So USAMO happened.

#1: What is this problem. I thought about the , trick (due to something I can't reveal atm) but saw the resulting equation and passed on the opportunity because I thought nothing would make of it. Whoops. So instead I only did the case where and ; it still took three pages, and I found half the set of total solutions, so clearly I should get half the points for this problem right?
#2: Was scared to complex bash, then homothety and "all you have to do is construct a parallelogram" made it really easy. Except I made a minor typo (basically I wrote instead of in multiple places but it doesn't affect what I wanted to prove at all since things "cancelled out" anyway) so I'm not sure whether I'm getting a 6 or a 7 though.
#3: Combo + #3 = didn't attempt
#4: This problem scarred my soul. It took me an hour just to figure out the answer in the case, and then a half hour to realize the general case was easy and that I had been thinking the wrong way all that time. But then I realized I had to prove that the row and column sum invariant implied equivalency, and this is where I died. I eventually stumbled on an induction argument but I had zero time to write it so it's quite sketchy. (And also it turns out I mis-applied stars and bars so my final answer is wrong as well. This should only be -1, but still, combo is bad.)
#5: Worked on this for 30 minutes, got nowhere.
#6: Didn't attempt.

1-3/7/0/1-3/0/0=8-13 projected (but leaning more toward the 8). USAMO is fat.

Modulo ARML it looks like my years as a high school math contest contestant are over