Problem 6. Vasya stood at the bus stop for some time. During this time, one bus and two trams passed by. After some time, Spy came to the same stop. While he was sitting there, 10 buses passed by. What is the minimum number of trams that could have passed during this time? Both buses and trams run at equal intervals, and buses run at an interval of 1 hour. [9 points] (M.A. Akhmanova)
Answer: 4 trams.

Solution. First, let's provide an example where four trams passed during the Spy's observation time: suppose buses run at 9:00, 10:00, ..., and trams run with an interval of 1 hour and 58 minutes - at 10:01, 11:59, 13:57, 15:55, 17:53, 19:51, 21:49, ... Then Vasya could have been standing from 10:01 to 11:59, and the Spy could have been observing from 12:00 to 21:00.

![](https://cdn.mathpix.com/cropped/2024_05_06_a29d822a105b57015a7cg-7.jpg?height=197&width=1227&top_left_y=1149&top_left_x=114)

Fig. 7

Now let's prove that during the Spy's observation time, fewer than four trams could not have passed.

To do this, we will prove that the tram interval is less than two hours. Indeed, if the interval is at least two hours, then Vasya stood for at least two hours, and during this time at least two buses passed, which contradicts the condition.

On the other hand, while the Spy was observing, ten buses passed. Between the 1st and 3rd buses, two hours passed, so at least one tram passed during this time. Similarly, between the 3rd and 5th buses, between the 5th and 7th buses, and between the 7th and 9th buses, at least three more trams passed. Therefore, the Spy saw at least four trams.

## INFORMATION ON ENROLLMENT IN $5-8$ GRADES

with advanced mathematics study in 2008

| School | Phone | Address | Grades | Terms |
| :---: | :---: | :---: | :---: | :---: |
| 2 | $137-17-69$  $137-69-31$ | Fotieva St., 18  (m. "Oktyabrskaya") | 7,8 | application acceptance from January 31 |
| 54 | $245-99-72$  $245-54-25$ | Dovatora St., 5  (m. "Sportivnaya") | 8 | March to May |
| 57 | $291-85-72$  $291-54-58$ | Mal. Znamensky per.,  7/10, bld. 5  (m. "Borovitskaya") | 8 | from March 26, Wednesdays at 16 |
| 179 | $692-48-51$ | Bol. Dmitrovka St., 5/6,  bld. 7 (m. "Okhotny Ryad") | $6,7,8$ | March |
| 192 | $137-33-55$ |  |  |  |
| $137-72-85$ | Leninsky Prospekt 34-A  (m. "Leninsky Prospekt") | 5,$7 ;$  additional in 6, 8 | March to May, Fridays at 16 |
| 218 | $976-19-85$ | Dmitrovskoye Shosse, 5a | $6-8$ | registration for interviews from March 17 |
| 444 | $465-60-52$ | Nizh. Pervomayskaya St.,  14, m. "Pervomayskaya" | 8 | March |
| 1514 | $131-80-38$  $131-80-33$ | Krupeiskaya St., 12  (m. "University") | $5 ;$  additional in 8 | from February;  additional March to April |
| 1543 | $433-16-44$ | 26 Baku Commissars St., 3, bld. 5  (m. "Yugo-Zapadnaya") | 8 | April |
| Intel-  lectual | $445-52-10$ | Kremenchugskaya St., 13  (m. "Park Pobedy") | 5,$7 ;$  additional in 6, 8 | registration from February,  exams from late March |

Information provided by the schools to the MCCME. Published free of charge.

Detailed information on enrollment in these and other classes on the website www.mccme.ru

The 6th oral mathematics olympiad for $6-7$ grades will take place on March 16, 2008.

Students who have received a diploma or certificate are invited.