Consider a finite set of balls each of which weighs at most 10 Kg.It's known that every arbitrary decomposition of these balls into two groups results in that exactly one of the groups has a maximum total weight of 10 Kg.Find the maximum weight of all balls(Total weight of all balls summed up).
Choices: $20 , 25 , 30 , 35 , 40$
No idea...
Assume that the two groups are required to be non-empty (else $(10)$ is the only solution).
Note that $(10,10,10)$ works, so it will suffice to show that there is no group that adds to greater than $30$.
Suppose that we had a group that added to $N>30$. In that case, each ball must weigh exactly $10$. Pf: suppose we had a ball of weight $W<10$. Then isolating that ball gives one group of weight $W$ and another of weight greater than $20$, a contradiction.
Thus we can assume that every ball in the group weighs exactly $10$. But if there were more than $3$ of them we could could take $2$ in one group and the rest in the other, again a contradiction.