2013 AIME II Problems
| 2013 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
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Instructions
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to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.Contents
Problem 1
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.
Problem 2
Positive integers 
and 
satisfy the condition
![\[\log_2(\log_{2^a}(\log_{2^b}(2^{1000}))) = 0.\]](http://latex.artofproblemsolving.com/6/6/a/66aed219de83a569530092031e044fd5e845b808.png)
Find the sum of all possible values of 
.
Problem 3
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 
.
Problem 4
In the Cartesian plane let 
and 
. Equilateral triangle 
is constructed so that 
lies in the first quadrant. Let 
be the center of 
. Then 
can be written as 
, where 
and 
are relatively prime positive integers and 
is an integer that is not divisible by the square of any prime. Find 
.
