2013 AIME II Problems/Problem 3

A large candle is $119$ 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 $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$ -th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$ . Find $10h$ .

Solution

We find that $T=10(1+2+\cdots +119)$ . From Gauss' formula, we find that the value of T is $10(7140)=71400$ . The value of $\frac{T}{2}$ is therefore $35700$ . We find that $35700$ is $10(3570)=10\cdot \frac{k(k+1)}{2}$ , so $3570=\frac{k(k+1)}{2}$ . As a result, $7140=k(k+1)$ , which leads to $0=k^2+k-7140$ . We notice that $k=84$ , so the answer is $10(119-84)=\boxed{350}$ .

2013 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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2013 AIME II Problems/Problem 3

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