Two locations $A$ and $B$ are 225 kilometers apart. The price of a metric ton of coal is 3 crowns 75 fillers in $A$ and 4 crowns 25 fillers in $B$, and the transportation cost is 8 fillers per ton and per kilometer. At which point on the line $AB$ will the coal cost the same, whether we bring it from $A$ or $B$? - Furthermore, show that along the entire line $AB$, the coal is most expensive precisely at this point.
The price of coal at point $C$, which is $x$ kilometers from $A$ and $225-x$ kilometers from $B$, is on one hand

$$
3.75 + 0.008 x
$$

and on the other hand

$$
4.25 + 0.008 (225 - x)
$$

where the coal can be brought to a single price whether from $A$ or $B$:

$$
\begin{gathered}
3.75 + 0.008 x_{0} = 4.25 + 1.8 - 0.008 x_{0} \\
0.016 x_{0} = 2.3 \\
x_{0} = 143.75 \text{ km}
\end{gathered}
$$

The price of coal at distances $x_{0}$ is given by the expression under 2), and the price differences to the right and left of $x = x_{0}$ are as follows:

and

$$
\begin{array}{cc}
0.008 (x_{0} - x) & x_{0} \leq x < x_{0} \\
0.008 (x - x_{0}) & x_{0} < x \leq 225
\end{array}
$$

which are positive in both cases, and thus the price at $x = x_{0}$, or 4 crowns 90 fillers, is indeed the highest.

Berzenczey Domokos, Déva; Debreczeni State Higher Real School, 7th grade; Jankovich György, Losoncz; Kutlik Igor, Pozsony; Pollák Sándor, Győr; Schiller Jenő and Szabó Gusztáv, Győr; Sztrapkovits István, S.-A.-Ujhely; Visnya Aladár and Ernő, Pécs. Imre János, Nyíregyháza