2005 AMC 10B Problem 4

Problem 4 of 25EasierAlgebra

For real numbers

a

and

b,

define

a \diamond b = \sqrt{a^2 + b^2}.

What is the value of

(5 \diamond 12) \diamond ((-12) \diamond (-5))?

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Solution

Each inner expression evaluates to

5 \diamond 12 = \sqrt{5^2 + 12^2} = 13

and similarly

(-12) \diamond (-5)

= \sqrt{144 + 25} = 13.

Then

13 \diamond 13 = \sqrt{13^2 + 13^2} = 13\sqrt{2}.

Thus, D is the correct answer.

Concepts: custom operation · radical

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