Problem 3. One hundred carousel seats are arranged in a circle at equal intervals. Each is painted yellow, blue, or red. Seats of the same color are placed consecutively and numbered $1,2,3, \ldots$ clockwise. The blue seat № 7 is opposite the red № 3, and the yellow № 7 is opposite the red № 23. Find how many yellow, blue, and red seats there are on the carousel. $[6$ points] (
Answer. Yellow 34, blue 20, red 46.

Solution. There are 19 seats between the 3rd and 23rd red seats, so there are also 19 seats between the 7th blue and the 7th yellow seats. This includes the first six yellow seats, and thus $19-6=13$ blue seats with numbers greater than 7. Therefore, the total number of blue seats is $7+13=20$.

![](https://cdn.mathpix.com/cropped/2024_05_06_ddc2d9f47d3072b1b452g-3.jpg?height=507&width=460&top_left_y=554&top_left_x=497)

Traversing half a circle clockwise from the 3rd red seat to the 7th blue seat, we pass $(100-2): 2=49$ seats (excluding these two). Six of them are blue, and the remaining $49-6=43$ are red. Therefore, the total number of red seats is $3+43=46$. The number of yellow seats is $100-20-46=34$.