As you may know, there is a thread for Problem 7 that lists all the important results discovered by CrowdMath for that problem. I thought it might be helpful to have a similar thread for the rest of the problems. Please add all important discoveries that have been made that are topical to Problem 8 here. Make sure to say where the theorems were proved and by whom. There are also threads for Problems 1-7 and 9. Thanks!