## Problem 143

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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?
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 gypa&egrave;tes sont l&agrave;.
zwuupeape
Posts: 189
Joined: Tue Jun 09, 2009 6:11 pm

### 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 ?? =\
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 gypa&egrave;tes sont l&agrave;.
zwuupeape
Posts: 189
Joined: Tue Jun 09, 2009 6:11 pm

### 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.
zwuupeape
Posts: 189
Joined: Tue Jun 09, 2009 6:11 pm

### Re: Problem 143

Ahh !! I realized my mistake!!

This is extremely obscure ...
deng
Posts: 1
Joined: Sat May 07, 2011 8:42 pm

### 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
kosiu_drumev
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!!!

Last edited by kosiu_drumev on Sat Jul 09, 2011 9:50 pm, edited 2 times in total.
hk
Posts: 10991
Joined: Sun Mar 26, 2006 10:34 am
Location: Haren, Netherlands

### Re: Problem 143

Jamie
Posts: 11
Joined: Sat Feb 28, 2009 9:40 am

### 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.
TripleM
Posts: 382
Joined: Fri Sep 12, 2008 3:31 am

### 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.
thedoctar
Posts: 74
Joined: Fri Apr 15, 2011 11:57 am
Location: Sydney, Australia

### Re: Problem 143

I've posted a question on the solved forum. Could anyone please answer it?
4x Intel(R) Core(TM) i3-2330M CPU @ 2.20GHz

fabas indulcet fames
hk
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Location: Haren, Netherlands

### Re: Problem 143

thedoctar
Posts: 74
Joined: Fri Apr 15, 2011 11:57 am
Location: Sydney, Australia

### 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) i3-2330M CPU @ 2.20GHz

fabas indulcet fames
hk
Posts: 10991
Joined: Sun Mar 26, 2006 10:34 am
Location: Haren, Netherlands

### Re: Problem 143

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).
thedoctar
Posts: 74
Joined: Fri Apr 15, 2011 11:57 am
Location: Sydney, Australia

### 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) i3-2330M CPU @ 2.20GHz

fabas indulcet fames
karluk
Posts: 2
Joined: Wed Jul 10, 2013 1:14 am

### 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
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.
TheEvil
Posts: 84
Joined: Sun Nov 13, 2011 10:38 am
Location: Szeged, Hungary

### 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 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.
karluk
Posts: 2
Joined: Wed Jul 10, 2013 1:14 am

### Re: Problem 143

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.
hk