MUMC 2010!

[the.watchman]

Hehe, I stole the question sheet, here are the questions:

(1) Let s be the number of snaps required to break an chocolate bar (ie. it consists of nm squares) into individual squares (of size ). Each snap can only be applied to one disconnected piece at a time. Find all possible values of s.

(2) Find all the arrangements of the numbers around a circle so that the difference between any two numbers next to each other is at most 2.

(3) For positive integers , if , prove that .

(4) Prove that among any 18 consecutive positive three-digit numbers, there must be at least one number divisible by the sum of its digits.

(5) Let N be a number whose digits when read from left to right form a strictly increasing sequence. The numbers 234 and 13479 are examples of such numbers. Find all possible values of the sum of the digits of 9N.

(6) Let ABCD be a convex quadrilateral (a quadrilateral is convex if and only if the two diagonals intersect at a point that is in the interior of the quadrilateral). Let diagonals AC and BD meet at the point X and let AB meet CD at Y. Suppose that triangles ABC, BCD, AXD have equal area. Prove that triangles YBC and ABD have equal area.

(7) In the equation: ,
both c and n are integers, and p is a prime. Are there any solutions to this equation? If so, find them. If not, prove that there are no solutions.

( A polyhedron is a geometric solid in three dimensions with flat faces and straight edges. If the faces are painted black and white, such that every white face is surrounded by black faces (surrounding faces must share an edge with the faces they surround), but the total area of the white faces exceeds the total area of the black faces, prove that one can't inscribe a sphere in the polyhedron.

[luken93]

q1.
This is correct isn't it?

[the.watchman]

Eh, I got 1, 2 and 4, got half answers for 3 and 5 (including some made up premises...), wrote bs for 6 and 7 and didn't touch 8.  Spent the last ten minutes writing out the "What a piece of work is a man!" speech on my last page.

Colin Lu got all 8 questions fully.  =)

Wow, that guy must be amazing...
I almost did as much as you, but my techniques were horrid (almost resorted to 'proof by elimination of impossible cases'...)

q1. [tex](m \times n) - 1 = s
This is correct isn't it?

Absolutely

[luken93]

Eh, I got 1, 2 and 4, got half answers for 3 and 5 (including some made up premises...), wrote bs for 6 and 7 and didn't touch 8.  Spent the last ten minutes writing out the "What a piece of work is a man!" speech on my last page.

Colin Lu got all 8 questions fully.  =)

Wow, that guy must be amazing...
I almost did as much as you, but my techniques were horrid (almost resorted to 'proof by elimination of impossible cases'...)

q1. [tex](m \times n) - 1 = s
This is correct isn't it?

Absolutely

woohoo, that was my goal, get one question right

[the.watchman]

I am interested to know:
Considering that I knew the answer for one of the questions, but had no idea about the process, would I get any marks for just putting a bogus explanation and the answer? I thought of doing that, but in the end I just left that question blank...

[luken93]

I am interested to know:
Considering that I knew the answer for one of the questions, but had no idea about the process, would I get any marks for just putting a bogus explanation and the answer? I thought of doing that, but in the end I just left that question blank...

well thats what i thought, but then if they say prove that im guessing you need some sort of equation or similar to prove it?
also, was q5 all sum to equal 9? i showed 3 random numbers and explained why i used them, however i doubt i will get full marks.

[the.watchman]

I am interested to know:
Considering that I knew the answer for one of the questions, but had no idea about the process, would I get any marks for just putting a bogus explanation and the answer? I thought of doing that, but in the end I just left that question blank...

also, was q5 all sum to equal 9? i showed 3 random numbers and explained why i used them, however i doubt i will get full marks.

That's the exact question I was thinking of
It's pretty obvious that it's 9, but how to prove it for all cases?
So I attempted a full proof and failed dismally

[luken93]

I am interested to know:
Considering that I knew the answer for one of the questions, but had no idea about the process, would I get any marks for just putting a bogus explanation and the answer? I thought of doing that, but in the end I just left that question blank...

also, was q5 all sum to equal 9? i showed 3 random numbers and explained why i used them, however i doubt i will get full marks.

That's the exact question I was thinking of
It's pretty obvious that it's 9, but how to prove it for all cases?
So I attempted a full proof and failed dismally

wat did that collin lu write then evangelion?

[EvangelionZeta]

I am interested to know:
Considering that I knew the answer for one of the questions, but had no idea about the process, would I get any marks for just putting a bogus explanation and the answer? I thought of doing that, but in the end I just left that question blank...

also, was q5 all sum to equal 9? i showed 3 random numbers and explained why i used them, however i doubt i will get full marks.

That's the exact question I was thinking of
It's pretty obvious that it's 9, but how to prove it for all cases?
So I attempted a full proof and failed dismally

wat did that collin lu write then evangelion?

No idea - didn't ask.  D:

[AzureBlue]

Hehe, I stole the question sheet, here are the questions:

Thanks Wow, 5/8 are proof questions and no chessboard/domino combinatorics question this year for the first time in ages. Actually looks like a quite accessible paper though, I'll sit the paper this weekend and see how I go.

Also, was the question sheet released at 1pm (ie. you can take it home anytime after 1pm today) or is it released in 3 days? LOL for AMO, the questions are banned for discussion for a whole month 

[luken93]

Hehe, I stole the question sheet, here are the questions:

Thanks Wow, 5/8 are proof questions and no chessboard/domino combinatorics question this year for the first time in ages. Actually looks like a quite accessible paper though, I'll sit the paper this weekend and see how I go.

Yeh compared to last year it seemed alot easier to understand, but alot harder to justify you reasons

[appianway]

I don't think it was different mathematically (although I didn't sit last year's senior so I can't compare), but rather, you didn't need much prior knowledge. You didn't need theorems or to know much about factorial-esque stuff.

[AzureBlue]

I don't think it was different mathematically (although I didn't sit last year's senior so I can't compare), but rather, you didn't need much prior knowledge. You didn't need theorems or to know much about factorial-esque stuff.

Correct. That tends to be the case with MUMCs - it's all about logic, you don't need to know obscure theorems or anything, whereas you sometimes do for AMOs or harder ones, though most things are able to be manipulated by simpler but useful theorems and a touch of ingenuity!

[luken93]

What are the prizes anyway, is it purely on the top 10/20/30 etc or is it the amount of marks?
Also, do you get a certificate of partic, credit, distinc etc?
Only 2 kids did it at our school, but dont wanna accept a "participation" award at assembly ahaha

[kyzoo]

I love proof questions ^.^, but only when I have the time like in the MUMC

I'm pretty sure I got full solutions for 2, 3, and 4.

I got the wrong answer for question 1 LOL, misinterpreted question >.<. I only realised this after talking to 3 other ppl who did the comp (Colin Lu, Evangelion, and a person from another school).

I got the answer for question 5, but like, I must have set up the equation wrong since Colin Lu told me the sum of the digits worked out to be 9, and I got something with like 20 variables.

I tried doing question 6, made some progrss, but ultimately didn't get the answer

I didn't touch question 7 or 8

...

Overall, I think I performed the same as I did last year.