HSC 2015 MX2 Marathon ADVANCED (archive) (2 Viewers)

lita1000 · Sep 13, 2015

Re: HSC 2015 4U Marathon - Advanced Level

Okay, but in that case the question doesn't belong in the advanced thread, should be in the normal 4U thread.

lita1000 said:
Dan why did u break the question up into parts? It might've been a nice question if u didn't spoon feed (like the HSC unfortunately). Here's a hard one:

$\\ $Show that for $a,b,c >0: (1-a)^2+(1-b)^2+(1-c)^2 \geq \frac{(1-a^2)(1-b^2)c^2}{(ab+c)^2}+\frac{(1-c^2)(1-b^2)a^2}{(bc+a)^2}+\frac{(1-a^2)(1-c^2)b^2}{(ac+b)^2}$

dan964 said:
because I can. also it was originally two separate questions. (parts 3-5 were originally separate question but required that the first two parts be done anyway). Hold on I'll edit it.

and i'll see if I can do your question, just maybe not straight away at this time of night

Bump

dan964 · Sep 13, 2015

Re: HSC 2015 4U Marathon - Advanced Level

lita1000 said:
I reckon you should even take out the part where they say "Show that V= blah blah blah".

lita1000 said:
Okay, but in that case the question doesn't belong in the advanced thread, should be in the normal 4U thread.

why? I will edit it down if you really want to. But this thread is for Q16 material of which this question is, even if it leads you a little bit. (The first two parts yes do belong in the other thread, but because they are part of the one question, I'd thought I'd be nice and leave them in).

I know you want a challenging question. I will edit it if you must when LATEX is stuffing up. (I will edit out the specific expression. This is a middle question, harder than last years 2014 HSC question but probably easier than some others.

And yes, people can also attempt yours... if they want a even more difficult one.

Physicklad · Sep 23, 2015

Re: HSC 2015 4U Marathon - Advanced Level

One hundred people line up to board an airplane. Each has a boarding pass with
assigned seat. However, the first person to board has lost his boarding pass and takes a random
seat. After that, each person takes the assigned seat if it is unoccupied, and one of unoccupied
seats at random otherwise. What is the probability that the last person to board gets to sit in
his assigned seat?

absolute fire

dan964 · Sep 23, 2015

Re: HSC 2015 4U Marathon - Advanced Level

Physicklad said:
One hundred people line up to board an airplane. Each has a boarding pass with
assigned seat. However, the first person to board has lost his boarding pass and takes a random
seat. After that, each person takes the assigned seat if it is unoccupied, and one of unoccupied
seats at random otherwise. What is the probability that the last person to board gets to sit in
his assigned seat?

absolute fire

can I assume there are 100 seats on the plane?

Physicklad · Sep 23, 2015

Re: HSC 2015 4U Marathon - Advanced Level

yep there are 100 seats soz

Drsoccerball · Sep 23, 2015

Re: HSC 2015 4U Marathon - Advanced Level

I think this is the answer:

$\frac{99! +98! +...+2!+1!}{100!}$

dan964 · Sep 23, 2015

Re: HSC 2015 4U Marathon - Advanced Level

i am probably wrong, but I got
1/2

Physicklad · Sep 23, 2015

Re: HSC 2015 4U Marathon - Advanced Level

dan964 said:
i am probably wrong, but I got
1/2

nice work it is 1/2. solution pls

dan964 · Sep 23, 2015

Re: HSC 2015 4U Marathon - Advanced Level

Physicklad said:
nice work it is 1/2. solution pls

Physicklad · Sep 23, 2015

Re: HSC 2015 4U Marathon - Advanced Level

haha nice

this one's beautiful but it still makes no sense to me-->

"Look at the situation when the k’th passenger enters. Neither of the previous passengers
showed any preference for the k’th seat vs. the seat of the first passenger. This in particular is
true when k = n. But the n’th passenger can only occupy his seat or the first passenger’s seat.
Therefore the probability is 1/2."

lita1000 · Sep 25, 2015

Re: HSC 2015 4U Marathon - Advanced Level

lita1000 said:
Dan why did u break the question up into parts? It might've been a nice question if u didn't spoon feed (like the HSC unfortunately). Here's a hard one:

$\\ $Show that for $a,b,c >0: (1-a)^2+(1-b)^2+(1-c)^2 \geq \frac{(1-a^2)(1-b^2)c^2}{(ab+c)^2}+\frac{(1-c^2)(1-b^2)a^2}{(bc+a)^2}+\frac{(1-a^2)(1-c^2)b^2}{(ac+b)^2}$

Bump

dan964 · Sep 25, 2015

Re: HSC 2015 4U Marathon - Advanced Level

lita1000 said:
Bump

It also surfices to show that this (1-a)(1-b)+(1-b)(1-c)+(1-b)(1-c) is bigger than the RHS, which might be easier or harder

RealiseNothing · Sep 26, 2015

Re: HSC 2015 4U Marathon - Advanced Level

Physicklad said:
haha nice

this one's beautiful but it still makes no sense to me-->

"Look at the situation when the k’th passenger enters. Neither of the previous passengers
showed any preference for the k’th seat vs. the seat of the first passenger. This in particular is
true when k = n. But the n’th passenger can only occupy his seat or the first passenger’s seat.
Therefore the probability is 1/2."

It's just saying that when the n'th passenger enters, the only possible seats left will be his own or the first person's. Both cases are equally likely so it's just 1/2.

awesome-0_4000 · Sep 26, 2015

Re: HSC 2015 4U Marathon - Advanced Level

Edit: The first (b)(iii) should be labeled (b)(ii) and the coefficient in front of the integral expression for a_n should be \frac{1}{\pi}.

psyc1011 · Sep 26, 2015

Re: HSC 2015 4U Marathon - Advanced Level

Are you posting your homework?

awesome-0_4000 · Sep 26, 2015

Re: HSC 2015 4U Marathon - Advanced Level

psyc1011 said:
Are you posting your homework?

No, I wrote the question.

InteGrand · Sep 29, 2015

Re: HSC 2015 4U Marathon - Advanced Level

$Let $ABC$ be a triangle with a point $D$ on $AB$, $E$ on $BC$ and $F$ on $AC$. It is given that $AD=BE=CF$ and that $\triangle DEF$ is equilateral.$

$Show that $\triangle ABC$ is equilateral.$

el_manu · Sep 30, 2015

Re: HSC 2015 4U Marathon - Advanced Level

I am a dumb kent.

InteGrand · Sep 30, 2015

Re: HSC 2015 4U Marathon - Advanced Level

el_manu said:
Easy. How is this 4u Advanced Level? Are you trolling? I dont even do 4u and i got it.

you prove the 3 corner triangles are congruent through SSS and then that means by matching angles of congruent triangles the corner angles are all equal, and since the sum of the angles of a triangle = 180 degrees, each angle is 180/3 = 60 degrees. Which is an equilateral tringle

How did you get the SSS?

el_manu · Sep 30, 2015

Re: HSC 2015 4U Marathon - Advanced Level

InteGrand said:
How did you get the SSS?

my bad made a wrong assumption to start off with

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