Difficult and beautiful 1 (open problem)

[ins-]

no one had solved ?!

[e2e4]

ins- писал(а):

e2e4, please, don't underestimate this problem. You may be right but even some proffesors failed with solving.

I can only can quote:

незваный гость писал(а):

Can you prove, it did not happen for lack of interest?

I saw many similar problems in collections of olympic problems for scholars, and in collections of entrance examinations on mathematical facultys of some technical institutes.
It is not an easy task, also. It is more complex, than standart school problems, of course. But it is also not the professor's level task .

ins- писал(а):

I also don't think your lemma is true. Try the following:

1. Choose one equaliteral triangle UVW.
2. Extend UV, VW, WU rays to the points A, B, C.
3. Extend UV, VW, WU to the meeting points M, N, P with the sides of ABC.

I think there are infinitely many triangles and you cannot warrant the rotation angles are all equals.

You think, or you can proof? I tryed to proof my statement, if in my proof is any errors, please, point it. A turn "I think" is not a mathematical argument.

I will try today to translate a proof to english, with some addon, to make it more clear. The only objection, that I saw, was TOTAL's "Расстояние не выбирается произвольно" - the length is not choosing random. But I think, it is evident from the task's condition.

[TOTAL]

e2e4 писал(а):

I will try today to translate a proof to english, with some addon, to make it more clear. The only objection, that I saw, was TOTAL's "Расстояние не выбирается произвольно" - the length is not choosing random. But I think, it is evident from the task's condition.

Очевидно, Вы неверно понимаете условие. От вершин треугольника на сторонах отложили одинаковые расстояния, соответствующий треугольник внутри оказался правильным. Вот так надо понимать условие. Вы же требуете, чтобы при откладывании любых одинаковых расстояний получался правильный треугольник. Для Вашего неверного понимания условия доказывать вообще ничего не нужно, так как при откладывании нулевого расстояния правильным внутренним треугольником оказывается сам исходный треугольник.

[ins-]

You are right - I cannot prove or disprove it is due to a lack of interest - but in math forums I saw - that problem was seen between 3000 and 4000 times. So I can think at least 3-4 people tried to solve it but they failed.

Your sources of problems are very interesting if they contain problems with the same level of difficulty.

I know "I think" or "I feel" is not a math argument. It is the reason I mentioned the construction steps.
If you play with that construction, using math drawing software you will see what I want to say. If you wish I'll try to prove my point of view.

Yes, it will be interesting to see your proof in English with corrections mentioned by TOTAL. To solve such problem a man should be very skilled matematician (in my opinion).

I friend of mine said equations from guaravpatil in mathlinks.ro also lead to correct solution but used in different way. But there are two cases
1) he is lost in calculations,
2) he is right and he don't have enough time.

[ins-]

Please, check the following solution. I think it is almost correct, but I want someone else to check it.

http://www.mathlinks.ro/Forum/viewtopic ... 86#1180286

also, please translate in "normal" language what 3) and 4) means.

[ins-]

http://www.mathlinks.ro/viewtopic.php?t=214127 I think this is very interesting.