1984 AHSME Problems/Problem 16
Contents
[hide]Problem
The function satisfies for all real numbers . If the equation has exactly four distinct real roots, then the sum of these roots is
Solution
Let one of the roots be . Also, define such that . Thus, we have and . Therefore, we have , and is also a root. Let this root be . The sum . Similarly, we can let be a root and define such that , and we will find is also a root, say, , so . Therefore, .
Solution 2
The graph of this function is symmetric around 2. Therefore, two roots will be greater than and the other two roots will be less than . These four roots are symmetric around , so the average of the four roots is . Then, the sum is .
See Also
1984 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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