3. The image below represents a Japanese fan with 10 ribs. The angle between two adjacent ribs (which we will assume to have no thickness) measures $15^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_ce85c4bb4de565e05ebdg-2.jpg?height=420&width=619&top_left_y=2183&top_left_x=730)
a) How many ribs does a Japanese fan have if, at the same opening (the same angle between the extreme ribs) as the fan in the image, the measure of the angle between two adjacent ribs is $11^{\circ} 15^{\prime}$?

b) How many types of Japanese fans have the property that, when opened to a semicircle (the angle between the extreme ribs extended), the measure of the angle between two adjacent ribs, expressed in degrees, is an integer between 16 and $32?$

Dorina Rapcea
## Solution.

a) The opening of the fan, measured in degrees, is $15^{\circ}(10-1)=135^{\circ}$. (1p) A fan with the property that, at the same opening as the one in the image, the measure of the angle between two adjacent ribs is $11^{\circ} 15^{\prime}$, has $\frac{135^{\circ}}{11^{\circ} 15^{\prime}}+1=13$ ribs.

b) Since a straight angle has a measure of $180^{\circ}$, the measure in degrees of the angle between two adjacent ribs should be a divisor of the number 180. (1p)

The set of divisors of 180 between 16 and 32 is $\{18,20,30\}$. (2p)

Therefore, there are 3 types of Japanese fans that have the property mentioned in the statement. (1p)