At Easter, the boys arrived at a family where 4 girls are being raised, each with a full bottle of cologne. Each boy poured half of the cologne on the girl he liked the most, 5 drops on each of the other 3 girls, and the remainder on the mother. What is the minimum number of drops of cologne that were in each bottle, if no girl received more than 3 bottles' worth of cologne, but the mother did?
Let $f$ be the number of mischievous boys, and $n$ be the number of drops in a bottle. Then a boy sprayed $\frac{n}{2}-15$ drops of cologne on his mother. Therefore,

\[
\begin{gathered}
f \cdot\left(\frac{n}{2}-15\right)>-3 n, \text{ i.e., } \\
\frac{6 n}{n-30} < f < \frac{6 n}{n-30} + 50. \text{ Thus, } n > 50. \text{ So, at least 51 drops of cologne are needed in a bottle. But our reasoning is not reversible, so further refinement is needed.}
\end{gathered}
\]

If $n>50$, then $\frac{6 n}{n-30} < f < \frac{6 n}{n-30} + 50$ with respect to the $f > \frac{6 n}{n-30} + 50$ estimate.

The value $f=15$ is possible if $n>50$. If there were 15 boys, then there was a girl who was chosen by at least four of them. She received at least 2 bottles of cologne from them, and 5 drops from the others, so she received at least $2 n + 11 \cdot 5$ drops of cologne. This is not more than 3 bottles, i.e., $3 n$ drops of cologne, if $n \leq 55$.

Thus, with $f=15$, at least 55 drops are needed. This is already achievable if four boys sprayed three girls and three boys sprayed one girl.

We need to check if $50 < n \leq 55$ and $f < 15$ can yield a better result. If $f \leq 13.5$, then for 13 or fewer boys, $n \leq 55$ (indeed, in this case $n \leq 56$), so such a reduction does not provide a better result.

If $f=14$, then from (1) $14 > \frac{6 n}{n-30}$, so $n > 52.5$. Thus, if $n \leq 53$, (1) is satisfied. Moreover, it is still possible that no girl is chosen by more than 4 boys - for example, four boys sprayed three girls, and two boys sprayed one girl - and thus the most perfumed girl received $4 \cdot \frac{2}{n} + 10 \cdot 5 = 2 n + 50$ drops of cologne, which is less than three bottles if $n \leq 53$.

Therefore, under the given conditions, there were at least 53 drops of cologne in a bottle, and this is possible if 4 boys sprayed 3 girls and 2 boys sprayed one girl.