Let $N=\{1,2,3,\ldots, n\}$.  
We sum all the elements of every nonempty subset of $N$.  
Which sum(s) appears most often? (Let's call this sum a *champion*).  
Using a simple pigeonhole argument a champion must appear at least $\frac{2^n-1}{T_n}$ times. ($T_n$ denotes the $n$-th triangular number).  
It seems that the *champion* should be somewhere around $T_n/2$ but I cannot prove it.  
   
Am I missing something obvious here?