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HSC 2015 MX2 Marathon (archive) (1 Viewer)

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kawaiipotato

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Re: HSC 2015 4U Marathon



















Is this correct method? My notation and method seems weird
 
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Paradoxica

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Re: HSC 2015 4U Marathon



















Is this correct method? My notation and method seems weird
That is correct. I believe the standard notation to express a string of digits is to place a horizontal bar above the entire string.

Next question.

Given , find the minimum value of

Hence, determine the minimum value of by induction or other means.
 
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Drsoccerball

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Re: HSC 2015 4U Marathon

That is correct. I believe the standard notation to express a string of digits is to place a horizontal bar above the entire string.

Next question.

Given , find the minimum value of

Hence, determine the minimum value of by induction or other means.














Note: I don't know why you would suggest induction...
 
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kawaiipotato

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Re: HSC 2015 4U Marathon

That is correct. I believe the standard notation to express a string of digits is to place a horizontal bar above the entire string.
So like a conjugate bar for complex numbers?
 

Paradoxica

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Re: HSC 2015 4U Marathon

So like a conjugate bar for complex numbers?
Code:
\overline{a_0a_1a_2a_3\dots a_{n-2}a_{n-1}a_n}
Yes. e.g. is an "n" digit number in a given base
Note: I don't know why you would suggest induction...
I was a bit rushed for time to leave cabra library as I was writing this question, induction seemed like a way to do it at the time. You could say I made an...
inductive
guess :tongue:
 
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kawaiipotato

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Re: HSC 2015 4U Marathon

Let u = pi/2 - x for both integrals
LHS integral = int from -pi/2 to pi/2 of cos^8 x dx
> 0 as it is even.
RHS integral = int from -pi/2 to pi/2 of sin(4pi - 8x) = -sin8x dx
= 0 as it is odd
Holds true
 

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RealiseNothing

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Re: HSC 2015 4U Marathon

Prove that if is a positive integer such that , then at least one of , or is composite.
n has to be odd, and since 2 and 3 are co-prime three consecutive odd numbers must take all possible values mod 3. This means there is exactly one of the values equal to 0 mod 3 and hence composite.
 
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Re: HSC 2015 4U Marathon

That's exactly what I did but the q asked to show that the answers Pi/4 then verify it using t formula
 

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Re: HSC 2015 4U Marathon

That's exactly what I did but the q asked to show that the answers Pi/4 then verify it using t formula
I'm struggling to see a way to solve it via t-method, at least not without another substitution (I don't know what additional sub would work anyway)
 

Drsoccerball

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Re: HSC 2015 4U Marathon

I'm struggling to see a way to solve it via t-method, at least not without another substitution (I don't know what additional sub would work anyway)
When you reach the step use the t method

But instead of
 

kawaiipotato

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Re: HSC 2015 4U Marathon

Don't think this would fall under any topic in 4U but I found a nice math/logic puzzle

 
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kawaiipotato

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Re: HSC 2015 4U Marathon

No since everyone has a different answer we can deduce that the remaining 9 people must match up to 0,1,..., 8 handshakes each
 
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