k(k+1-i)i^2=\frac{(k+1)k(k+1)(2k+1)}{6}-\frac{k^2(k+1{)}^2}{4}=\frac{k(k+1)2(k+2)}{12}$.