2017 AMC 12A Problems/Problem 25
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[hide]Problem
The vertices of a centrally symmetric hexagon in the complex plane are given by For each , , an element is chosen from at random, independently of the other choices. Let be the product of the numbers selected. What is the probability that ?
Solution
It is possible to solve this problem using elementary counting methods. This solution proceeds by a cleaner generating function.
We note that both lie on the imaginary axis and each of the have length and angle of odd multiples of , i.e. . When we draw these 6 complex numbers out on the complex plane, we get a crystal-looking thing. Note that the total number of ways to choose 12 complex numbers is . Now we count the number of good combinations.
We first consider the lengths. When we multiply 12 complex numbers together, their magnitudes multiply. Suppose we have of the numbers ; then we must have . Having will take care of the length of the product; now we need to deal with the angle.
We require . Letting be , we see that the angles we have available are , where we must choose exactly 8 angles from the set and exactly 4 from the set . If we found a good combination where we had of each angle , then the amount this would contribute to our count would be . We want to add these all up. We proceed by generating functions.
Consider The expansion will be of the form . Note that if we reduced the powers of mod and fished out the coefficient of and plugged in (and then multiplied by ) then we would be done. Since plugging in doesn't affect the 's, we do that right away. The expression then becomes where the last equality is true because we are taking the powers of mod . Let denote the coefficient of in . Note . We use the roots of unity filter, which states where . In our case , so we only need to find the average of the 's. We plug in and take the average to find the sum of all coefficients of . Plugging in makes all of the above zero except for and . Averaging, we get . Now the answer is simply
Alternate solution
By changing to , we can give a bijection between cases where and cases where , so we'll just find the probability that and divide by in the end. Multiplying the hexagon's vertices by doesn't change , and switching any with doesn't change the property , so the probability that remains the same if we only select our 's at random from Since and , we must choose exactly times to make . To ensure is real, we must either choose times, times, or both and times. This gives us a total of good sequences , and hence the final result is
See Also
2017 AMC 12A (Problems • Answer Key • Resources) | |
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