1984 AHSME Problems/Problem 16
Contents
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 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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