AIME 2013 II, 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's 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 $10(119-84)=\fbox{350}$.

Answer: 350