5. Every school from a certain region sent 3 students to the competition. Andrej, Blaž, and Žan represented the same school. When all the competitors lined up to receive their starting numbers, Andrej noticed that there were as many competitors in front of him as there were behind him. Behind him were also his two classmates: Blaž was 19th in line, and Žan was 28th. How many schools participated in the competition?

## 53rd Mathematical Competition for High School Students in Slovenia

Selection Competition, April 1, 2009

## Problems for 2nd Year Students
I/5. Let $x$ be the number of contestants before Andrej. Then there are also $x$ contestants after Andrej, making the total number of contestants $2x + 1$, which is odd. Since Andrej is ahead of Blaž, who is in 19th place, there can be at most 17 contestants before Andrej, so $x \leq 17$. Therefore, the total number of contestants is at most $2x + 1 \leq 2 \cdot 17 + 1 = 35$. Since Žan is in 28th place, there are at least 28 contestants. Each school sent 3 contestants, so the total number of contestants must be divisible by 3. Between 28 and 35, only 30 and 33 are divisible by 3, but only 33 is odd. Therefore, there were 33 contestants, which means 11 schools participated.

Conclusion that the total number of contestants is odd .............................................................................

Conclusion that there are at most 17 contestants before Andrej .................................... 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-08.jpg?height=49&width=1633&top_left_y=1643&top_left_x=220)

There are at least 28 contestants in total .......................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-08.jpg?height=60&width=1633&top_left_y=1735&top_left_x=220)

11 schools participated ............................................................................................