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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All AMC 10 Problems and Solutions |
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