Discussion on Bangladesh National Math Camp
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*Mahi*
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by *Mahi* » Sun Apr 01, 2012 1:07 pm
Phlembac Adib Hasan wrote:I have come from school only a few minutes ago and solved this problem first.
Solution :
This is only half the case.What if $k=2^x>m$?
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Phlembac Adib Hasan
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by Phlembac Adib Hasan » Sun Apr 01, 2012 3:16 pm
I am not realizing.Will you please explain?I have showed for every odd number>1 and for all the even numbers other than powers of two we can express them as sum of some consecutive numbers.Will you please give an example which part I am missing?
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*Mahi*
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by *Mahi* » Sun Apr 01, 2012 3:18 pm
sourav das wrote:Most positive integers can be expressed as a sum of two or more consecutive
positive integers. For example, $24 = 7 + 8 + 9$ and $51 = 25 + 26$. A
positive integer that cannot be expressed as a sum of two or more consecutive
positive integers is therefore interesting. What are all the interesting
integers?
The red part.
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Niloy Da Fermat
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by Niloy Da Fermat » Sun Apr 01, 2012 3:28 pm
i still don't get it.we are selecting $ m $ st $ m $ is always greater than $ k $.isn't it?
kame......hame.......haa!!!!
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*Mahi*
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by *Mahi* » Sun Apr 01, 2012 3:31 pm
Niloy Da Fermat wrote:i still don't get it.we are selecting $ m $ st $ m $ is always greater than $ k $.isn't it?
Then try choosing $m,k$ from $24=3 \times 8 = 3 \cdot 2^3 =(2 \cdot 1+1) 2^3 = (2 \cdot 2 -1) 2^3$
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Phlembac Adib Hasan
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by Phlembac Adib Hasan » Sun Apr 01, 2012 4:15 pm
Why?Here I'll take $m=8$,$k=1$.I am still not realizing.
I said I can find some $k$ and $m$ for every even number except the powers of two.,But I am not giving any exact value.I think you are pointing to this:How can it be true for $10$?It can't be expressed as sum of five consecutive integers.But notice that I also wrote something about the even case.
Here $10=2.5=2.(2.3-1)$.Hence we have to take four numbers.So I can take $m>k$.
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Ehsan - Posts:6
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by Ehsan » Sun Apr 01, 2012 4:44 pm
It's the problem 2.4.15 from The Art And Craft of Problem Solving.
There it was said, "algebra can be used, but so can pictures."
Try solving by picture, it's much interesting and simple.
Every even integer is a sum of two primes.
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Masum
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by Masum » Sun Apr 01, 2012 5:18 pm
You can actually conclude that if a number can be expressed as a sum of some consecutive numbers in $d(n)$ ways, then $d(n)$ is the number of odd divisors of $n$. That also makes a sense why powers of $2$ won't satisfy the condition.(reverse thinking)
One one thing is neutral in the universe, that is $0$.