BdMO 2021 National Camp Discussion Thread

Discussion on Bangladesh National Math Camp
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Pro_GRMR
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BdMO 2021 National Camp Discussion Thread

Unread post by Pro_GRMR » Fri Apr 23, 2021 4:27 pm

Hey, Campers!

This thread's purpose is:-

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1. To post our solutions to the problem sets. (Using Latex is easier this way)
2. To check each other's solutions and learn from them.
3. To share hints about the problems.
Good Practices to follow:-

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1. Use [hide][/hide] to hide your hint/solution.
2. Describe what you're posting.
"When you change the way you look at things, the things you look at change." - Max Planck

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Pro_GRMR
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Example

Unread post by Pro_GRMR » Fri Apr 23, 2021 4:43 pm

This is an example.
I'm sharing a hint for problem 1 of IMO Prep Pset-1.

Problem-
Find, with proof, all positive integers $n$ for which $2^n+12^n+2011^n$ is a perfect square.
Hint-
Did you know that all odd squares are $1$ modulo 8?
"When you change the way you look at things, the things you look at change." - Max Planck

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MrCriminal
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Re: Example

Unread post by MrCriminal » Fri Apr 23, 2021 4:53 pm

Pro_GRMR wrote:
Fri Apr 23, 2021 4:43 pm
This is an example.
I'm sharing a hint for problem 1 of IMO Prep Pset-1.

Problem-
Find, with proof, all positive integers $n$ for which $2^n+12^n+2011^n$ is a perfect square.
Hint-
Did you know that all odd squares are $1$ modulo 8?
Finding the magic number is tough for newbies like me ):

Dustan
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Re: Example

Unread post by Dustan » Fri Apr 23, 2021 5:17 pm

Pro_GRMR wrote:
Fri Apr 23, 2021 4:43 pm
This is an example.
I'm sharing a hint for problem 1 of IMO Prep Pset-1.

Problem-
Find, with proof, all positive integers $n$ for which $2^n+12^n+2011^n$ is a perfect square.
Hint-
Did you know that all odd squares are $1$ modulo 8?
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Pro_GRMR
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Re: Example

Unread post by Pro_GRMR » Fri Apr 23, 2021 6:59 pm

Dustan wrote:
Fri Apr 23, 2021 5:17 pm
MrCriminal wrote:
Fri Apr 23, 2021 4:53 pm
Finding the magic number is tough for newbies like me ):
This is how I did it:
We see that $1$ obviously works.
Then show that the perfect square must be odd and so, $n$ will have to be even, If $n \geq 2$. If not then $2^n+12^n+2011^n\not\equiv1$(mod $8$)
And then we do the same thing as Dustan did in Case 1. And so, there exists only one such $n$ and it is $1$
"When you change the way you look at things, the things you look at change." - Max Planck

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Anindya Biswas
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Re: BdMO 2021 National Camp Discussion Thread

Unread post by Anindya Biswas » Fri Apr 23, 2021 9:21 pm

When $n>1$
First take mod $4$. It gives us $n$ even
Then consider mod $3$. It gives us $n$ odd
The rest should be clear.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Asif Hossain
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Re: BdMO 2021 National Camp Discussion Thread

Unread post by Asif Hossain » Sun Apr 25, 2021 10:44 am

Next problem Pls :)
Hmm..Hammer...Treat everything as nail

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