Problem:
A large candle is centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes seconds to burn down the first centimeter from the top, seconds to burn down the second centimeter, and seconds to burn down the -th centimeter. Suppose it takes seconds for the candle to burn down completely. Then seconds after it is lit, the candle's height in centimeters will be . Find .
Solution:
Let equal the number of centimeters burned since the candle was lit. The time required to burn down centimeters when is an integer is seconds. Therefore and .
To find the number of centimeters, , the candle has burned down in seconds, set equal to and note that this implies . Thus , and . The requested product is .
The problems on this page are the property of the MAA's American Mathematics Competitions