Problem 143

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shadowx360
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Problem 143

Post by shadowx360 »

I need someone to clarify for me a point about problem 143. It says in the beginning that the triangle has no angle greater than 120 degrees. Does that mean, in trying to obtain the answer, I must calculate the degrees of the triangles and make sure none is over 120 degrees?

So if I make sure that a,b,c,p,q,r aren't negative, and are positive integers, the sum of p,q, and r are not greater than 120000 , then the triangle is a Toricelli triangle? Or do I have to add in code to make sure that the angles are greater than 120 degrees?
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daniel.is.fischer
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Re: Problem 143

Post by daniel.is.fischer »

You should be able to know from your construction that all your triangles have all angles less than 120 degrees. If you don't know that, a check would be appropriate.
Il faut respecter la montagne -- c'est pourquoi les gypaètes sont là.
zwuupeape
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Re: Problem 143

Post by zwuupeape »

I was sure I managed to solve this one. But I get wrong answer!

I have a total of 508 triangles for the original limit L = 120000. Total sum is 8 digits, starts with a 3.

If I lower it to 12000 I get only 38 triangles, total sum 251752.

What's wrong ?? =\
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daniel.is.fischer
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Re: Problem 143

Post by daniel.is.fischer »

No idea. That's what I get, too.
Il faut respecter la montagne -- c'est pourquoi les gypaètes sont là.
zwuupeape
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Re: Problem 143

Post by zwuupeape »

Taking twice the limit I get a total of 1069 different triangles, with p + q + r at 133050063.
If I take the sum of the corresponding a,b,c instead of p,q,r with the original limit I get a total of a+b+c = 56990134.

The smallest p + q + r there is the one in the example 784.
Next one is p + q + r = 1029.
Largest p + q + r = 119952.
zwuupeape
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Re: Problem 143

Post by zwuupeape »

Ahh !! I realized my mistake!!

This is extremely obscure ...
deng
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Re: Problem 143

Post by deng »

I got exactly the same result as you did. Would you please
give a little hint about the mistake: is the 508 triangles too
large or too small?

Thanks
kosiu_drumev
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Re: Problem 143

Post by kosiu_drumev »

I have a question not exactly about the prblem 143, as it was given. I am looking to find a proof that this point realy gives a minimum sum. Please, if anyone can help with usefull link or something ...

... My special thanks to hk ...
The Hofman's proof is realy remarkable!!!

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Last edited by kosiu_drumev on Sat Jul 09, 2011 9:50 pm, edited 2 times in total.
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hk
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Re: Problem 143

Post by hk »

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Jamie
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Re: Problem 143

Post by Jamie »

Is one such triangle (43, 147, 152), with a r+s+t of 185? I seem to be getting it for some reason (and as far as I can confirm it works), but the above posts suggest that it is not.
TripleM
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Re: Problem 143

Post by TripleM »

The sum is indeed minimised with p+q+r=185, and a, b, and c are all integral, but those facts alone aren't quite sufficient conditions for it to be a Torricelli triangle.
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thedoctar
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Re: Problem 143

Post by thedoctar »

I've posted a question on the solved forum. Could anyone please answer it?
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hk
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Re: Problem 143

Post by hk »

Answered.
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thedoctar
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Re: Problem 143

Post by thedoctar »

I have another question, in regards to the solution. I've edited my original post to put the question in there, as to not waste archived posts.
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hk
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Re: Problem 143

Post by hk »

I'm afraid my reply could become rather lengthy and probably could mean a few question-answer rounds.
Perhaps you can PM or email me, after a bit of rethinking what your actual problem with the pdf is.
(One suggestion that might help: throw overboard all geometrical considerations and consider the derivation as purely algebraic).
(I'd love to help you, having been a math teacher myself).
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thedoctar
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Re: Problem 143

Post by thedoctar »

Okay, I'll probably PM you on the weekend once I've had a good read of the solution. School often keeps me busy during the week.
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karluk
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Re: Problem 143

Post by karluk »

There appears to be a minor discrepancy between the statement of problem 143 and the accompanying diagram. The statement of the problem says
XB = q, and XC = r
for any point X in triangle ABC. But the diagram labels TC = q and TB = r for the Torricelli point. In order to be consistent with the statement of the problem, it should be TC = r and TB = q instead.

I gather that this is probably not the reason problem 143 has stumped so many people, though.
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TheEvil
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Re: Problem 143

Post by TheEvil »

karluk wrote:There appears to be a minor discrepancy between the statement of problem 143 and the accompanying diagram. The statement of the problem says
XB = q, and XC = r
for any point X in triangle ABC. But the diagram labels TC = q and TB = r for the Torricelli point. In order to be consistent with the statement of the problem, it should be TC = r and TB = q instead.

I gather that this is probably not the reason problem 143 has stumped so many people, though.

The problem states that for any point inside the triangle (let's call it X) we have XB=r,...
Then we change to the optimal solution which we call T. It is just one of the possible X's.
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karluk
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Re: Problem 143

Post by karluk »

The problem states that for any point inside the triangle (let's call it X) we have XB=r
Not true. The problem states that "XB=q" for an arbitrary point X in the triangle, not "XB=r". It's the diagram that accompanies the problem statement that labels "TB=r" for the specific Toricelli point in the triangle. That label is not consistent with the problem statement, and I was pointing out the discrepancy, hoping that one of the admins would take the trouble to make a correction so that the diagram agrees with the problem statement.
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hk
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Re: Problem 143

Post by hk »

Changed.
Do you know that nobody noticed this for six years?
I hope you will be able to solve the problem now within a few minutes.
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