2022 AMC 8

Complete problem set with solutions and individual problem pages

Problem 2 Easy

Consider these two operations:

a \blacklozenge b = a^2 - b^2

a \bigstar b = (a - b)^2

What is the output of (5 \, \blacklozenge \, 3) \, \bigstar \, 6?

  • A.

    -20

  • B.

    4

  • C.

    16

  • D.

    100

  • E.

    220

Answer:D

We can find a general solution to any ((a \, \blacklozenge \, b) \, \bigstar \, c).

~~~~((a \, \blacklozenge \, b) \, \bigstar \, c)

=((a^2-b^2) \, \bigstar \, c)

=(a^2-b^2-c)^2

=a^4+b^4-(a^2)(b^2)-2(a^2)(c)-(b^2)(a^2)+2(b^2)(c)+c^2

=5^4+3^4-(5^2)(3^2)-2(5^2)(6)-(3^2)(5^2)+2(3^2)(6)+6^2

=625+81-225-300-225+108+36

=\boxed{\textbf{(D) } 100}

Related Problems
Problem 1 2022 AMC 8
Problem 3 2022 AMC 8
Problem 4 2022 AMC 8
Problem 5 2022 AMC 8
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