13.359. Find two two-digit numbers with the following property: if to the larger of the sought numbers a zero is appended on the right and then the smaller number, and to the smaller number the larger number is appended on the right and then a zero, then from the two five-digit numbers thus obtained, the first, when divided by the second, gives a quotient of 2 and a remainder of 590. In addition, it is known that the sum, composed of twice the larger sought number and three times the smaller, is equal to 72.
Solution.

Let $x, y$ be the required numbers. According to the problem, we have the system

$$
\left\{\begin{array}{l}
1000 x+y=2(1000 y+10 x)+590, \\
2 x+3 y=72,
\end{array} \text { from which } x=21, y=10\right.
$$

Answer: 21 and 10.

13. 360. A cyclist departs from $A$ to $B$. The distance from $A$ to $B$ is 60 km; the cyclist's speed is constant. Without stopping in $B$, he continues back with the same speed, but after one hour of travel from $B$, he makes a 20-minute stop. After this, he continues his journey, increasing his speed by 4 km/h. Within what range is the cyclist's speed $V$ if it is known that the time spent on the return trip from $B$ to $A$ is no more than the time spent on the trip from $A$ to $B$?

Solution.

Let $V$ be the cyclist's speed.

$$
\text { According to the problem, } \frac{60}{V} \geq 1+\frac{20}{60}+\frac{60-1 \cdot V}{V+4} \text {, from which } 0<V \leq 20 \text { (km/h). }
$$

Answer: $\quad 0<V \leq 20$ km/h

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1221.jpg?height=483&width=571&top_left_y=485&top_left_x=90)

Fig. 13.18