2021 AMC 10A Fall Problem 16

Question

Consider the function

f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|

Its graph is symmetric about which of the following? (Here

\lfloor x \rfloor

is the greatest integer not exceeding

x.

)

the

y

-axis

the line

x=1

the origin

the point

\left(\dfrac{1}{2},0\right)

the point

(1,0)

Compare

f(1-x)

with

f(x)

The relation is point symmetry, not line symmetry

For every real

x,

\begin{aligned} &f(1-x)=|\lfloor 1-x\rfloor| \\ &\quad {}-|\lfloor x\rfloor|=-f(x). \end{aligned}

Equivalently, replacing

x

by

\frac{1}{2}+t

gives

f\!\left(\frac{1}{2}-t\right)=-f\!\left(\frac{1}{2}+t\right).

This is point symmetry about

\left(\frac{1}{2},0\right).

Thus, D is the correct answer.

Problem 16 in other years