Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m-1}$, such that $ \displaystyle n = \sum_{i=1}^{m-1} a_i(m-a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m-1}$ may not distinct and $ 1 \leq a_i \leq m-1$.
Problem
Source: China TST 2004 Quiz
Tags: algebra unsolved, algebra
hxy09
18.04.2009 15:25
I have to admit that it is indeed an awful and ugly problem.The answer is $ 2,3,5,6,7,8,13,14,15,17,19,21,23,26,27,30,47,51,53,55,61$ The solution was extremely complicated and required much calculation. But we can easily prove there only exist finitely $ n$ satisfy the condition by Lagrange 4-square Theorem.
FelixD
02.05.2010 00:32
After having solved this problem today, I thought exactly the same xD But IMHO, this problem isn't even hard, just much to much to compute
gauman
13.03.2012 15:20
hxy09 wrote: I have to admit that it is indeed an awful and ugly problem.The answer is $ 2,3,5,6,7,8,13,14,15,17,19,21,23,26,27,30,47,51,53,55,61$ The solution was extremely complicated and required much calculation. But we can easily prove there only exist finitely $ n$ satisfy the condition by Lagrange 4-square Theorem. $a_i =1$ then every square sastify!!!!!