2013 AIME II Problems
|2013 AIME II (Answer Key)|
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Suppose that the measurement of time during the day is converted to the metric system so that each day has metric hours, and each metric hour has metric minutes. Digital clocks would then be produced that would read just before midnight, at midnight, at the former AM, and at the former PM. After the conversion, a person who wanted to wake up at the equivalent of the former AM would set his new digital alarm clock for , where , , and are digits. Find .
Positive integers and satisfy the condition Find the sum of all possible values of .
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 .
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 .
In equilateral let points and trisect . Then can be expressed in the form , where and are relatively prime positive integers, and is an integer that is not divisible by the square of any prime. Find .
Find the least positive integer such that the set of consecutive integers beginning with contains no square of an integer.
A group of clerks is assigned the task of sorting files. Each clerk sorts at a constant rate of files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in hours and minutes. Find the number of files sorted during the first one and a half hours of sorting.
A hexagon that is inscribed in a circle has side lengths , , , , , and in that order. The radius of the circle can be written as , where and are positive integers. Find .