2023 AMC 12B Problems/Problem 22
Contents
Problem
A real-valued function has the property that for all real numbers
and
Which one of the following cannot be the value of
Solution 1
Substituting we get
Substituting
we find
This gives
Plugging in
implies
, so answer choice
is impossible.
~AtharvNaphade
Solution 2
First, we set and
.
Thus, the equation given in the problem becomes
\[
f(0) + f(0) = 2 f(0) \cdot f(0) .
\]
Thus, or 1.
Case 1: .
We set .
Thus, the equation given in the problem becomes
\[
2 f(a) = 0 .
\]
Thus, for all
.
Case 2: .
We set .
Thus, the equation given in the problem becomes
Thus, for any ,
Therefore, an infeasible value of is
\boxed{\textbf{(E) -2}}.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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 |
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