Consider the following problem:
You are given a multiset (a set with repetitions allowed) of $2n+1$ real numbers, say $S = \{r_1, \dots, r_{2n+1}\}$.
These numbers are such that for every $k$, the multiset $S - \{r_k\}$ can be split into two multisets of size $n$ each, such that the sum of numbers in one multiset is same as the sum of numbers in the other.
Show that all the numbers must be equal.( i.e. $r_{i} = r_{j}$)
Please stop reading further if you want to try and solve this problem.
Spoiler:
Now this problem can easily be solved using Linear Algebra. We have a set of $2n+1$ linear equations, which corresponds to a matrix equation $Ar = 0$. It can be shown that $A$ has rank at least $2n$ which implies the result.
The question is, is there any solution to this problem which does not involve any linear algebra?