Problem 143
Forum rules
As your posts will be visible to the general public you
are requested to be thoughtful in not posting anything
that might explicitly give away how to solve a particular problem.
This forum is NOT meant to discuss solution methods for a problem.
See also the topics:
Don't post any spoilers
Comments, questions and clarifications about PE problems.
As your posts will be visible to the general public you
are requested to be thoughtful in not posting anything
that might explicitly give away how to solve a particular problem.
This forum is NOT meant to discuss solution methods for a problem.
In particular don't post any code fragments or results.
Don't start begging others to give partial answers to problems
Don't ask for hints how to solve a problem
Don't start a new topic for a problem if there already exists one
Don't start begging others to give partial answers to problems
Don't ask for hints how to solve a problem
Don't start a new topic for a problem if there already exists one
See also the topics:
Don't post any spoilers
Comments, questions and clarifications about PE problems.

 Posts: 4
 Joined: Mon May 25, 2009 6:21 pm
Problem 143
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?
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?
 daniel.is.fischer
 Posts: 2400
 Joined: Sun Sep 02, 2007 11:15 pm
 Location: Bremen, Germany
Re: Problem 143
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.
Re: Problem 143
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 ?? =\
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 ?? =\
 daniel.is.fischer
 Posts: 2400
 Joined: Sun Sep 02, 2007 11:15 pm
 Location: Bremen, Germany
Re: Problem 143
No idea. That's what I get, too.
Il faut respecter la montagne  c'est pourquoi les gypaètes sont là.
Re: Problem 143
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.
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.
Re: Problem 143
Ahh !! I realized my mistake!!
This is extremely obscure ...
This is extremely obscure ...
Re: Problem 143
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
give a little hint about the mistake: is the 508 triangles too
large or too small?
Thanks

 Posts: 6
 Joined: Wed Jun 02, 2010 10:43 pm
Re: Problem 143
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!!!
... My special thanks to hk ...
The Hofman's proof is realy remarkable!!!
Last edited by kosiu_drumev on Sat Jul 09, 2011 9:50 pm, edited 2 times in total.
Re: Problem 143
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.
Re: Problem 143
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.
Re: Problem 143
I've posted a question on the solved forum. Could anyone please answer it?
4x Intel(R) Core(TM) i32330M CPU @ 2.20GHz
fabas indulcet fames
fabas indulcet fames
Re: Problem 143
Answered.
Re: Problem 143
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.
4x Intel(R) Core(TM) i32330M CPU @ 2.20GHz
fabas indulcet fames
fabas indulcet fames
Re: Problem 143
I'm afraid my reply could become rather lengthy and probably could mean a few questionanswer 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).
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).
Re: Problem 143
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.
4x Intel(R) Core(TM) i32330M CPU @ 2.20GHz
fabas indulcet fames
fabas indulcet fames
Re: Problem 143
There appears to be a minor discrepancy between the statement of problem 143 and the accompanying diagram. The statement of the problem says
I gather that this is probably not the reason problem 143 has stumped so many people, though.
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.XB = q, and XC = r
I gather that this is probably not the reason problem 143 has stumped so many people, though.
Re: Problem 143
karluk wrote:There appears to be a minor discrepancy between the statement of problem 143 and the accompanying diagram. The statement of the problem saysfor 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.XB = q, and XC = r
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.
Re: Problem 143
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.The problem states that for any point inside the triangle (let's call it X) we have XB=r
Re: Problem 143
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.
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.