HSC 2013 MX2 Marathon (archive) (1 Viewer)

obliviousninja · Oct 21, 2013

Re: HSC 2013 4U Marathon

http://snag.gy/tqGMd.jpg

im unsure about part 4, i gave the question to heroic, but i think he is afk

bottleofyarn · Oct 21, 2013

Re: HSC 2013 4U Marathon

obliviousninja said:
http://snag.gy/tqGMd.jpg

im unsure about part 4, i gave the question to heroic, but i think he is afk

Hrmm. So from iii) with c = 0:
$$Let the roots be $ 2, \alpha, mi, -mi$
$d^2 = bd(0) - eb^2 = -eb^2$
$$but a, b, c, d, e all integers and therefore real, $ d = \pm b\sqrt{-e}$
$and $ e = 2\beta m^2 $ must be even (since e is an integer, $\beta $ is real, m is real)$
$\therefore $ d is even (since b is integer, e even), as required$$

Hope there aren't any logical flaws here.

Sy123 · Oct 21, 2013

Re: HSC 2013 4U Marathon

bottleofyarn said:
It's terribly inelegant IMO. Here's one:
$$For a monic cubic, the discriminant is $ {(\alpha - \beta)^2 (\beta - \gamma)^2 (\gamma - \alpha)^2$
$$Prove that there are three distinct real zeroes iff $ \Delta > 0$

Hopefully this is a valid proof

$Take a monic quadratic$

$x^2 + ax + b=0 \ \ $with roots$ \ \ x_1 \ \ x_2$

$$This quadratic has distinct real roots if and only if$ \ \ a^2 - 4b > 0$

$\therefore \ \ (x_1 + x_2)^2 - 4x_1 x_2 = (x_1 - x_2)^2 > 0 \ \ \ \fbox{1}$

$Take a monic cubic$

$p(x) = (x-\alpha)(x-\beta)(x-\gamma)$

$This polynomial has distinct real roots if and only if$

$The quadratic with roots$

$\alpha, \gamma \ \ $has$ \ \ \triangle > 0$

$\alpha , \beta \ \ $has$ \ \ \triangle > 0$

$\beta , \gamma \ \ $has$ \ \ \triangle > 0$

$Since the three discriminants above are$

$(\alpha - \gamma)^2 \ , \ (\alpha - \beta)^2 \ , \ (\beta - \gamma)^2 \ \ \Rightarrow \fbox{1}$

$respectively$

$Therefore the multiplication of all these roots$

$p(x) = (x-\alpha)(x-\beta)(x-\gamma) = (x-\alpha)(x^2 - (\beta + \gamma)x + \beta \gamma)$

$= (x-\beta)(x^2 - (\alpha + \gamma)x + \alpha \gamma)$

$= (x-\gamma)(x^2 - (\alpha + \beta)x + \alpha \beta)$

$Therefore since all discriminants of the above quadratics must be greater than zero, therefore the roots are distinct and real if and only if$

$(\alpha - \beta)^2 (\beta - \gamma)^2 (\gamma - \alpha)^2> 0$

ForOneEon · Oct 21, 2013

Re: HSC 2013 4U Marathon

bottleofyarn said:
Hrmm. So from iii) with c = 0:
$$Let the roots be $ 2, \alpha, mi, -mi$
$d^2 = bd(0) - eb^2 = -eb^2$
$$but a, b, c, d, e all integers and therefore real, $ d = \pm b\sqrt{-e}$
$and $ e = 2\beta m^2 $ must be even (since e is an integer, $\beta $ is real, m is real)$
$\therefore $ d is even (since b is integer, e even), as required$$

Hope there aren't any logical flaws here.

Thanks, I was trying to work through that myself. Forgot to notice your fourth line of argument.

ForOneEon · Oct 21, 2013

Re: HSC 2013 4U Marathon

Sorry for being selfish or anything, but can someone post up a mechanics question? I'm dying to get more practice in that topic.
Here's a conics question for those who want it:

$Find the locus of the point of intersection of perpendicular tangents to the curve y = x + \left (\frac{1}{x} \right )$

Sy123 · Oct 21, 2013

Re: HSC 2013 4U Marathon

$$Prove for real$ \ \ a,b,c,d$

$|ab+cd| \leq \sqrt{a^2+c^2} \sqrt{b^2+d^2}$

RealiseNothing · Oct 21, 2013

Re: HSC 2013 4U Marathon

Sy123 said:
$$Prove for real$ \ \ a,b,c,d$

$|ab+cd| \leq \sqrt{a^2+c^2} \sqrt{b^2+d^2}$

$b^2c^2 + a^2d^2 \geq 2\sqrt{a^2b^2c^2d^2} = 2abcd$

$b^2c^2 + a^2d^2 + (a^2b^2 +c^2d^2) \geq 2abcd + (a^2b^2 +c^2d^2)$

$(a^2+c^2)(b^2+d^2) \geq (ab+cd)^2$

$\sqrt{a^2+c^2}\sqrt{b^2+d^2} \geq |ab+cd|$

seanieg89 · Oct 21, 2013

Re: HSC 2013 4U Marathon

I tried a few different ways of approaching Carrots problem but yeah, the best one can do it seems is find an expression in terms of alpha (where alpha is the positive solution in (0,pi/2) to tan(t)-t=2*pi).

asianese · Oct 21, 2013

Good luck 4u kiddies for tomorrow. Just remember it is just an exam - you won't die.

Make it count and do your best.

(If you get to take a copy of your exam please scan it and upload! We are eager to see what the bos has served up this year!)

asianese · Oct 21, 2013

Good luck 4u kiddies for tomorrow. Just remember it is just an exam - you won't die.

Make it count and do your best.

(If you get to take a copy of your exam please scan it and upload! We are eager to see what the bos has served up this year!)

asianese · Oct 22, 2013

I'm sure you all went brilliantly. I want to see bos state rank!!

Sy123 · Oct 22, 2013

Re: HSC 2013 4U Marathon

barbernator said:
good stuff sy, I challenge you to keep this running for the whole year.

Challenge complete?

Makematics · Oct 22, 2013

Re: HSC 2013 4U Marathon

RIP. Time to start 2014 marathon ?

RealiseNothing · Oct 22, 2013

Re: HSC 2013 4U Marathon

We didn't make 100 pages

Sy123 · Oct 22, 2013

Re: HSC 2013 4U Marathon

Makematics said:
RIP. Time to start 2014 marathon ?

I'd rather just add on 2014 stuff here tbh

RealiseNothing said:
We didn't make 100 pages

=(

Menomaths · Oct 22, 2013

Re: HSC 2013 4U Marathon

Solve cos2(x) - sin(x) = 1 for 0<= x <= 2pi
As the result of an experiment, a curve of the form y = f(x) is drawn. It is suspected that f(x) is expressible in the form f(x) = (1+ax)n where “a”
is real and “n” is a positive integer. For values of x so small that third and higher powers can be neglected: y = 1-6x+16x^2 is practically identical
with the given curve. Assuming the curves coincide for these values of x, what are the values of “a” and “n”?

hit patel · Oct 23, 2013

Re: HSC 2013 4U Marathon

Guys how did it go? The test?

hayabusaboston · Oct 23, 2013

Re: HSC 2013 4U Marathon

hit patel said:
Guys how did it go? The test?

Was pretty easy for most, but also very easy to make a lot of silly mistakes.

hecticated · Oct 23, 2013

Re: HSC 2013 4U Marathon

thanks for this! im doing 4u for 2014, so hopefully this will help

hit patel · Oct 23, 2013

Re: HSC 2013 4U Marathon

RealiseNothing said:
We didn't make 100 pages

Well 96 pages for the 2014 boyz.

Sy123 said:
Challenge complete?

Yeah. My challenge to u sy123: keep it alive and till the end of 2014 HSC at 200pages

Sy123 said:
I'd rather just add on 2014 stuff here tbh

yep.
=(

hayabusaboston said:
Was pretty easy for most, but also very easy to make a lot of silly mistakes. But you will never get more thana raw 90 in such a test hit patel

I am god of careless mistakes and mind blanks during exams. so much so of a god that coz of me other people that sit nearby me make it too. ahahaha, i remodelled ur quote.

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