My brain is still dizzing but Let me show my approach (See if you can fix my problem in my proof :/)
(1,55)= (1,2) U [2,4) U [4,8) U [8,55)
IF ANY OF THE THREE NUMBERS FALL IN (1,2) OR [2,4) OR [4,8) THEN WE ARE DONE.
BUT IF AT LEAST 4 NUMBERS FALL IN [8,55)....(Shut Down)
Pigeonhole Problem
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Hmm..Hammer...Treat everything as nail
- Mehrab4226
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- Location:Dhaka, Bangladesh
Re: Pigeonhole Problem
I will only say 5 numbers must be in [8,55). As my solution says $a_6 >8$ and you can give the exact argument and show (1,8) can have at most 5 points. So the rest are in [8,55).
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
- gwimmy(abid)
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Re: Pigeonhole Problem
This is my solution:
$[1,55) = [1,1] U (1,2) U [2,3) U [3,5) U [5, 8) U [8,13) U [13,21) U [21,34) U [34,55)$
Now if we limit ourselves to picking one element from set at most, then we can have $n$ numbers such that no number satisfies $a+b>c$. But we have to pick $10$ numbers from $9$ sets and by PHP, $2$ numbers would come from a single set. So, we will end up with $3$ numbers $a,b,c$ satisfying the inequality.
$[1,55) = [1,1] U (1,2) U [2,3) U [3,5) U [5, 8) U [8,13) U [13,21) U [21,34) U [34,55)$
Now if we limit ourselves to picking one element from set at most, then we can have $n$ numbers such that no number satisfies $a+b>c$. But we have to pick $10$ numbers from $9$ sets and by PHP, $2$ numbers would come from a single set. So, we will end up with $3$ numbers $a,b,c$ satisfying the inequality.
- Mehrab4226
- Posts:230
- Joined:Sat Jan 11, 2020 1:38 pm
- Location:Dhaka, Bangladesh
Re: Pigeonhole Problem
Quite Elegantly Donegwimmy(abid) wrote: ↑Tue Apr 06, 2021 12:02 pmThis is my solution:
$[1,55) = [1,1] U (1,2) U [2,3) U [3,5) U [5, 8) U [8,13) U [13,21) U [21,34) U [34,55)$
Now if we limit ourselves to picking one element from set at most, then we can have $n$ numbers such that no number satisfies $a+b>c$. But we have to pick $10$ numbers from $9$ sets and by PHP, $2$ numbers would come from a single set. So, we will end up with $3$ numbers $a,b,c$ satisfying the inequality.
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