2021 Fall AMC 10A Problems/Problem 16

The graph of $f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$ .)

$\textbf{(A) }$ the $y$ -axis $\qquad \textbf{(B) }$ the line $x = 1$ $\qquad \textbf{(C) }$ the origin $\qquad \textbf{(D) }$ the point $\left(\dfrac12, 0\right)$ $\qquad \textbf{(E) }$ the point $(1,0)$

Solution 1

Since $f(1-x)=|\lfloor 1-x \rfloor|-|\lfloor x \rfloor|=-f(x)$ , $f(x)$ is symmetric about the point $\left(\dfrac12, 0\right)$ $\box{\textbf{(D) } } \text{the point} \left(\dfrac12, 0\right)$ (Error compiling LaTeX. Unknown error_msg)

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2021 Fall AMC 10A Problems/Problem 16

Solution 1