Difference between revisions of "2020 USAMO Problems/Problem 1"
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+ | ==Problem 1== | ||
+ | Let <math>ABC</math> be a fixed acute triangle inscribed in a circle <math>\omega</math> with center <math>O</math>. A variable point <math>X</math> is chosen on minor arc <math>AB</math> of <math>\omega</math>, and segments <math>CX</math> and <math>AB</math> meet at <math>D</math>. Denote by <math>O_1</math> and <math>O_2</math> the circumcenters of triangles <math>ADX</math> and <math>BDX</math>, respectively. Determine all points <math>X</math> for which the area of triangle <math>OO_1O_2</math> is minimized. | ||
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+ | ==Solution== | ||
+ | [[File:2020 USAMO 1.png|400px|right]] | ||
+ | Let <math>E</math> be midpoint <math>AD.</math> Let <math>F</math> be midpoint <math>BD \implies</math> | ||
+ | <cmath>EF = ED + FD = \frac {AD}{2} + \frac {BD}{2} = \frac {AB}{2}.</cmath> | ||
+ | <math>E</math> and <math>F</math> are the bases of perpendiculars dropped from <math>O_1</math> and <math>O_2,</math> respectively. | ||
+ | |||
+ | Therefore <math>O_1O_2 \ge EF = \frac {AB}{2}.</math> | ||
+ | |||
+ | <cmath>CX \perp O_1O_2, AX \perp O_1O \implies \angle O O_1O_2 = \angle AXC</cmath> | ||
+ | <math>\angle AXC = \angle ABC (AXBC</math> is cyclic) <math>\implies \angle O O_1O_2 = \angle ABC.</math> | ||
+ | |||
+ | Similarly <math>\angle BAC = \angle O O_2 O_1 \implies \triangle ABC \sim \triangle O_2 O_1O.</math> | ||
+ | |||
+ | 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> | ||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
==Video Solution== | ==Video Solution== | ||
https://www.youtube.com/watch?v=m157cfw0vdE | https://www.youtube.com/watch?v=m157cfw0vdE | ||
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+ | {{MAA Notice}} |
Latest revision as of 22:53, 18 October 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
Video Solution
https://www.youtube.com/watch?v=m157cfw0vdE
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