Difference between revisions of "2021 AMC 12A Problems/Problem 7"
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==Problem== | ==Problem== | ||
| − | + | What is the least possible value of <math>(xy-1)^2+(x+y)^2</math> for real numbers <math>x</math> and <math>y</math>? | |
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| + | <math>\textbf{(A)} ~0\qquad\textbf{(B)} ~\frac{1}{4}\qquad\textbf{(C)} ~\frac{1}{2} \qquad\textbf{(D)} ~1 \qquad\textbf{(E)} ~2</math> | ||
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==Solution== | ==Solution== | ||
Expanding, we get that the expression is <math>x^2+2xy+y^2+x^2y^2-2xy+1</math> or <math>x^2+y^2+x^2y^2+1</math>. By the trivial inequality the minimum value for this is <math>(D) 1</math>, which can be achieved at <math>x=y=0</math>. ~aop2014 | Expanding, we get that the expression is <math>x^2+2xy+y^2+x^2y^2-2xy+1</math> or <math>x^2+y^2+x^2y^2+1</math>. By the trivial inequality the minimum value for this is <math>(D) 1</math>, which can be achieved at <math>x=y=0</math>. ~aop2014 | ||
Revision as of 14:57, 11 February 2021
Contents
Problem
What is the least possible value of 
for real numbers 
and 
?
Solution
Expanding, we get that the expression is 
or 
. By the trivial inequality the minimum value for this is 
, which can be achieved at 
. ~aop2014
Note
See problem 1.
See also
| 2021 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 6 |
Followed by Problem 8 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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