The fast, middle and slow three cars start from the same place at the same time and catch up with the cyclist in front along the same road at the same speed. It took the three cars 6 minutes, 10 minutes and 12 minutes to catch up with the cyclist in front. Now we know that the speed of express train is 24 kilometers per hour, and the speed of bus is 20 kilometers per hour. How many kilometers does the local train run?

24 * 6/60 = 2.4km，20 * 10/60 = 10/3km。

10/3-2.4 =14/15km.

Cyclist's speed: (14/15)/(10/60-6/60) =14 km/h.

The cyclist is in front of the fast, medium and slow trains (24- 14)*6/60= 1 km, and the fast, medium and slow trains catch up at the same time.

Speed difference between local train and cycling:1/(12/60) = 5 km/h.

So the speed of the local train is:14+5 =19 km/h.

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Suppose that the starting point of the first fast, middle and slow three festivals is one kilometer away from the cyclist, the speed of the cyclist is V, the speed of the express train is V 1, the speed of the middle shift is V2, and the speed of the local train is V3. From the meaning of the question, we can get:

(1) The express train took 6 minutes to catch up with the cyclist: a+6V = 6V 1.

(2) Catch up with the cyclist in the middle train 10 minutes: a+10V = 10V2.

(3) The local train 12 minutes to catch up with the cyclist: a+12V = 12V3.

Substituting V 1=24 and V2=20 into (1) and (2), we can get a = 60 and V = 14.

Substitute A = 60 and V = 14 into equation (3) to get V3 = 19.