A water jug is empty, and its center of gravity is above the inside bottom of the jug. Water is poured into the jug until the center of gravity of the jug and water (considered together) is as low as possible. Explain why this center of gravity must lie at the surface of the water.
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If P is the center of gravity of the system, then as long as P is above the surface of the water, it will fall as the surface rises. When P reaches the surface, adding any more water will raise the water level and hence the level of P.
From Edward Barbeau, Murray Klamkin, and William Moser, Five Hundred Mathematical Challenges, 1995.