Pigeonhole Problem
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10 real numbers are chosen between 1 and 55 prove that there are at least three numbers forming a triangle.
(Combinatorics e hathekhori)
(Combinatorics e hathekhori)
- Mehrab4226
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Re: Pigeonhole Problem
I couldn't solve this using PHP.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
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-Henri Poincaré
- Anindya Biswas
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Re: Pigeonhole Problem
Pretty satisfying solution, I don't know if I can apply PHP here, but this was a really nice solution (I never thought fibonacci sequence would come up here in this question)
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
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Re: Pigeonhole Problem
An analogous problem was posted in aops PHP wiki page maybe manhattan something olympiad which was solved by dividing the interval and applying php. But this problem was in php chapter that's why i thought php might work though i have a stern belief that it can be solved by php same way as in the aops. (Though I am a nooB :/ )
Hmm..Hammer...Treat everything as nail
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Re: Pigeonhole Problem
a mistake numbers are on open interval (1,55). (Still your proof is right)
Hmm..Hammer...Treat everything as nail
- Mehrab4226
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Re: Pigeonhole Problem
Then the selected numbers are not necessarily distinct! They can have repetition too.Asif Hossain wrote: ↑Tue Feb 09, 2021 5:50 pma mistake numbers are on open interval (1,55). (Still your proof is right)
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré
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Re: Pigeonhole Problem
I didn't understand the arguement Let there are $10$ numbers such that no three of them forms a triangle, in other words, there is no three of them having the property $a+b>c$, thus all have the property $a+b \leq c$ does it imply the triangle inequality?(As triangle inequality is a+b>c, b+c>a, c+a>b. does a+b>c imply all of them??)
Hmm..Hammer...Treat everything as nail
- Mehrab4226
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Re: Pigeonhole Problem
Asif Hossain wrote: ↑Tue Feb 09, 2021 8:31 pm
I didn't understand the argument Let there are $10$ numbers such that no three of them forms a triangle, in other words, there is no three of them having the property $a+b>c$, thus all have the property $a+b \leq c$ does it imply the triangle inequality?(As triangle inequality is a+b>c, b+c>a, c+a>b. does a+b>c imply all of them??)
Oh sorry, I should have mentioned that $c$ is the largest of them. Then we only have to worry about $a+b > c$ As the others are true no matter what a,b,c are, where c is the largest.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré
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Re: Pigeonhole Problem
Mehrab4226 wrote: ↑Tue Feb 09, 2021 8:36 pmAsif Hossain wrote: ↑Tue Feb 09, 2021 8:31 pm
I didn't understand the argument Let there are $10$ numbers such that no three of them forms a triangle, in other words, there is no three of them having the property $a+b>c$, thus all have the property $a+b \leq c$ does it imply the triangle inequality?(As triangle inequality is a+b>c, b+c>a, c+a>b. does a+b>c imply all of them??)
Oh sorry, I should have mentioned that $c$ is the largest of them. Then we only have to worry about $a+b > c$ As the others are true no matter what a,b,c are, where c is the largest.
But would that hold if repitition is allowed??
Hmm..Hammer...Treat everything as nail
- Mehrab4226
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Re: Pigeonhole Problem
Yes. $a_1 >1,a_2 > 1$ so $a_1+a_2 \leq a_3 > 2$ and so on.Asif Hossain wrote: ↑Tue Feb 09, 2021 8:56 pmBut would that hold if repitition is allowed??Mehrab4226 wrote: ↑Tue Feb 09, 2021 8:36 pmOh sorry, I should have mentioned that $c$ is the largest of them. Then we only have to worry about $a+b > c$ As the others are true no matter what a,b,c are, where c is the largest.Asif Hossain wrote: ↑Tue Feb 09, 2021 8:31 pm
I didn't understand the argument Let there are $10$ numbers such that no three of them forms a triangle, in other words, there is no three of them having the property $a+b>c$, thus all have the property $a+b \leq c$ does it imply the triangle inequality?(As triangle inequality is a+b>c, b+c>a, c+a>b. does a+b>c imply all of them??)
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré