2016 AMC 10A Problems/Problem 17
Contents
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, , so
Multiplying both sides of the inequality by , we have
and by the distributive property,
Subtracting on both sides of the inequality gives us
Therefore, , so the least possible value of is . The sum of the digits of is .
Pattern
Let N = , p(N) = 1 Let N = , p(N) = \frac{1}{11}15 So as we notice, starting from n = , p(n) can be written in the form \frac{x}{N+1}$, where x starts at 1 when N = 10, and increases 1 every multiple of 5.
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.