Apple Transporting

Step one: First you want to make 3 trips of 1,000 apples 333 miles. You will be left with 2,001 apples and 667 miles to go.

Step two: Next you want to take 2 trips of 1,000 apples 500 miles. You will be left with 1,000 apples and 167 miles to go (you have to leave an apple behind).

Step three: Finally, you travel the last 167 miles with one load of 1,000 apples and are left with 833 apples in CITY B.

$$ In order to get maximum number apples when arriving at City $B$ , the truck must load maximum number of apples towards to City $B$ . In this case, the truck must load $1000$ apples. Due to tax, the truck driver should be clever so the steps he should do are:

1. Loading the truck with $1000$ apples, going to cover $x$ km towards $B$ (let's say this spot as point $P$ ), putting down $(1000-x)$ apples at point $P$ , and going back to $A$ to load the other $1000$ apples.
2. Next, he carries the truck with $1000$ apples and goes to point $P$ . When arriving at point $P$ , the number of apples will be $(1000-x)$ apples. He then takes $x$ apples at point $P$ . Now, the number of apples at point $P$ is $(1000-2x)$ apples. He continues traveling with carrying $1000$ apples and covers $y$ km (let's say this spot as point $Q$ ). After covering $y$ km, the number of apples will be $(1000-y)$ apples. He then puts down $(1000-y)$ apples at point $Q$ , and goes back to $A$ to load another $1000$ apples.
3. The last step, he carries the truck with $1000$ apples and goes to point $P$ . When arriving at point $P$ , the number of apples will be $(1000-x)$ apples. He then takes another $x$ apples. Now, the number of apples at point $P$ is $(1000-3x)$ apples. He continues traveling with carrying $1000$ apples and goes to point $Q$ . When arriving at point $Q$ , the number of apples will be $(1000-y)$ apples. He then takes all the apples at point $Q$ . Now, the number of apples that is carried by the truck is $(2000-2y)$ apples. For the final travel, the truck will cover $(1000-y-x)$ km to reach City B.

The number of apples that is left at point $P$ should be minimum. We set $1000-3x=0$ . Therefore $$ $x=\left\lfloor \frac{1000}{3} \right\rfloor = 333.$ $$ The number of apples that is carried by the truck at point $Q$ should be maximum. We set $2000-2y=1000$ . Therefore $$ $y=\left\lfloor \frac{200-1000}{2} \right\rfloor= 500.$ $$ Thus, the maximum number of apples that is left when arriving at $B$ is $$ $2000-2y-(1000-y-x)=1000-y+x=1000-500+333=\boxed{833}$ $$ Anyway, during the process, the truck driver will leave an apple at point $P$ . If I were the truck driver, I would eat that apple instead of leaving it offhand. LOL. $$ $\text{\# }\mathbb{Q.E.D.}\text{ \#}$