SMO: How stupidity cancels out ingenuity

OK... so, amid all the confusion over whose answers are correct and whose are wrong, I took some time to compile everything and put it up here... these answers are either obtained by the majority or just my own answers...

1) 93640 (My own answer, but likely to be correct)
Method: Split into different cases and WHACK.

2) 3 (Again, my own answer, not sure if correct, but sounds correct enough =P)
Method: Hard to describe, but there are 45 pairs of students, so 90 problems solved, but each student solves only 3, so only 30 problems actually, so means you triple-count at least 1 problem, so answer is 3. Convincing?

3) 66666 (Stupid me...)
Method: Just WHACK. Stupid me, can win $100 MAC dollars in <15mins, can't do this correctly...

4) 99999 (So lame...)
Method: Prime factorization...

5) 2008 (Confirmed)
Method: I whacked it out, but you can use recursive sequences... anyway, p(x)=x^2+10x+8.

6) 4 (I proved it just now, but stupid me, read the question wrongly)
Method: Connect CD, DE. Triangles ACB, ADC and AED are similar.

7) 135 (Proven, and I read the question wrongly again)
Method: Sine rule, and all your double angle formulas. Anyway, AD=6, the rest is easy.

8) 2 (Proven, by me again)
Method: Angles subtended by same chord are equal. Using angle-chasing, BD=ID, hence angle IBQ is right angle. Then BQ*IB = IQ*BP for obvious reasons and so on...

9) 1 (Proven as well)
Method: Triangles DEC and CEA are similar. Triangles ABC and ACD are also similar. Done.

10) 99 (Wei Quan said, seconded by me... after SMO... another question wrong again...)
Method: Listing the worse-case scenario. Which is {1, 12, 23, 34, 45, 56, 67, 78, 89, 99}. Done.

11) 2645 (Confirmed)
Method: Split into different cases that involve different positions of '4'. Then use P&C.

12) 3024 (Confirmed)
Method: P&C, girls 1st, guys later...

13) 1611 (Confirmed)
Method: The same old thing. Split into different cases and WHACK.

14) 162 (Not too sure, but should be confirmed)
Method: Same as question 13...

15) 11250 (Confirmed, ARGH I got it right, then I bloody freaking @#$%&F***ING!!! CHANGED it. Which shows the exact meaning of the title of this post... )
Method: Prime factorization. Twice.

16) 500 (Confirmed, but stupid me, added 1 for fun, now got 1 mark deducted)
Method: The general solution for this problem is to... go read your past year papers. OK, Zhe Wei says it's Legendre's theorem. Looks like he did mug.

17) 67 (May have calculation error, but otherwise should be confirmed)
Method: Binomial expansion, obviously. Find a4, find a3, solve.

18) 19 (Confirmed)
Method: Binomial expansion again. nCk + nC(k+1) = (n+1)C(k+1), and nCn = (n+1)C(n+1) = 1. So find a4, find a3, solve.

19) 6024 (Another example which shows what the title of this post means... I got the method, then I got careless, stupid me)
Method: Expand (a+b+c+d+e)^3. Simplify. Also, be reminded that -a = b+c+d+e. Done.

20) 10 (Confirmed)
Method: WHACK. The sequence repeats itself... Every now and then *evil grin*

21) 40312 (Not confirmed)
Method: Are the only numbers which do not satisfy the criteria 12345678, 23456781, 34567812, 45678123, 56781234, 67812345, 78123456 and 81234567? If so I'm done. If not I'm dead.

22) 94 (Not confirmed, but likely to be correct)
Method: Didn't exactly prove it, but it's quite clear that that the function is symmetric about the line x=2005. Hence f(x) = A(x-2005)^4 + B (x-2005)^2 + C. Find A, B and C, whack the rest and solve.

23) ??? (Anyone has any idea?)
Method: Programming perhaps?

24) 521 (Confirmed by Zhe Wei and Wei Quan)
Method: Find f(1), f(0) and f(-1). Remember that if f(x) passes through (a,b) then f-inverse(x) passes through (b,a). Whack and solve.

25) 2287 (Confirmed)
Method: List them down and whack them out.

So... your stupid C-rep always finds his mistakes AFTER he hands in his paper... now that he's at home, he can "go home cry"...

Alternatively, he should bang his head on the wall hard for being such a careless boy... Did 23, 7 wrong...

You know, it feels just like knowing the fact that you didn't win any award for the beautiful SIMC poster that you had worked on for a month...

Jackie