Preliminary results: 
-Scores are the product of your math score (0, 1, 2, 3, 4, 5, 6, or 7) and your style score (.3, .4, .5, .6, .7, .8, .9. or 1).
-All problems are equally weighted.
-X.X means not received

                       #1    #2    #3    Total
1st: Casey Mihaloew  | 5.6 | 4.8 | 5.6 | 16.0
2nd: Daniel Li       | 6.3 | 1.8 | 6.3 | 14.4
3rd: Akshar Wunnava  | 6.3 | 1.8 | 5.6 | 13.7
4th: Brian Hamrick   | 6.3 | 1.6 | 5.6 | 13.5
5th: Bruce Sun       | 6.3 | 0.7 | 5.6 | 12.6
6th: Divya Garg      | 4.9 | 1.2 | 5.6 | 11.7
8th: Oleg Lazarev    | 4.9 | 1.4 | 4.9 | 11.2
8th: Keeyoung Lee    | 5.6 | X.X | 5.6 | 11.2
8th: Emma Pierson    | 4.9 | 1.4 | 4.9 | 11.2
10th: Luke Cheng     | 5.6 | 0.5 | 4.9 | 11.0
11th: Sam Rush       | 6.3 | 1.8 | X.X | 8.1
12th: Renjie You     | 6.3 | 1.2 | X.X | 7.5
13th: Aryan Khojandi | 5.6 | 1.6 | X.X | 7.2

Notes on partial scoring:
#1)
+2 pts for mentioning trivial inequality
+2 pts for mentioning that differences of squares increase
+3 pts for mentioning squares are 0 or 1 in mod 3 or mod 4 
+4 pts for correct sequence
#2)
a) 2 pts total
+1 pt for showing divisibility by 20 or 50
+1 pt for any useful factorization
+1 pt for handwavy attempt at induction
-style pts for excessive casework
b) 5 pts total
+1 pt for mentioning difference of squares
+4 pts for the factorization
#3)
+1 pt for mentioning that cyclic quadrilaterals have supplementary angles
+1 pt for mentioning angle chasing

Notes not on partial scoring:
-Overall, there were several high scores. Keep up the good work.
-Note that I was more lenient with style scores this time. Clearer and more rigorous proofs are appreciated.
#1)
-It's better to just give a proof by example. Just construct a sequence of length n and prove that it works.
-Be concise! You don't have to purposely lengthen a proof that should be extremely short.
-Some people who turned in other problems didn't turn in a #1, which is quite strange. 
#2)
-Most people looked at the units' digit and did a bunch of casework. You guys should learn how to use mods. 
-Also, you don't have to show your work when expanding polynomials or multiplying numbers.
-Splitting into divisible by 4 and divisible by 25 seemed to be the most popular solution for part a.
#3)
-Pretty much everybody got a 0 or a 7 on this problem, since it was basically testing if you knew how to deal with cyclic quadrilaterals.
-When you write out an angle chase, it's better to write the whole chase at once. See the solutions for details.
-Submitting just the given diagram won't ever get you any points. Make some observations to try to get partial credit.