2013 AIME II Problems
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Suppose that the measurement of time during the day converted to the metric system so that each day has 10 metic hours, and each metric hour has 100 metric minutes. Digital clocks would then be produced that would read 9:99 just before midnight, 0:00 at midnight, 1:25 at the former 3:00 AM, and 7:50p at the former 6:00. After the conversion, a person who wanted to wake up at the equivalent to the former 6:36AM would set his new digital alarm clock for A:BC, where A, B, and C are digits. Find 100A +10B + C.
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 .