This week's Friday challenge is the following problem aimed at high school students. A solution to this problem will be posted next week.
Let there be several numbers written on a blackboard. It is allowed to erase any two of them, which are not simultaneously equal to 0, say $a$ and $b$ and replace them with $a - b/2$ and $b+a/2$. Is it possible after several of such steps to arrive at the original numbers?
Hint: Check what happens to the sum of squares of these numbers.