Consider the \(4!\) possible permutations of the numbers \(1,2,3,4\). Which of those permutations keep the expression \(x_1x_2+x_3x_4\) the same?
Show that if \(1+3+5+7+...+97+99=50^2\), then \(1+3+5+7+...+97+99+101=51^2\).
Prove that for all positive integers \(n\) there exists a partition of the set of positive integers \(k\le2^{n+1}\) into sets \(A\) and \(B\) such that \[\sum_{x\in A}x^i=\sum_{x\in B}x^i\] for all integers \(0\le i\le n\).