2021 AMC 10A Spring Problem 9

Question

What is the least possible value of

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

for real numbers

x

and

y?

0

\dfrac{1}{4}

\dfrac{1}{2}

1

2

Expand the expression and look for cancellation

After expansion, every term except the constant is nonnegative

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Expanding, we get

\begin{gathered} x^2y^2 - 2xy + 1 + x^2 + 2xy + y^2 \\ = x^2y^2 + x^2 + y^2 + 1. \end{gathered}

Note that every square must be non-negative. Therefore, the minimum value is when all the terms except

1

are

0,

making the sum

1.

This is attainable when

x = y = 0.

Thus, D is the correct answer.

Problem 9 in other years