From point A, 100 planes (the flagship and 99 additional planes) take off simultaneously. With a full tank of fuel, a plane can fly 1000 km. In flight, planes can transfer fuel to each other. A plane that has transferred fuel to others makes a gliding landing. How should the flight be organized so that the flagship flies as far as possible?
Let's describe the optimal fuel exchange procedure. Initially, 100 aircraft take off with full tanks. As soon as possible, one of the aircraft distributes its fuel to the others, after which 99 aircraft have full tanks, and the freed-up aircraft lands. By this point, the aircraft will have flown \(1000 \cdot 1 / 100\) km. Similarly, when the opportunity arises, another aircraft distributes its fuel to the remaining aircraft (by this point, the aircraft will have flown another \(1000 \cdot 1 / 99\) km), and so on. In the final step, the fuel is transferred to the flagship. As a result of this process, the flagship will fly \(1000 \cdot \left(1 + \frac{1}{2} + \ldots + \frac{1}{100}\right) \approx 5187\) km.

Now let's demonstrate the optimality of the procedure described above. Let's take the fuel tank of an aircraft as a unit of fuel, the time it takes for an aircraft to fly 1000 km as a unit of time, and 1000 km as a unit of distance. If there are \(N\) aircraft in the air, the fuel consumption rate is \(N\). If the total volume of fuel in the tanks is \(V\), then there are at least \(\lceil V \rceil\) aircraft in the air, and thus the fuel consumption rate is at least \(\lceil V \rceil\). Now, let's forget about the aircraft and remember only that the fuel consumption rate is no less than the upper integer part of the volume of fuel. We need the fuel to burn as long as possible. However, it is not advantageous to burn the fuel faster than at the rate of \(\lceil V \rceil\). Indeed, if we burned the fuel faster than at the rate of \(\lceil V \rceil\) for some time, then by burning the fuel at the rate of \(\lceil V \rceil\) for the same period, we would have burned less fuel, which means that the same volume of fuel would last longer.