Difference between revisions of "2022 AIME II Problems/Problem 12"
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~Steven Chen (www.professorchenedu.com) | ~Steven Chen (www.professorchenedu.com) | ||
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+ | https://youtu.be/4qiu7GGUGIg | ||
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+ | ~MathProblemSolvingSkills.com | ||
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==See Also== | ==See Also== | ||
{{AIME box|year=2022|n=II|num-b=11|num-a=13}} | {{AIME box|year=2022|n=II|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 21:13, 10 January 2023
Contents
Problem
Let and
be real numbers with
and
such that
Find 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
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.