HSC 2013 MX2 Marathon (archive) (1 Viewer)

[Carrotsticks]

Re: HSC 2013 4U Marathon

A stick of some length L is split into two smaller sticks (not necessarily equal). The larger of these two are split into two more sticks at some arbitrary point. As a result, there are three sticks in total.

Show that the probability that the three sticks form a triangle is 3ln(2) - 2.

God damn it Sy123, your dp confuses me (but my Santa hat > your top hat).

 

[Sy123]

Re: HSC 2013 4U Marathon

There you go Sy123

Well for i, I managed to simply brute force my way through it by:



Then just doing loads of algebra eventually yielding me the answer. However if this is the only way to do this question (by letting z=x+iy), then I will be a bit disappointed heh. So is there a good/clever way to do this?

I will try the geometric means later.

But as for part b. As k approaches l, the circle will become the perpendicular bisector of z_1 and z_2.

 

[Sy123]

Re: HSC 2013 4U Marathon

A stick of some length L is split into two smaller sticks (not necessarily equal). The larger of these two are split into two more sticks at some arbitrary point. As a result, there are three sticks in total.

Show that the probability that the three sticks form a triangle is 3ln(2) - 2.

God damn it Sy123, your dp confuses me (but my Santa hat > your top hat).

Can the sticks cross over each other, or must the sticks make a perfect triangle without any extra bits sticking out?

(Also my slowpoke is more classy (and manly)) than yours :s)

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

No extra bits sticking out.

 

[RealiseNothing]

Re: HSC 2013 4U Marathon

A stick of some length L is split into two smaller sticks (not necessarily equal). The larger of these two are split into two more sticks at some arbitrary point. As a result, there are three sticks in total.

Show that the probability that the three sticks form a triangle is 3ln(2) - 2.

God damn it Sy123, your dp confuses me (but my Santa hat > your top hat).

Nice question, I'm giving it a go and have an idea what to do, but let me make a variation question:

A stick of some length L is split into two smaller sticks (not necessarily equal). The smaller of these two are split into two more sticks at some arbitrary point. As a result, there are three sticks in total.

Find the probability that the sticks form a triangle.

 

[jyu]

Re: HSC 2013 4U Marathon

0

 

[Sy123]

Re: HSC 2013 4U Marathon

0

By Triangle Inequality yes?

 

[RealiseNothing]

Re: HSC 2013 4U Marathon

By Triangle Inequality yes?

Yes. It's how you do carrot's question I think. Basically what's the probability that the sum of the two shorter sides cut is larger than the longest side.

 

[Sy123]

Re: HSC 2013 4U Marathon

Yes. It's how you do carrot's question I think. Basically what's the probability that the sum of the two shorter sides cut is larger than the longest side.

That is what I originally thought, however I was not able to do much with the 3 inequalities that I had and no numbers.

 

[RealiseNothing]

Re: HSC 2013 4U Marathon

That is what I originally thought, however I was not able to do much with the 3 inequalities that I had and no numbers.

Same, I got stuck doing it that way. I'll try it again tomorrow if no one has got it by then, I really need to get back to my chemistry assignment lol.

 

[Fus Ro Dah]

Re: HSC 2013 4U Marathon

That is what I originally thought, however I was not able to do much with the 3 inequalities that I had and no numbers.

Same, I got stuck doing it that way. I'll try it again tomorrow if no one has got it by then, I really need to get back to my chemistry assignment lol.

A clue: We simplify the problem by considering the triangle OAB, where O is the origin and A and B are the x and y intercepts, respectively, of the line x+y=k. The set of all possible trisections into triangles exists in triangle OAB. Generally, questions such as these are done using geometric probability, which usually involves finding the area between a curve and a line or possibly the ratio between the area of a shape that lies entirely within another area representing the total set.

 

[jyu]

Re: HSC 2013 4U Marathon

3ln2-2 is small. One would think the prob is higher.

 

[Fus Ro Dah]

Re: HSC 2013 4U Marathon

3ln2-2 is small. One would think the prob is higher.

3ln2-2 is correct. I worked it out just then.

 

[Trebla]

Re: HSC 2013 4U Marathon

Damn these Slowpokes everywhere...

 

[cutemouse]

Re: HSC 2013 4U Marathon

Damn these Slowpokes everywhere...

Long time no see. Out of the closet?

 

[seanieg89]

Re: HSC 2013 4U Marathon

A stick of some length L is split into two smaller sticks (not necessarily equal). The larger of these two are split into two more sticks at some arbitrary point. As a result, there are three sticks in total.

Show that the probability that the three sticks form a triangle is 3ln(2) - 2.

God damn it Sy123, your dp confuses me (but my Santa hat > your top hat).

Hmmm...I got 2*log(2)-1. Are you sure your answer is correct?

 

[Sy123]

Re: HSC 2013 4U Marathon






Who needs integration by parts when you have this :s

 

[cutemouse]

Re: HSC 2013 4U Marathon

Well for i, I managed to simply brute force my way through it by:



Then just doing loads of algebra eventually yielding me the answer. However if this is the only way to do this question (by letting z=x+iy), then I will be a bit disappointed heh. So is there a good/clever way to do this?

Maybe you can use the fact that |z|^2 = z zbar...

 

[jyu]

Re: HSC 2013 4U Marathon

3ln2-2 is correct. I worked it out just then.

What you have worked out is not the answer to the question. Your answer is for forming a right angled triangle. Now I see why the probability is so low.

 

[RealiseNothing]

Re: HSC 2013 4U Marathon

A stick of some length L is split into two smaller sticks (not necessarily equal). The larger of these two are split into two more sticks at some arbitrary point. As a result, there are three sticks in total.

Show that the probability that the three sticks form a triangle is 3ln(2) - 2.

God damn it Sy123, your dp confuses me (but my Santa hat > your top hat).

3ln2-2 is correct. I worked it out just then.

Hmmm...I got 2*log(2)-1. Are you sure your answer is correct?

What you have worked out is not the answer to the question. Your answer is for forming a right angled triangle. Now I see why the probability is so low.

This is going to be good.