498. A courier departs from $A$ and on the first day travels 10 li, and each subsequent day travels $\frac{1}{4}$ li more. After three days, another courier departs from city $B$, located 40 li behind city $A$, and travels in the same direction, traveling seven li on the first day, and each subsequent day traveling $\frac{2}{3}$ li more. How many days after the departure of the first courier will both couriers meet?

## Mannheim Problem.
498. Sturm's Solution. The required number of days is $x$. The distance traveled by the first courier is the sum of the terms of an arithmetic progression, the extreme terms of which are 10 and $10+\frac{x-1}{4}$, i.e., it is equal to $\left(20+\frac{x-1}{4}\right) \frac{x}{2}$, or $\frac{(79+x) x}{8}$. The second courier is on the road until meeting the first for $x-3$ days and travels

$$
\left[14+\frac{(x-4) \cdot 2}{3}\right] \cdot \frac{x-3}{2}
$$

or

$$
\frac{(17+x)(x-3)}{3} \text { leagues. }
$$

According to the problem,

$$
\frac{(79+x) x}{8}-\frac{(17+x)(x-3)}{3}-40=0
$$

or

$$
5 x^{2}-125 x+552=0
$$

from which

$$
x_{1}=5.72 \ldots, x_{2}=19.27 \ldots
$$

It was assumed that $x$ is an integer, so the roots of the equation do not seem to satisfy the condition. However, it can be shown that the integer parts of the roots (5 and 19) indicate that there were two meetings: one after 5 days, and the second after 19 days.

Firstly, if $a$ is the distance traveled by the first courier, and $b$ is the distance traveled by the second courier increased by 40 leagues, then, assuming that the first is on the road for an integer number $x$ days, and the second for an integer number $x-3$ days, we have the identity:

$$
5 x^{2}-125 x+552=24(b-a)
$$

This is obvious, as equation (2) is derived from (1) by changing the signs of all terms and multiplying them by 24. Now, substituting 5 and then 6 for $x$ in the left-hand side of the second equation, since the smaller root $5.72 \ldots$ lies between these numbers, the result of the first substitution is $>0$, and the second is $b$. Thus, the first meeting, as stated, took place between the fifth and sixth day. Similarly, we can prove that the second meeting took place after 19 days. The possibility of this second meeting is easily understood, as the second courier, increasing his speed more than the first, will meet him after being overtaken initially by the first. This is confirmed by determining after how many days both couriers have the same speed. We find the number of days to be 13, which lies between 5 and 19.

Mannheim, the well-known French geometer.