Problem 6.6. Several oranges (not necessarily of equal weight) were picked from a tree. When they were weighed, it turned out that the weight of any three oranges taken together is less than $5 \%$ of the total weight of the remaining oranges. What is the smallest number of oranges that could have been picked?
Answer: 64.

Solution. 64 oranges could have been, for example, if all of them had the same mass $m$, since $\frac{3 m}{61 m}<0.05$.

Suppose there were no more than 63 oranges. Consider the three heaviest oranges with a total mass of $M$. All the other oranges (no more than 60) can be divided into no more than 20 groups, each containing no more than 3 oranges. Obviously, the total mass in each group does not exceed $M$, so the total mass in all groups does not exceed $20 M$. Since $\frac{M}{20 M}=0.05$, the three heaviest oranges account for no less than $5\%$ of the total mass of the other oranges. Contradiction.