Difference between revisions of "2024 AMC 12B Problems/Problem 17"
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Revision as of 00:55, 14 November 2024
Problem 17
Integers 
and 
are randomly chosen without replacement from the set of integers with absolute value not exceeding 
. What is the probability that the polynomial 
has 
distinct integer roots?
.
Solution 1
-10<= a, b <= 10 , a,b has 21 choices per Vieta, x1x2x3 = -6, x1 + x2+ x3 = -a , x1x2+ x2x3 + x3x1 = b
Case: (1) (x1,x2,x3) = (-1,-1,6) , b = 13 not valid
(2) (x1,x2,x3) = (-1,1,6) , b = -1, a=-6 valid
(3) (x1,x2,x3) = ( 1,2,-3) , b = -7, a=0 valid
(4) (x1,x2,x3) = (1,-2,3) , b = -7, a=2 valid
(5) (x1,x2,x3) = (-1,2,3) , b = 1, a=4 valid
(6) (x1,x2,x3) = (-1,-2,-3) , b = 11 invalid
probability = 
= 
See also
| 2024 AMC 12B (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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
