Difference between revisions of "2022 AIME II Problems/Problem 12"
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Revision as of 13:30, 15 January 2023
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)
Video Solution
~MathProblemSolvingSkills.com
Video Solution
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.