Missandei starts from Gulltown towards Oldtown by the shortest path, driving at the maximum permissible speed. From Oldtown, she drives at a speed of 10 km/hr towards Lannisport. When Missandei starts from Gulltown, Varys starts at the same time from Lannisport to Oldtown along the shortest path, always driving at the maximum permissible speed.
If they don’t stop anywhere, at what point will they meet?

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

Path taken by Missandei is GO, OL at a speed of 25 kmph and 10 kmph. Path taken by Varys is LO at a speed of 20 kmph.

Missandei reaches O 2hrs after he start. In the mean time, Varys travells 20 * 2 = 40km.

Now, both are travelling in opposite direction and 20 km far. 

The speed of Varys is 20 and that of Missandei is 10 km.

The time taken for them to meet is 20/30 = 2/3 hr.

In 2/3 hr, travels Missandei 6.67 km and Varys travels 13.33 km.

So, they will meet at 6.67 km south of Oldtown.