Difference between revisions of "2016 AMC 10A Problems/Problem 17"
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Revision as of 11:24, 5 February 2016
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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2016 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
