$a_1<a_2<\cdots <a_n$ 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!
Dat cool problem
- Phlembac Adib Hasan
- Posts:1016
- Joined:Tue Nov 22, 2011 7:49 pm
- Location:127.0.0.1
- Contact:
Welcome to BdMO Online Forum. Check out Forum Guides & Rules
Re: Dat cool problem
Hint 1 (easy):
${\color{White} {\text{For the minimum such series, the averages are } 1^2, 2^2, \cdots (n-1)^2}}$
Hint 2:
${\color{White} {\text{Split the rangesto the limit, and prove no further improvement is possible }}}$
${\color{White} {\text{For the minimum such series, the averages are } 1^2, 2^2, \cdots (n-1)^2}}$
Hint 2:
${\color{White} {\text{Split the rangesto the limit, and prove no further improvement is possible }}}$
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
- Samiun Fateeha Ira
- Posts:23
- Joined:Sat Aug 24, 2013 7:08 pm
- Location:Dhaka, Bangladesh
Re: Dat cool problem
I think the term was $\frac{{a_{i-1}}^2+{a_i}^2}{2}$ in our exam!
- Phlembac Adib Hasan
- Posts:1016
- Joined:Tue Nov 22, 2011 7:49 pm
- Location:127.0.0.1
- Contact:
Re: Dat cool problem
Edited. I think it's alright now.
Welcome to BdMO Online Forum. Check out Forum Guides & Rules