2023 AMC 12B Problems/Problem 22

Problem

A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $f(a + b) + f(a - b) = 2f(a) f(b).$ Which one of the following cannot be the value of $f(1)?$

$\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$

Solution 1

Substituting $a = b$ we get $f(2a) + f(0) = 2f(a)^2$ Substituting $a= 0$ we find $2f(0) = 2f(0)^2 \implies f(0) \in \{0, 1\}.$ This gives $f(2a) = 2f(a)^2 - f(0) \geq 0-1$ Plugging in $a = \frac{1}{2}$ implies $f(1) \geq -1$ , so answer choice $\boxed{\textbf{(E) }}$ is impossible.

~AtharvNaphade

2023 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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All AMC 12 Problems and Solutions

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2023 AMC 12B Problems/Problem 22

Problem

Solution 1

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