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]} = (\frac {O_1 O_2} {AB})^2 \ge (\frac {EF} {AB})^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> |
==Video Solution== | ==Video Solution== | ||
https://www.youtube.com/watch?v=m157cfw0vdE | https://www.youtube.com/watch?v=m157cfw0vdE |
Revision as of 15:33, 15 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