Question

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 $\frac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h.$ Find $10h.$

Answer

Solution:

Burning the first $x$ centimeters takes $10(1 + 2 + \cdots + x) = 5x(x+1)$ seconds, so $T = 5 \cdot 119 \cdot 120 = 71400$ and $\frac{T}{2} = 35700.$

Setting $5x(x+1) = 35700$ gives $x(x+1) = 7140 = 84 \cdot 85,$ so at time $\frac{T}{2}$ the candle has burned down exactly $84$ centimeters. Its height is $h = 119 - 84 = 35,$ and $10h = 350.$

2013 AIME II Problem 3

Solution:

Problem 3 in Other Years