Results (top 5):
-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

Rank |    Name        |  #1 |  #2 |  #3 | Total
1.0  | Brian Hamrick  | 4.9 | X.X | 5.6 | 10.5
2.0  | Akshar Wunnava | 5.6 | 4.2 | X.X | 9.9
3.0  | Daniel Li      | 6.3 | X.X | X.X | 6.3
4.0  | Luke Cheng     | 5.6 | X.X | X.X | 5.6
5.5  | Oleg Lazarev   | 4.9 | X.X | X.X | 4.9
5.5  | Casey Mihaloew | 4.9 | 0.0 | 0.0 | 4.9

Notes: 
-If all 20 or so of you worked together, your combined score would be 16.1! Congratulations, that's pretty good, as these problems were nontrivial.
-It happens that the only math scores given in this contest were 0,1,6,7.
#1:
-Remember that inductions need base cases. For this problem, some of you tried to make your base case n=11 and ended up doing a lot of unnecessary computation. Start with smaller numbers! The inequality holds for n>2. I just put n>10 so that different approaches could work out without you having to make additional cases for small numbers.
-Remember to write inequalities FORWARDS, even if you solved it backwards. That means start with true statements, and make logical steps to arrive at the desired result.
#2: 
-Not much progress here...
-Note that CEYX is a cyclic quad, which solves the problem pretty quickly.
-If you use coordinates, make sure to show your work. Realize that obviously since the problem statement is true, of course using coordinates will eventually work out... but you have to show all your nasty algebra in your proof.
#3:
-Note that the inequality is not symmetric in its variables, so you cannot assume that a>=b>=c. However, it is cyclic, so you can assume a>=b,c.
-The substitution a=x+y, b=y+z, c=z+x is natural for inequalities involving the sides of a triangle, and it definitely works for this problem.
-This problem has many, many solutions. The one included in the solutions just happens to be the most elegant, albeit the least motivated. I did not come up with it.