10566 - Crossed Ladders

All about problems in Volume 105. If there is a thread about your problem, please use it. If not, create one with its number in the subject.

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BiK
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10566 - Crossed Ladders

Post by BiK » Thu Oct 09, 2003 1:13 am

I thought that I have good background Mathematics and that this should be an easy problem in Geometry. Just the opposite, this proved wrong. I'm totally stumped. Please help me to solve the problem.

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Post by Larry » Thu Oct 09, 2003 2:52 am

binary search.

titid_gede
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Post by titid_gede » Thu Oct 09, 2003 5:39 am

binary search? what is the relationship between this problem and binary seach? isnt it totally math? please give me enlightment.. :) :) :)
i havent solved this problem, and i'm very weak in math.
thanks before...
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sharklu2000
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Post by sharklu2000 » Thu Oct 09, 2003 6:47 am

It means that you continues trying the middle of the large number and the small number until you get the root.
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Observer
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Post by Observer » Thu Oct 09, 2003 11:29 am

This method is usually referred to as "bisection method".
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Farid Ahmadov
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Post by Farid Ahmadov » Thu Oct 09, 2003 12:22 pm

Hi. I don't think that I am weak in math. But I don't understand why I am wrong?

I found a formula and can prove it.
Do you see the picture below?

Image

It is my prove. Isn't it right and why?
Thanx.
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Observer
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Post by Observer » Thu Oct 09, 2003 2:43 pm

Plz ignore me. :D
Last edited by Observer on Fri Oct 10, 2003 5:14 pm, edited 1 time in total.
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BiK
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Post by BiK » Thu Oct 09, 2003 5:10 pm

2Farid:

How do you get r/c = z*z?

hewei
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Post by hewei » Thu Oct 09, 2003 5:37 pm

Hi, Farid Ahmadov!
There was a fault in your proof:
From 6th line: a/b = a1/b1 = z
To 7th line: r/c = c/r1/ = z*z

The process 6th -> 7th requires the angle you noted in the graph to be right, i.e. 90 degrees.

That's why the formula is even not suitable for the sample, isn't it?

Farid Ahmadov
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Post by Farid Ahmadov » Thu Oct 09, 2003 8:54 pm

Sorry. I am not correct. My mind was somewhere near 90 degrees.
Only 7th formula is not correct. r/c = c/r1 = z*z*d
d = sin(90+alfa+beta). I have to find what it is.
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BiK
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Post by BiK » Thu Oct 09, 2003 10:37 pm

Thanks to all. After all finding an equation and solving it through binary search was enough. However a mathematician always tries to find some nice looking formula which maybe is not possible in this problem (our mistake).

Farid Ahmadov
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Post by Farid Ahmadov » Fri Oct 10, 2003 4:05 pm

Hi BiK. You are right. I know how to solve it with binary search. But I want to solve it matematically. And it is possible because l depends on only x,y and c. I just have to find dependence between them.
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hewei
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Post by hewei » Sat Oct 11, 2003 3:45 am

Well, as far as I see, to find a nice formula is as difficult as to solve the equation below:

(when x is not equal to y)
(x*x - l*l) * sqrt(y*y - l*l) - (y*y - l*l) * sqrt(x*x - l*l) = c*(x*x - y*y)

but, when x equals y, the problem becomes obviously trivial:
l = sqrt(x*x - 4*c*c)

fqsxr
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help:wrong answer

Post by fqsxr » Sun Oct 12, 2003 9:14 am

I've found an idea:
solve sqrt(x^2-l^2)*sqrt(y^2-l^2)/(sqrt(x^2-l^2)+sqrt(y^2-l^2))-c=0
and I've written an program for solving this
but the online judge told me wrong answer
Who can tell me how to debug this? Can I see the test datas?

marian
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Post by marian » Sun Oct 12, 2003 9:34 pm

Mathematical solution to this problem can be found here: http://mathworld.wolfram.com/CrossedLaddersProblem.html

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