2011 AMC 12A Problems/Problem 14
Contents
Problem
Suppose and are single-digit positive integers chosen independently and at random. What is the probability that the point lies above the parabola ?
Solution
If lies above the parabola, then must be greater than . We thus get the inequality . Solving this for gives us . Now note that constantly increases when is positive. Then since this expression is greater than when , we can deduce that must be less than in order for the inequality to hold, since otherwise would be greater than and not a single-digit integer. The only possibilities for are thus , , and .
For , we get for our inequality, and thus can be any integer from to .
For , we get for our inequality, and thus can be any integer from to .
For , we get for our inequality, and thus can be any integer from to .
Finally, if we total up all the possibilities we see there are points that satisfy the condition, out of total points. The probability of picking a point that lies above the parabola is thus
Video Solution
https://www.youtube.com/watch?v=u23iWcqbJlE ~Shreyas S
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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All AMC 12 Problems and Solutions |
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