The capital city, King’s Landing, located 40 km to the south of Gulltown on the road connecting Gulltown to Winterfell, did not have a straight road, connecting to Meereen. Now, a new expressway is being built to connect these two towns by a straight road.

What should be the maximum speed limit allowed on this expressway so that it cuts down the travel time, from Meeren to King’s Landing, from the fastest possible route through the road network shown in the diagram, by 20 minutes?

We are told that a car always travelling at the maximum permissible speed, and always taking the shortest route takes 1 hour to reach Oldtown from Meereen, 2 hours to reach Gulltown from Oldtown, and 45 minutes to reach Winterfell from Gulltown.

So, distance between Oldtown and Meereen is 100*1 = 100km

Distance between Gulltown and Oldtown is 25*2 = 50km.

Distance between Winterfell and Gulltown. is 120*3/4 = 90km.

Let the distance between Meereen and Winterfall be x.

Using the pythagoras theorem, $$(100+50)^2=90^2+x^2$$

x = 120 km

For convience, we are refering the vertices as the starting letters of towns.

Using the similar triangle rule, MOL is similar to MGW.

So, $$\frac{MO}{ML}=\frac{MG}{MW}$$

$$\frac{100}{ML}=\frac{150}{120}$$

ML = 80 km.

LW = 40 km.

$$\frac{MO}{LO}=\frac{MG}{WG}$$

$$\frac{100}{LO}=\frac{150}{90}$$

LO = 60 km

The direct path from M to K will $$120^2+50^2 = 130^2$$.

The shortest time taking path from Meereen to King's landing. Through M0GK, time taken = $$\frac{100}{100}+\frac{50}{25}+\frac{40}{120}\ =1+2+\frac{1}{3}=3.33$$

Through MWK, time taken = $$\frac{120}{120}+\frac{50}{120}\ =1.42$$

So, the second case that is MWK is the fastest route. Time taken will be 1 hr 25 mins.

If the travel time cuts down by 20 mins, the new time will be 1 hr 5 mins.So, it should take 13/12 hrs.

speed = $$\frac{130}{\frac{13}{12}}=120 kmph$$