Okay...

*First I would like to say that I JUST REALIZED THAT THE PLACE WAS AOPS-LANDIA. :)*

1. The cyclist has traveled on the hilly road to Beast Island from Aopslandia and then back to Aopslandia, which means that the cyclist went uphill once at a rate of 20km per hour and went downhill once at a rate of 30km per hour.

2. Let the distance between Aopslandia and Beast Island be D (kilometers), and let the time needed to travel from Aopslandia to Beast Island be T1 (hours). Let the time needed to travel back from Beast Island to Aopslandia be T2 (hours).

3. The formula for average speed is Total Distance divided by Total Time. In this case, the total distance is D+D=2D, and the total time is T1+T2 hours, so the average speed will be (2D)/(T1+T2). Time equals distance over speed, so we can rewrite the expression into (2D)/(D/20)+(D/30). The least common multiple of 20 and 30 is 60, so the expression will turn into (2D)/(3D/60+2D/60)=(2D)/(5D)/60. Let's apply the fraction rule a/b/c=(a*c)/b turning the expression into (2D*60)/(5D)=(120D)/(5D). 120/5=24, so the cyclist's average speed during the entire round trip is 24km per hour.

**Answer: 24km per hour **