Dat cool problem

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Phlembac Adib Hasan
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Dat cool problem

Unread post by Phlembac Adib Hasan » Wed Oct 01, 2014 8:20 pm

$a_1<a_2<\cdots <a_n$ are positive integers such that $\dfrac{a_{i-1}^2+a_i^2}2$ is a perfect square for every positive integer $i<n$. Find the minimum value of $a_n$ in terms of $n$.

$\small \textbf{Some personal comments:}$ We were given this amazing problem in an exam of the SSC camp. I enjoyed every minute I spent to solve it, albeit it took considerably long time. I can never recall a problem that I have solved in the very last week. But my brain still remembers this one! This proves how much it liked this problem. :) So, everyone, have fun solving it!
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*Mahi*
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Re: Dat cool problem

Unread post by *Mahi* » Wed Oct 01, 2014 9:57 pm

Hint 1 (easy):
${\color{White} {\text{For the minimum such series, the averages are } 1^2, 2^2, \cdots (n-1)^2}}$
Hint 2:
${\color{White} {\text{Split the rangesto the limit, and prove no further improvement is possible }}}$
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Samiun Fateeha Ira
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Re: Dat cool problem

Unread post by Samiun Fateeha Ira » Wed Oct 01, 2014 11:24 pm

I think the term was $\frac{{a_{i-1}}^2+{a_i}^2}{2}$ in our exam!

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Phlembac Adib Hasan
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Re: Dat cool problem

Unread post by Phlembac Adib Hasan » Thu Oct 02, 2014 8:59 am

Edited. I think it's alright now.
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