2022 AIME II Problems/Problem 12
Problem
Let and be real numbers with and such thatFind the least possible value of
Solution
Denote is an ellipse whose center is and foci are and . is an ellipse whose center is and foci are and .
Since is on , the sum of distance from to and is equal to twice the semi-major axis of this ellipse, .
Since is on , the sum of distance from to and is equal to twice the semi-major axis of this ellipse, .
Therefore, is the sum of the distance from to four foci of these two ellipses. To minimize this, must be the intersection point of the line that passes through and , and the line that passes through and .
The distance between and is .
The distance between and is .
Hence, , i.e. .
The straight line connecting the points and has the equation . The straight line connecting the points and has the equation . These lines intersect at the point . This point satisfies both equations for . Hence, is possible.
Therefore,
~Steven Chen (www.professorchenedu.com)
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Video Solution
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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