2010 AMC 10B Problem 4

Problem 4 of 25EasierAlgebra

For a real number

x,

define

\heartsuit(x)

to be the average of

x

and

x^2.

What is the value of the following expression?

\heartsuit(1)+\heartsuit(2)+\heartsuit(3)

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Solution

We have

\heartsuit(1) = \dfrac{1 + 1^2}{2} = 1,

\heartsuit(2) = \dfrac{2 + 2^2}{2} = 3,

and

\heartsuit(3) = \dfrac{3 + 3^2}{2} = 6.

Then

1 + 3 + 6 = 10.

Thus, C is the correct answer.

Concepts: custom operation · mean

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.