Difference between revisions of "2020 USAMO Problems/Problem 1"
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The area of <math>\triangle OO_1O_2</math> is minimized if <math>CX \perp AB</math> because | The area of <math>\triangle OO_1O_2</math> is minimized if <math>CX \perp AB</math> because | ||
<cmath>\frac {[OO_1O_2]} {[ABC]} = \left(\frac {O_1 O_2} {AB}\right)^2 \ge \left(\frac {EF} {AB}\right)^2 = \frac {1}{4}.</cmath> | <cmath>\frac {[OO_1O_2]} {[ABC]} = \left(\frac {O_1 O_2} {AB}\right)^2 \ge \left(\frac {EF} {AB}\right)^2 = \frac {1}{4}.</cmath> | ||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
==Video Solution== | ==Video Solution== | ||
https://www.youtube.com/watch?v=m157cfw0vdE | https://www.youtube.com/watch?v=m157cfw0vdE |
Revision as of 11:49, 16 September 2022
Problem 1
Let be a fixed acute triangle inscribed in a circle
with center
. A variable point
is chosen on minor arc
of
, and segments
and
meet at
. Denote by
and
the circumcenters of triangles
and
, respectively. Determine all points
for which the area of triangle
is minimized.
Solution
Let be midpoint
Let
be midpoint
and
are the bases of perpendiculars dropped from
and
respectively.
Therefore
is cyclic)
Similarly
The area of is minimized if
because
vladimir.shelomovskii@gmail.com, vvsss