2024 AMC 12B Problems/Problem 16

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

so answer 4

~luckuso