Without giving too much away, I want to say that I’m concerned that about 3/4 of the values k can that from the range [1,M^2*N^2] cannot possibly have optimal solutions. Since k is randomly chosen, how does this affect the scoring?

Whoever is the closest will get a better score. Remember that scoring is all relative, so the lowest scoring person will get a 1, and everybody else will be scaled appropriately. It’s entirely possible there are no exact solutions for many values of K.

So the scoring is done against other people?

Oh, sorry, I thought you already knew what a challenge problem was. Well, if you read on the contest page, you can find this paragraph

“Each contest will have one min/max tie breaker problem, where the

best solution will receive one point and all other solutions will receive a fraction based how close they come to the best solution”

This problem is the challenge problem. You can see on the scoreboard that there are fractional scores, which are coming from this problem.