2022 AIME II Problems/Problem 12
Contents
Problem
Let and be real numbers with and such thatFind the least possible value of
Solution
Denote .
Because , is on an ellipse whose center is and foci are and .
Hence, the sum of distance from to and is equal to twice the major axis of this ellipse, .
Because , is on an ellipse whose center is and foci are and .
Hence, the sum of distance from to and is equal to twice the major axis of this ellipse, .
Therefore, is the sum of the distance from to four foci of these two ellipses.
To make this minimized, is 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, .
Therefore,
~Steven Chen (www.professorchenedu.com)
Video Solution
~MathProblemSolvingSkills.com
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 | ||
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