Two couriers set off from two locations, $A$ and $B$; the first towards $B$, the second towards $A$. At the point of meeting, the first had traveled 12 miles more than the second. If they continue their journey at their original speeds, the first will reach their destination 9 days after the meeting, while the second will reach their destination 16 days after the meeting. How far apart are the two locations from each other?
Suppose the two couriers meet $x$ days later, then the first one takes $x+9$ days, and the second one takes $x+16$ days to complete the route $AB=y$.

The first courier covers $\frac{y}{x+9}$ miles in one day, and $\frac{xy}{x+9}$ miles in $x$ days; the second courier covers $\frac{xy}{x+16}$ miles in $x$ days, and the sum of the two distances equals the entire route. In symbols:

$$
\begin{aligned}
& \frac{xy}{x+9}+\frac{xy}{x+16}=y \\
& \frac{x}{x+9}+\frac{x}{x+16}=1
\end{aligned}
$$

From this,

$$
\begin{gathered}
x(x+16)+x(x+9)=(x+9)(x+16) \\
2x^2+25x=x^2+25x+144 \\
x^2=144 \\
x=12 \text{ days. }
\end{gathered}
$$

The distances covered by the first and second couriers in 12 days are $\frac{y}{2}+6$ miles and $\frac{y}{2}-6$ miles, respectively. These distances are inversely proportional to each other, as are the speeds of the couriers, that is,

$$
(y+12):(y-12)=28: 21
$$

or

$$
\begin{gathered}
28y-12 \times 28=21y+21 \times 12 \\
7y=12 \times 49 \\
y=12 \times 7=84
\end{gathered}
$$

(Geist Emil, real gymnasium VII. class, Györ).

The problem was also solved by: The ag.hitv.ev. high school VIII. class, Budapest; Dörnyei Károly, Győr; Friedmann Bernát, S.-A.-Ujhely; Galter János, Sz. Udvarhely; Grünhut Béla, Pécs; Kohn Márkus, Pécs; Rumpler Ernő, Győr; Visnya Aladár and Weisz Lipót, Pécs.