A friend asked me this:
A woman gave a different number of apples to each of her five children. Any three of her children together received more apples than the remaining two children. What is the least number of apples that she could have given?
I did some 'trial and error' and I think the answer is $5$. I checked this for the cases where the least number of apples were $1, 2, 3, 4$ and $5$ and in each of those cases I chose the number of apples given to the other children to be $(x+1)$, $(x+2)$, $(x+3)$, $(x+4)$ where $x$ the least number of apples given. The reason why I chose them is because if I had chosen bigger numbers, then I'll have less chance that the sum of the lowest $3$ numbers is more than the sum of the remaining $2$ numbers (the highest).
As you can see that my line of reasoning is in no way rigorous. So my question is whether there is a way to mathematically (and without trial-and-error) solve this?