1984 AMC 12 Problem 16

Attempt Problem 16 of the 1984 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1984 AMC 12 solutions, or check the answer key.

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16.

The function

f(x)

satisfies

f(2+x)=f(2-x)

for all real numbers

x.

If the equation

f(x)=0

has exactly four distinct real roots, then the sum of these roots is

0

2

4

6

8

The equation makes the graph symmetric about

x=2

Pair each root

2+r

with its reflected root

2-r

The relation shows that every root

2+r

is paired with

2-r.

Four distinct roots therefore form two such pairs, and each pair has sum

4.

The sum of all four roots is

2\cdot4=8.

Therefore, the correct answer is E.

Problem 16 in other years