Difference between revisions of "1976 AHSME Problems/Problem 30"
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− | a + b + c = 12 | + | a + b + c = 12 |
− | ab + ac + bc = 44 | + | |
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abc = 48. | abc = 48. | ||
Revision as of 02:36, 15 June 2020
Problem 30
How many distinct ordered triples satisfy the equations
Solution
The first equation suggests the substitution , , and . Then , , and . Substituting into the given equations, we get
a + b + c = 12
ab + ac + bc = 44
abc = 48.
Then by Vieta's formulas, , , and are the roots of the equation which factors as Hence, , , and are equal to 2, 4, and 6 in some order.
Since our substitution was not symmetric, each possible solution leads to a different solution , as follows:
\[
Hence, there are solutions in . The answer is (E).