Difference between revisions of "2013 AIME II Problems"
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+ | ==Problem 4== | ||
+ | In the Cartesian plane let <math>A = (1,0)</math> and <math>B = \left( 2, 2\sqrt{3} \right)</math>. Equilateral triangle <math>ABC</math> is constructed so that <math>C</math> lies in the first quadrant. Let <math>P=(x,y)</math> be the center of <math>\triangle ABC</math>. Then <math>x \cdot y</math> can be written as <math>\tfrac{p\sqrt{q}}{r}</math>, where <math>p</math> and <math>r</math> are relatively prime positive integers and <math>q</math> is an integer that is not divisible by the square of any prime. Find <math>p+q+r</math>. | ||
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+ | [[2013 AIME II Problems/Problem 4|Solution]] |
Revision as of 18:01, 4 April 2013
2013 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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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 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 .