Someone wants to pay 682 forints using 10, 5, and 1 forint banknotes. How many of each type of banknote should they pay? How many of each type of banknote should they give if the number of 10 forint banknotes given is equal to the number of 5 and 1 forint banknotes given.
According to the first part of the task,

$$
10 x+5 y+z=682
$$

and since the coefficient of $z$ is the positive unit, the solution of the equation is as follows:

$$
\begin{gathered}
x=u \\
y=v \\
z=682-10 u-5 v
\end{gathered}
$$

If we add the following to the equation under 1):

$$
x=y+z
$$

which expresses the condition stated in the second part of the task, and substitute the value of $x$ thus obtained into 1), we get the following equation:

$$
15 y+11 z=682
$$

From this,

$$
\begin{gathered}
z=\frac{682-15 y}{11}=62-y-\frac{4 y}{11} \\
\frac{4 y}{11}=u, \quad y=\frac{11 u}{4}=2 u+\frac{3 u}{4} \\
\frac{3 u}{4}=v, \quad u=\frac{4 v}{3}=v+\frac{v}{3} \\
\frac{v}{3}=t, \quad v=3 t \\
u=4 t, \quad y=11 t, \quad z=62-15 t, \quad x=62-4 t
\end{gathered}
$$

Thus, the solutions are

$$
\begin{gathered}
x=62-4 t \\
y=11 t \\
z=62-15 t
\end{gathered}
$$

where $t$ can be $1, 2, 3,$ and $4$.

Then

$$
\begin{aligned}
& x=58, \quad 54, \quad 50 \text { and } 46 \\
& y=11, \quad 22, \quad 33 \text { and } 44 \\
& z=47, \quad 32, \quad 17 \text { and } 2
\end{aligned}
$$

(Fleischner Illés, 5th class of the national rabbinical training institute (fögymn. VIII.o. t., Budapest))

The task was also solved by: Baruch Jenő and Suták Sándor, fögymn. VIII. o. t., Nyíregyháza; ifj. Breznyik János, lyc. t., Selmeczbánya; Fried Ármin, föreálisk. VII. o. t., Déva; Heymann Tivadar, för. VIII.o. t., Györ; Hirschler Ârmin, föreálisk. VIII. o. t., Györ; ifj. Imre János főgymn. VIII. o. t., Nyiregyháza; Pilczer Ignácz, fögymn. VIII. o. t., Kaposvár; Prónai Gyözö, fögymn. VIII. o. t., Beszterczebánya; Schulhof Gábor, föreálisk. VII. o. t., Pécs; Visnya Aladár, fóreálisk. VII. o.t., Pécs.