2021 AMC 12A Problem 7

Problem 7 of 25EasierAlgebra

What is the least possible value of

(xy - 1)^2 + (x + y)^2

for real numbers

x

and

y?

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Solution

Expanding,

\begin{aligned} &(xy-1)^2 + (x+y)^2 \\ &= x^2y^2 - 2xy + 1 + x^2 \\ &\quad {}+ 2xy + y^2 \\ &= x^2y^2 + x^2 + y^2 + 1. \end{aligned}

This factors as

(x^2 + 1)(y^2 + 1).

Each factor is at least

1,

so the product is at least

1,

with equality when

x = y = 0.

Thus, the correct answer is D.

Concepts: algebraic manipulation · factoring · optimization

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