2016 AMC 10A Problems/Problem 17
Problem
Let be a positive multiple of . One red ball and green balls are arranged in a line in random order. Let be the probability that at least of the green balls are on the same side of the red ball. Observe that and that approaches as grows large. What is the sum of the digits of the least value of such that ?
Solution
Let . Then, consider blocks of green balls in a line, along with the red ball. Shuffling the line is equivalent to choosing one of the positions between the green balls to insert the red ball. Less than of the green balls will be on the same side of the red ball if the red ball is inserted in the middle block of balls, and there are positions where this happens. Thus, . Solving the inequality gives , so the least value of is . The sum of the digits of is .
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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