TJUSAMO CONTEST #2 RESULTS - Thursday, November 2nd, 2006

I. The raw scores:
Each problem is worth a total of seven math points. The score for each problem will also be multiplied by a style score, which is a multiple of 0.1 between 0 and 1. An X signifies that the solution for the problem was not received. If you receive a 0, then your styled score was replaced by your style multiplier.
ID--|-1-math/styled-|-2-math/styled-|-3-math/styled-|-total-math/styled---
A1--|------0/0.6----|------0/0.8----|------0/0.8----|--------- 0/0.0 -----
A2--|------5/3.5----|------1/0.7----|------X/X.X----|-(10th)-- 6/4.2 -----
A3--|------5/4.0----|------0/0.8----|------X/X.X----|-(11th)-- 5/4.0 -----
A4--|------1/0.7----|------X/X.X----|------X/X.X----|--------- 1/0.7 -----
B1--|------2/1.6----|------5/3.5----|------X/X.X----|-(8th)--- 7/5.1 -----
B2--|------6/4.8----|-----*7/4.2----|------5/4.0----|-(1st)---18/13.0-----
B3--|------1/0.8----|------0/0.7----|------0/0.8----|--------- 1/0.8 -----
C1--|------2/1.8----|-----*7/5.6*---|------X/X.X----|-(4th)--- 9/7.4 -----
C2--|------0/0.7----|------X/X.X----|------X/X.X----|--------- 0/0.0 -----
C3--|------0/0.6----|------0/0.7----|------X/X.X----|--------- 0/0.0 -----
C4--|------0/0.6----|------X/X.X----|------0/0.8----|--------- 0/0.0 -----
C5--|------5/3.5----|------0/0.8----|------0/0.8----|-(12th)-- 5/3.5 -----
C6--|------2/1.4----|------X/X.X----|------0/0.8----|--------- 2/1.4 -----
D1--|------1/0.7----|------2/1.4----|------X/X.X----|--------- 3/2.1 -----
D2--|------X/X.X----|------0/0.6----|------X/X.X----|--------- 0/0.0 -----
D3--|------2/1.8----|-----*7/4.9----|------2/1.6----|-(3rd)---11/8.3 -----
D4--|------0/0.7----|------X/X.X----|------X/X.X----|--------- 0/0.0 -----
D5--|------1/0.7----|------0/0.6----|------0/0.8----|--------- 1/0.7 -----
D6--|------3/2.7----|------2/1.6----|------X/X.X----|-(9th)--- 5/4.3 -----
E1--|------0/0.8----|------0/0.8----|------0/0.7----|--------- 0/0.0 -----
E2--|-----*7/6.3*---|------X/X.X----|------X/X.X----|-(5th)--- 7/6.3 -----
E3--|------X/X.X----|------5/3.0----|------X/X.X----|-(14th)-- 5/3.0 -----
E4--|------X/X.X----|------6/5.4----|------X/X.X----|-(7th)--- 6/5.4 -----
E5--|------2/1.4----|------0/0.8----|------0/0.9----|--------- 2/1.4 -----
F1--|------0/0.6----|------0/0.8----|------X/X.X----|--------- 0/0.0 -----
F2--|------1/0.7----|------1/0.7----|------0/0.8----|--------- 2/1.4 -----
F3--|------2/1.4----|------X/X.X----|------0/0.8----|--------- 2/1.4 -----
F4--|------0/0.8----|-----*7/5.6*---|------0/0.8----|-(6th)--- 7/5.6 -----
G1--|------2/1.8----|------X/X.X----|------0/0.7----|--------- 2/1.8 -----
G2--|------3/2.4----|------0/0.8----|------X/X.X----|-(15th)-- 3/2.4 -----
H1--|------2/1.6----|------2/1.6----|------X/X.X----|-(13th)-- 4/3.2 -----
H2--|------2/1.6----|------0/0.7----|------X/X.X----|--------- 2/1.6 -----
H3--|------2/1.6----|------X/X.X----|------X/X.X----|--------- 2/1.6 -----
I1--|------0/0.7----|------0/0.9----|------0/0.7----|--------- 0/0.0 -----
I2--|------X/X.X----|------0/0.8----|------X/X.X----|--------- 0/0.0 -----
I3--|------0/0.7----|------X/X.X----|------X/X.X----|--------- 0/0.0 -----
I4--|------1/0.6----|------X/X.X----|------X/X.X----|--------- 1/0.6 -----
J1--|------2/1.4----|------X/X.X----|------X/X.X----|--------- 2/1.4 -----
J2--|------2/1.6----|------0/0.8----|------0/0.8----|--------- 2/1.6 -----
J3--|------4/2.8----|------0/0.7----|-----*7/5.6*---|-(2nd)---11/8.4 -----
-Total contestants: 40
-Top 15 average: 5.62666


II. Score distributions:
Math|--#1--|--#2--|--#3--
7---|---1--|---4--|---1--
6---|---1--|---1--|---0--
5---|---3--|---2--|---1--
4---|---1--|---0--|---0--
3---|---2--|---0--|---0--
2---|--12--|---3--|---1--
1---|---6--|---2--|---0--
0---|--10--|--16--|--14--
X---|---4--|--12--|--23--

Note that for the following table, only solutions with nonzero math scores will be considered. The rest will be X's.
Style|--#1--|--#2--|--#3--
1.0--|---0--|---0--|---0--
0.9--|---5--|---1--|---0--
0.8--|---9--|---4--|---3--
0.7--|--11--|---6--|---0--
0.6--|---1--|---1--|---0--
0.5--|---0--|---0--|---0--
0.4--|---0--|---0--|---0--
0.3--|---0--|---0--|---0--
0.2--|---0--|---0--|---0--
0.1--|---0--|---0--|---0--
0.0--|---0--|---0--|---0--
X----|--14--|--28--|--37--


III. Overall awards:
-1st Place: [B2](13.0)
-2nd Place: [J3](8.4)
-3rd Place: [D3](8.3)
-Honorable Mentions(other people who solved a problem): [C1](7.4), [E2](6.3), [F4](5.6).

IV. Other positive awards:
-Best solution to problem #1(6.3): [E2]
-Best solutions to problem #2(5.6): [C1],[F4]
-Best solution to problem #3(5.6): [J3]
-Most interesting solution: [E5](problem #3)
-Clearest: [F4]
-Most concise: [E2]
-Best generalizations: [J3](although they weren't fully proven)
-Best rigor: [E4](but you missed a small step)
-Best diagram: [G2](this diagram disproved many people's solutions for 1-ii)

V. Not positive awards:
-Most pages submitted: [B2](9)
-Best BS: [B3](claimed answer was Yes, then attempted to prove answer was No)
-Most confusing: [A2](a lot of unnecessary casework)
-Messiest: [E3](over half of your letters are illegible)


VI. Comments:
-These results were compiled one day after the contest, as they will usually be. However, several internal factors delayed the publication date.
-The Princeton teams have been decided.
-Problem #1-i: The result is intuitive, but solutions must be rigorous. Several of you were denied points because your proof was wavy or did not prove all claims.
-Problem #1-ii: The only known way of doing this is with areas. Several of you tried to say something about lengths not matching up, but considering the following. Some proofs did not even use the information that both sides were rational, and would also apply to a 1 by sqrt(3) rectangle. Also, the following construction breaks some of your arguments. Put a triangle in the square with the hypotenuse on one of the square's sides, and rotate this triangle about the center of the square such that each side of the square is covered with a triangle. Now we have a smaller square in the middle. Repeat this process to cover this secondary square. As demonstrated, an irrational side of the triangle has been covered, and we have also reduced the size of the square.
-The fact that each score from 0 to 7 was given for problem #1 is purely coincidental.
-Problem #2: Noone found the intended solution, which involved making a bijection using a little number theory with the division algorithm.
-Problem #3: Any time you see a polynomial on an olympiad, try to factor it. The fact that this is a #3 does not mean it will use extremely advanced methods. Always try to think simply first.
-Note that just because can be false sometimes does not mean that it is false.
-FORMAT IS IMPORTANT. Leave margins, write legibly, put all the user and problem identification in the top right corner, and be concise! Check the first TJUSAMO lecture for more information and tips.