Three students $A, B$ and $C$ are traveling from a location on the National Highway No.$5$ on direction to Hanoi for participating the HOMC $2018$. At beginning, $A$ takes $B$ on the motocycle, and at the same time $C$ rides the bicycle. After one hour and a half, $B$ switches to a bicycle and immediately continues the trip to Hanoi, while $A$ returns to pick up $C$. Upon meeting, $C$ continues the travel on the motocycle to Hanoi with $A$. Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motocycle is $50$ km per hour and of the both bicycles are $10$ km per hour. Find the distance from the starting point to Hanoi.
1. **Initial Travel:**
   - A and B travel together on the motorcycle for 1.5 hours.
   - Speed of the motorcycle = 50 km/h.
   - Distance traveled by A and B in 1.5 hours:
     \[
     1.5 \times 50 = 75 \text{ km}
     \]
   - C travels on the bicycle for 1.5 hours.
   - Speed of the bicycle = 10 km/h.
   - Distance traveled by C in 1.5 hours:
     \[
     1.5 \times 10 = 15 \text{ km}
     \]

2. **B Continues on Bicycle:**
   - B switches to a bicycle and continues to Hanoi.
   - Let \( t \) be the total time of the journey.
   - Distance remaining for B to travel:
     \[
     d - 75 \text{ km}
     \]
   - Time taken by B to travel this distance:
     \[
     t - 1.5 \text{ hours}
     \]
   - Using the speed of the bicycle (10 km/h):
     \[
     10(t - 1.5) = d - 75
     \]

3. **A Returns to Pick Up C:**
   - A returns to pick up C.
   - Distance between A and C when A starts returning:
     \[
     75 - 15 = 60 \text{ km}
     \]
   - Let \( h \) be the time it takes for A to meet C.
   - Distance traveled by C in \( h \) hours:
     \[
     10h \text{ km}
     \]
   - Distance traveled by A in \( h \) hours:
     \[
     50h \text{ km}
     \]
   - They meet when the sum of the distances they travel equals 60 km:
     \[
     10h + 50h = 60
     \]
     \[
     60h = 60
     \]
     \[
     h = 1 \text{ hour}
     \]
   - Distance from the starting point where they meet:
     \[
     15 + 10 = 25 \text{ km}
     \]

4. **A and C Travel Together to Hanoi:**
   - A and C travel together on the motorcycle from the 25 km mark.
   - Time taken by A and C to travel the remaining distance:
     \[
     t - 2.5 \text{ hours}
     \]
   - Distance remaining for A and C to travel:
     \[
     d - 25 \text{ km}
     \]
   - Using the speed of the motorcycle (50 km/h):
     \[
     50(t - 2.5) = d - 25
     \]

5. **Solving the Equations:**
   - From B's travel equation:
     \[
     10(t - 1.5) = d - 75
     \]
     \[
     d = 10(t - 1.5) + 75
     \]
     \[
     d = 10t - 15 + 75
     \]
     \[
     d = 10t + 60
     \]
   - From A and C's travel equation:
     \[
     50(t - 2.5) = d - 25
     \]
     \[
     50t - 125 = d - 25
     \]
     \[
     50t - 100 = d
     \]
   - Equating the two expressions for \( d \):
     \[
     50t - 100 = 10t + 60
     \]
     \[
     40t = 160
     \]
     \[
     t = 4 \text{ hours}
     \]
   - Substituting \( t = 4 \) into the expression for \( d \):
     \[
     d = 10t + 60
     \]
     \[
     d = 10(4) + 60
     \]
     \[
     d = 40 + 60
     \]
     \[
     d = 100 \text{ km}
     \]

The final answer is \(\boxed{100 \text{ km}}\)