RESULTS: 2011 Diamond 7

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RESULTS: 2011 Diamond 7

Post by Chocola » Mon Oct 03, 2011 8:13 am

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1) Which arena has the second highest VD?
2) Which arena has the second lowest VD?
3) Let n be an integer. Prove that if 2n²+5n-12 is even, then n must be even. Your proof can be in any method that makes sense as long as it's different from the preceding ones.

Hint: Do not assume the conclusion. For example, "n is even" should not be anywhere except for the last line/sentence.
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Re: RESULTS: 2011 Diamond 7

Post by xcrossfacekillahx » Mon Oct 03, 2011 8:55 am

1. Arena 19
2. Arena 18
3. since p implies q (p->q)
let p be "2n²+5n-12 is even"
let q be "n is even"

use proof by contradiction
Assume 2n²+5n-12 is even but n is odd
since n is odd, 2n²+5n-12 would be odd because 2n² is even, 5n is odd and 12 is even. odd + even + even = odd
which is a contradiction to the original statement so n is even

But there might be a twist to it. Holo is treasurer.
Last edited by xcrossfacekillahx on Mon Oct 03, 2011 9:32 am, edited 4 times in total.
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Re: RESULTS: 2011 Diamond 7

Post by Bastion » Mon Oct 03, 2011 9:06 am

1) Which arena has the second highest VD?
10: Goko Ruri vs Sakagami Tomoyo (top)
25: Holo vs Aisaka Taiga (2nd)
2) Which arena has the second lowest VD?
2: Shana vs Katsura Hinagiku (top)
17: Misaka Mikoto vs Tachibana Kanade (2nd)
3) Let n be an integer. Prove that if 2n²+5n-12 is even, then n must be even. Your proof can be in any method that makes sense as long as it's different from the preceding ones.
Assume n is even. You didn't state that the proof even needed to be mathmatical.

Edited to include the first two questions, the third question only got an underline to emphasize a piece of the statement.
Assuming is a usable method.
Last edited by Bastion on Mon Oct 03, 2011 8:46 pm, edited 2 times in total.
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Re: RESULTS: 2011 Diamond 7

Post by Team Rocket Elite » Mon Oct 03, 2011 9:23 am

1) Arena 19
2) Arena 18

3) Working on this
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Re: RESULTS: 2011 Diamond 7

Post by RegalStar » Mon Oct 03, 2011 9:24 am

1. 19
2. 21
3. The original equation = (2n-3)(n+4). 2n is even and 3 is odd, so 2n-3 is always odd. 4 is even, so if n is odd then n+4 is also odd, and (2n-3)(n+4) would be odd. Thus, n must be even if (2n-3)(n+4) is even.

Don't like roundabout answers but since I'm not the first one to the obvious, gotta come up with something else.
Last edited by RegalStar on Mon Oct 03, 2011 2:31 pm, edited 1 time in total.
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Re: RESULTS: 2011 Diamond 7

Post by Yakuman » Mon Oct 03, 2011 12:00 pm

1) Arena 19
2) Arena 21

3) Let's take this apart. 2n² translates to SnS which is an acronym to Shakugan no Shana. 5n refers to the n in Shana's name who is currently fifth on the chart. -12 is Shana's point difference to Kanade and Mikoto on the chart. If this monster is even that means exactly one or all three parts of this thing has to be even in order to be true.
Since the 'n' in SnS refers to 'no', a word with two letters, it is even. The second 'n' is the fourth letter in Shana's name, so it's also even. If we would remove the point difference between her and the leaders, she would end up second on the chart, since that would mean that she defated Mikoto (the chance for beating Tenshi is insignificant). Two is also an even number.
Here you go, I proved that if 2n²+5n-12 is even then n is even too.
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Re: RESULTS: 2011 Diamond 7

Post by YagamiHayate » Mon Oct 03, 2011 1:53 pm

1) Arena 19
2) Arena 7
3) 2n2+5n-12 factorised is (2n-3)(n+4)
2n is even, so (2n-3) is odd.
Odd*Even = Even, so (n+4) is even, therefore n is even.


Edit: I just realised this is exactly what RegalStar already did. Uhh, I can't be bothered thinking of another method.
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Re: RESULTS: 2011 Diamond 7

Post by Alexander » Mon Oct 03, 2011 2:28 pm

1. Arena 19
2. Arena 18
WhateverShow
Given 2n²+5n-12

We have that -12=Even

Assuming n=Odd
2n²=Even
5n=Odd

Even+Odd-Even=Odd
Last edited by Alexander on Tue Oct 04, 2011 12:06 am, edited 1 time in total.
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Re: RESULTS: 2011 Diamond 7

Post by Jeffrey-sama » Mon Oct 03, 2011 2:59 pm

1) Which arena has the second highest VD?
Arena 19

2) Which arena has the second lowest VD?
Arena 2

3) Let n be an integer. Prove that if 2n²+5n-12 is even, then n must be even. Your proof can be in any method that makes sense as long as it's different from the preceding ones.
Seriously... why math? ._."
And, xcrossfacekillahx took my contradiction answer (just when I thought calculus at school was going to be useful).
I'll try to expand on his proof.

Proof

Code: Select all

Assume 2n² + 5n - 12 is even.
Suppose n is odd.
Lemma 1: If n² is even, n is also even
    Let n = 2k (even since any multiple of 2 is even)
    n² = (2k)²
    n² = 4k²
    n² = 2(2k²) -> even
    Q.E.D.
By Lemma 1, n² is odd if n is odd.
Thus, 2n² is even.

Lemma 2: The product of two odd numbers is odd.
    Let two odd numbers be (2n + 1) and (2m + 1)
    Multiplying them together would result in:
    (2n + 1)(2m + 1)
    = 4nm + 2n + 2m + 1
    = 2(2nm + n + m) + 1
    -> odd
    Q.E.D.
By Lemma 2, 5n is odd.

Lemma 3: The sum of and even and odd number is odd.
    Let an even number be 2n and an odd number be (2m + 1)
    Adding them together would result in:
    2n + 2m + 1
    = 2(n + m) + 1
    -> odd
    Q.E.D.
And so, 2n² + 5n - 12 -> even + odd - even.
By Lemma 3, 2n² + 5n - 12 is odd.
-> CONTRADICTION
Therefore, n must be even if 2n² + 5n - 12 is even.
Q.E.D.
Last edited by Jeffrey-sama on Mon Oct 03, 2011 3:25 pm, edited 2 times in total.
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Re: RESULTS: 2011 Diamond 7

Post by Eclairs » Mon Oct 03, 2011 3:04 pm

Q1 - 2nd largest VD : Arena 25 [ Taiga X Holo ]
Q2 - 2nd closest match : Arena 17 [ Railgun x Kanade ]
Q3 -
boring proof.Show
f(n) = 2n^2 + 5n - 12 - eq. 1
lets make a 2nd equation to define odd and even .
[ odd ] = 2x+1 , where -infinity < x < infinity , x is an integer. - eq. 2
[ even ] = 2x , where -infinity < x < infinity, x is an integer. - eq. 3
from the equation 1 : lets call 2n^2 = A . it can also be defined as 2x, where x = n^2, hence A is even ---- 1st premise
: lets call 5n = B, we can the redefine B as 2*[2n] + n, and lets redefine 2n as N, this gives B a new equation of B = 2N + n
: lets call 12 = C, C can easily be defined from equation 2 as C = 2*[6] , hence C is even ---- 2nd premise
From mathematics : [ even ] - [ even ] + [ even ] = [ even ] , [ even] - [ odd ] + [ even ] = [ odd ] = 2nd Premise
Proof : [2p] - [2q] + [ 2r] = 2[p-q+r] and it satisfy equation 2, where [p-q+r] = x
: [2k]+[2l+1]- [2m] =2[k+l-m] + 1 = 2[k+l-m]+1 and satisfy equation 3, where [k+l-m] = x
From the equation 1 ( simplified ) : A + B - C. For A + B - C to satisfy equation 2 : A, B and C must be [ even ], with A and C are already [even ] ,
We can the condense the proving into the B part.
B = 5n = 2[2n] + n = 2N + n. for B to be [ even ] , n must be [ even ], from the 2nd premise
< edit 1 : copied the wrong equation, it was supposed to be 2n^2 + 5n - 12, not 2n^2 - 5n + 12 : but the proof is still the same ;) >
Last edited by Eclairs on Mon Oct 03, 2011 6:29 pm, edited 3 times in total.
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Re: RESULTS: 2011 Diamond 7

Post by fujibayashi » Mon Oct 03, 2011 3:52 pm

1) Which arena has the second highest VD?
Arena 19
2) Which arena has the second lowest VD?
Arena 9
3) Let n be an integer. Prove that if 2n²+5n-12 is even, then n must be even. Your proof can be in any method that makes sense as long as it's different from the preceding ones.
SpoilerShow
2n² must even.
Even-12(even) =even.
If 2n²+5n-12 is even,then 5n = even.
5 times even → even
So n=even.
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Re: RESULTS: 2011 Diamond 7

Post by lihuazou » Mon Oct 03, 2011 4:32 pm

1) Arena 19
2) Arena 18
3) Because 2n²+4n-12 is even
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Re: RESULTS: 2011 Diamond 7

Post by maglor » Mon Oct 03, 2011 4:54 pm

lihuazou wrote:1) Arena 19
2) Arena 18
3) Because 2n²+4n-12 is even
@lihuazou : Next time, I think you should help KS and I grade these proofs.
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Re: RESULTS: 2011 Diamond 7

Post by amdrag » Mon Oct 03, 2011 6:09 pm

I. Arena 25 (Holo/Taiga)

II. Arena 16 (Yui/Hitagi)

III. We know these things to be true:

1) even + odd = odd
2) even + even = even
3) odd + odd = even
4) odd * even = even
5) odd * odd = odd
6) even * even = even

2n²+5n-12 (assume n is an integer)

2n² must be even, regardless of n's value. If n is odd, it's even * odd * odd, which must be even (#4). If n is even, it's even * even * even, which must be even (#6).

5n may be even or odd, depending on if n is odd. If n is odd, 5n is odd (#5). If n is even, 5n is even (#4).
-12 is even.

We know that 2n² and -12 are even, so their sums must be even (#2). All that leaves unknown is the value of 5n.

If 5n is odd, then the equation is even + odd + even, which must be odd (#1). We also know that if 5n is odd, then n is odd (see earlier), so if 2n²+5n-12 is odd, then n must be odd.

If 5n is even, then the equations is even + even + even, which must be even (#2). We also know that if 5n is even, then n is even (see earlier), so if 2n²+5n-12 is even, then n must be even.
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Re: RESULTS: 2011 Diamond 7

Post by Chocola » Mon Oct 03, 2011 7:16 pm

@xcrossfacekillahx: simple and lovely
@RegalStar: Very good, just what I had in mind when I made it factorable. However, your sequence of proof should to be: (2n-3)(n+4) is even, so (n+4) must be even since even/odd=even, and hence even-4=even. Using division makes you not assume what n is first, although it will work if you clarify by breaking up into case 1: n is even, case 2: n is odd since they are disjoint.
@Alexander: The first part of your proof is all you need, because it's proving my contrapositive: assume n is odd and arrive at being odd. The second part of your proof is wrong because it assumes n is even so delete it.

@Eclairs: I'll let maglor handle this...
@fujibayashi: good and concise
@lihuazou: the easiest proof, although you might want to clarify n = 2n²-4n+12 and the RHS is even.
@amdrag: maglor can handle this too...

Right now, everything simple has been taken except the contrapositive proof where you assume n = 2k+1 and arrive at 2n²+5n-12 = 2(~)+1 after lots of algebra. If someone wants to try it go ahead.
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Re: RESULTS: 2011 Diamond 7

Post by Deathscyther » Mon Oct 03, 2011 7:43 pm

1) arena 19
2) arena 6
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Re: RESULTS: 2011 Diamond 7

Post by BrawlRlz28 » Mon Oct 03, 2011 8:02 pm

1 Arena 5
2 Arena 18
3 Simple, if someone says its even, its most likely even. So N is even.....yeah I don't get it either.
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Re: RESULTS: 2011 Diamond 7

Post by Momento10 » Mon Oct 03, 2011 9:01 pm

1. 1
2. 2
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Re: RESULTS: 2011 Diamond 7

Post by Natsumi » Mon Oct 03, 2011 9:28 pm

1. Arena 25
2. Arena 4
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Re: RESULTS: 2011 Diamond 7

Post by blackbeard » Tue Oct 04, 2011 12:09 am

1.18
2.19
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