2016 AMC 12B Problem 2

Attempt Problem 2 of the 2016 AMC 12B 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 2016 AMC 12B solutions, or check the answer key.

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The harmonic mean of two numbers can be computed as twice their product divided by their sum. The harmonic mean of

1

and

2016

is closest to which integer?

2

45

504

1008

2015

Twice the product over the sum:

\dfrac{2\cdot1\cdot2016}{1+2016}

\dfrac{4032}{2017}

is just a hair under a whole number; round it

The harmonic mean is

\dfrac{2\cdot1\cdot2016}{1+2016}=\dfrac{4032}{2017}.

Since

\dfrac{2016}{2017}

is very close to

1,

this is just under

2,

so the closest integer is

2.

Thus, the correct answer is A.

Problem 2 in other years