Question 111
Martin hastily poured $n$ litres of water into $n$ bottles. Some bottles are therefore fuller than others. He then remembers that his task was to put exactly one litre of water into each bottle before closing its cap. Since he has not yet closed the caps, there is still time to fix his blunder.
To do so, he chooses a first bottle, pours part of its contents into another bottle if he wishes, and then caps this first bottle. He then chooses a second bottle (which may or may not be the one into which he partially emptied the first bottle), pours part of its contents into another bottle (but not into the first bottle, which is already capped) if he wishes, and then caps it. Then he proceeds in the same way with a third bottle, and so on, until all the bottles are capped.
Prove that, if Martin cleverly chooses the bottles he caps and those into which he partially empties them, he will indeed end up fixing his blunder.
Note: the bottles are oversized, and could each hold up to $n$ litres of water, which will prevent them from overflowing.
To do so, he chooses a first bottle, pours part of its contents into another bottle if he wishes, and then caps this first bottle. He then chooses a second bottle (which may or may not be the one into which he partially emptied the first bottle), pours part of its contents into another bottle (but not into the first bottle, which is already capped) if he wishes, and then caps it. Then he proceeds in the same way with a third bottle, and so on, until all the bottles are capped.
Prove that, if Martin cleverly chooses the bottles he caps and those into which he partially empties them, he will indeed end up fixing his blunder.
Note: the bottles are oversized, and could each hold up to $n$ litres of water, which will prevent them from overflowing.