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Difference between revisions of "2013 AIME II Problems"

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==Problem 1==
 
==Problem 1==
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==Problem 2==
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==Problem 3==
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A large candle is <math>119</math> 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 <math>10</math> seconds to burn down the first centimeter from the top, <math>20</math> seconds to burn down the second centimeter, and <math>10k</math> seconds to burn down the <math>k</math>-th centimeter.  Suppose it takes <math>T</math> seconds for the candle to burn down completely.  Then <math>\tfrac{T}{2}</math> seconds after it is lit, the candle's height in centimeters will be <math>h</math>.  Find <math>10h</math>.
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[[2013 AIME II Problems/Problem 3|Solution]]

Revision as of 14:54, 4 April 2013

2013 AIME II (Answer Key)
Printable version | AoPS Contest CollectionsPDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
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Problem 1

Problem 2

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

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