Solution:

At first look, it seems that the person cannot transfer any banana as the camel can carry at most 1000 bananas and going to eat all bananas when destination is reached.

The trick is to take an approach that the Camel need to divide the trips in parts and drops the bananas at intermediate point and then returns to starting point to pick up remaining bananas again.

Remember we have to carry most banana to the destination and Camel has to move 3 times load (3000 bananas) than his Capacity (1000 bananas) at the starting point ( Point A).

Part-1:

First we calculate how many banana's Camel will eat to transfer 3000 bananas by 1 km.

For that Camel will have to

1. Move forward with 1000 bananas by 1 km – Will eat up 1 banana in the way forward
2. Leave 998 banana after 1 km and return with 1 banana – will eat up 1 banana in the way back
3. Pick up the next 1000 bananas and move forward – Will eat up 1 banana in the way forward
4. Leave 998 banana after 1 km and return with 1 banana – will eat up 1 banana in the way back
5. Will carry the last 1000 bananas from point A(starting point) and move forward – will eat up 1 banana

Note: After point 5 the Camel does not need to return to point A again. 

So to move 3000 bananas by 1km, the Camel will eat up 5 bananas.

Now the logic is Camel will be eating 5 bananas to move all bananas by 1 km till there are 2000 bananas left, then after he will not need 5 bananas to move all bananas by 1 km as the load is now 2 times of his capacity-2000 bananas, not 3 times!(Not 5 then how many? we will calculate in part 2). So when there are 2000 bananas left it is end of the Part-1 travel.

Before going further, first we will calculate here that how many kms part-1 travel is? i.e 2000 bananas left meaning 1000 bananas have been eaten.

simple: 5 bananas → 1 km , 1000 bananas →  ? . Ans: 200 km

So to move all bananas by 200 km (part-1 travel) Camel would have eaten up 1000 bananas and is now left with 2000 bananas.

Part-2:

Now let's calculate how many banana's Camel will eat to transfer 2000 bananas by 1 km.

For that Camel will have to

1. Move forward with 1000 bananas by 1 km – Will eat up 1 banana in the way forward
2. Leave 998 banana after 1 km and return with 1 banana – will eat up this 1 banana in the way back
3. Pick up the next 1000 bananas and move forward – Will eat up 1 banana in the way forward

Note: After point 3 the Camel does not need to return to the starting point.

So to move 2000 bananas by 1km, the Camel will eat up 3 bananas.

Same way Camel will be eating 3 bananas to move all bananas by 1 km till there are 1000 bananas left. So we will calculate here that how many kms part-2 travel is? i.e 1000 bananas left and 1000 been eaten.

3 bananas → 1 km , 1000 bananas →  ? . Ans: 333.33 km

So to move all bananas by 333 km (part 2 travel) Camel would have eaten up 1000 bananas and is now left with 1000 bananas.

Part-3:

Now, for the last part, Part-3, the Camel only has to move forward. He has already covered 533 (200+333) out of 1000 km in Parts Part1 & Part2. Now he has to cover only 467 km and he has 1000 bananas.

He will eat up 467 bananas on the way forward, and at point B (Destination), the Camel will be left with only 533 Bananas.

Ans is : 533 Bananas.