B3. Ana and Beno are cycling on perpendicular cycling paths in the direction of the clock hands. Ana is cycling on the shorter cycling path, while Beno is on the longer one. The speeds of Ana and Beno are in the ratio $v_{\text {Ana }}: v_{\text {Beno }}=2: 7$. Both start simultaneously at point $S$ (see the figure, which also shows the lengths of the individual path segments). How many laps

![](https://cdn.mathpix.com/cropped/2024_06_07_a0d1506e8e2028d6d427g-05.jpg?height=305&width=671&top_left_y=193&top_left_x=1229)
does Ana and how many Beno complete when they first meet again at point $S$?

## 58th Mathematical Competition for High School Students in Slovenia  Regional Competition, April 2, 2014

## Problems for 2nd Year

Time for solving: 120 minutes. In section A, we will score a correct answer with two points, and deduct one point for an incorrect answer. Enter the answers for section A in the left table, leave the right table blank.
![](https://cdn.mathpix.com/cropped/2024_06_07_a0d1506e8e2028d6d427g-06.jpg?height=170&width=802&top_left_y=594&top_left_x=640)
B3. Ana's lap is $1000 \mathrm{~m}$ long, and Ben's is $1200 \mathrm{~m}$. Suppose they meet again at point $S$ after Ana has completed $n$ laps and Ben has completed $k$ laps. Ana completes $n$ laps in time $n \cdot \frac{1000 \mathrm{~m}}{v_{\text {Ana }}}$, and Ben completes $k$ laps in time $k \cdot \frac{1200 \mathrm{~m}}{v_{\text {Beno }}}$. These two times must be equal, so we have

$$
n \cdot \frac{1000 \mathrm{~m}}{v_{\mathrm{Ana}}}=k \cdot \frac{1200 \mathrm{~m}}{v_{\mathrm{Beno}}}
$$

From this equation, we can express

$$
\frac{n}{k}=\frac{1200 \mathrm{~m}}{1000 \mathrm{~m}} \cdot \frac{v_{\text {Ana }}}{v_{\text {Beno }}}=\frac{6}{5} \cdot \frac{2}{7}=\frac{12}{35}
$$

Since we are looking for the smallest possible $n$ and $k$, and the fraction $\frac{12}{35}$ is already in its simplest form, it follows that $n=12$ and $k=35$. Therefore, the first time they meet again at point $S$, Ana has completed 12 laps, and Ben has completed 35 laps.

Writing or using the length of Ana's path ($n \cdot 1000 \mathrm{~m}$) and the length of Ben's path ($k \cdot 1200 \mathrm{~m}$) from the start to the first meeting 1 point

Writing the equation $n \cdot \frac{1000 \mathrm{~m}}{v_{\text {Ana }}}=k \cdot \frac{1200 \mathrm{~m}}{v_{\text {Beno }}}$ 2 points

Expressing the ratio $\frac{n}{k}=\frac{12}{35}$ 1 point

Argued conclusion that $n=12$ and $k=35$

2 points (If the contestant just writes down the solution, 1 point is awarded.)

## Solutions for 2nd Year

In section A, a correct answer will be worth two points, while an incorrect answer will result in one point being deducted. To avoid a potential negative final score, each contestant is given 3 initial points.

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| $\mathrm{A}$ | $\mathrm{E}$ | $\mathrm{B}$ |

Justifications: