A boarding school installed a shower room for 100 students. How many shower heads are economical, if the boiler preheating takes 3 minutes for each head; in addition, it must also run during the shower. We allocate 12 minutes for each group.
I. Solution: Let the number of shower heads be denoted by $x$, and the total heating time by $T$. Then:

$$
3 x+12 \cdot \frac{100}{x}=T
$$

We are looking for the positive integer value of $x$ for which $T$ is the smallest possible.

$$
\begin{gathered}
3 x^{2}-T x+1200=0 \\
x_{1,2}=\frac{T \pm \sqrt{T^{2}-12^{2} \cdot 10^{2}}}{6}
\end{gathered}
$$

This gives a real or zero solution for $x$ only if the expression under the square root is positive, so the smallest possible value of $T^{2}$ is: $T^{2}=12^{2} \cdot 10^{2}$. From this, $T=120$ (since it cannot be negative). Substituting this into the equation for $x$ gives $x=20$, which is an integer, so under the conditions of the problem, 20 shower heads are the most economical.

Péter Szépfalusy (Szeged, "András Dugonics" Gymnasium VII. b. class)

II. Solution: If the number of shower heads is $x$ and the number of groups is $y$, then $x y=100$. The heating time is $n=3 x+12 y=3(x+4 y)$. By the arithmetic and geometric mean inequality,

$$
\frac{x+4 y}{2} \geq \sqrt{4 x y}=20
$$

Equality holds if $x=4 y$, in which case $x=20, y=5$ and $n=120$ minutes $=2$ hours. If there were a fractional group, then $x y>100$, so $\frac{x+4 y}{2}>20$, thus indeed 20 shower heads are the most economical.

László Gehér (Zalaegerszegi Gymnasium VII. class)

Note: Except for the first solution, the solvers assumed that in addition to $x$, $\frac{100}{x}$ is also an integer. Some started from this and tried all possible values of $x$. This method of solving is fortunate when it is ensured that a question can be answered by a finite number of trials, and thus trying these cases is a sure way to provide an answer. In our case, however, it is conceivable that the most economical solution might involve the last group not using all the showers. If we allow for a 4-minute preheating time, we already encounter such a case. The additional note to the second solution shows that this possibility could have been considered.