AMC 10A 2021 Problem 9 | Factorizing Problem

What is the least possible value of $(x y-1)^{2}+(x+y)^{2}$ for real numbers $x$ and $y$ ?

• 0
• $\frac{1}{2}$
• $\frac{1}{4}$
• 1
• 2

AMC 10A 2021 Problem 9

1

Expand the expression.

So, we get that the expression is $x^{2}+2 x y+y^{2}+x^{2} y^{2}-2 x y+1$ or $x^{2}+y^{2}+x^{2} y^{2}+1$.

Then the minimum value for this is 1 , which can be achieved at $x=y=0$. .

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