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2021 AMC 12A Spring Problem 7

Below is the professionally curated solution for Problem 7 of the 2021 AMC 12A Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Spring solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:algebraic manipulationfactoringoptimization

Difficulty rating: 1530

7.

What is the least possible value of (xy−1)2+(x+y)2(xy - 1)^2 + (x + y)^2 for real numbers xx and y?y?

00

14\dfrac14

12\dfrac12

11

22

Solution:

Expanding, (xy−1)2+(x+y)2=x2y2−2xy+1+x2+2xy+y2=x2y2+x2+y2+1. (xy-1)^2 + (x+y)^2 = x^2y^2 - 2xy + 1 + x^2 + 2xy + y^2 = x^2y^2 + x^2 + y^2 + 1. This factors as (x2+1)(y2+1).(x^2 + 1)(y^2 + 1).

Each factor is at least 1,1, so the product is at least 1,1, with equality when x=y=0.x = y = 0.

Thus, the correct answer is D.

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