Difference between revisions of "2022 AIME II Problems/Problem 12"
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==Solution== | ==Solution== |
Revision as of 13:01, 7 February 2023
Vishal has 5 slavicks. How much people does he own?
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 |
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