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 ?
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Solution:
Let , where is a positive integer. There are equally likely possible positions for the red ball in the line of balls. Number these from one end. The red ball will not divide the green balls so that at least of them are on the same side if it is in position . This includes positions. The probability that or more of the green balls will be on the same side is therefore .
Solving the inequality for yields . The value of corresponding to the required least value of is therefore , so . The sum of the digits of is .
The problems on this page are the property of the MAA's American Mathematics Competitions