Difference between revisions of "2021 USAMO Problems/Problem 1"
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Rectangles <math>BCC_1B_2,</math> <math>CAA_1C_2,</math> and <math>ABB_1A_2</math> are erected outside an acute triangle <math>ABC.</math> Suppose that<cmath>\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.</cmath>Prove that lines <math>B_1C_2,</math> <math>C_1A_2,</math> and <math>A_1B_2</math> are concurrent. | Rectangles <math>BCC_1B_2,</math> <math>CAA_1C_2,</math> and <math>ABB_1A_2</math> are erected outside an acute triangle <math>ABC.</math> Suppose that<cmath>\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.</cmath>Prove that lines <math>B_1C_2,</math> <math>C_1A_2,</math> and <math>A_1B_2</math> are concurrent. | ||
+ | |||
+ | ==Solution== | ||
+ | [[File:2021 USAMO 1.png|400px|right]] | ||
+ | Let <math>D</math> be the second point of intersection of the circles <math>AB_1B</math> and <math>AA_1C.</math> Then | ||
+ | <cmath>\angle ADB = 180^\circ – \angle AB_1B,\angle ADC = 180^\circ – \angle AA_1C \implies</cmath> | ||
+ | <cmath>\angle BDC = 360^\circ – \angle ADB – \angle ADC =</cmath> | ||
+ | <cmath>= 360^\circ – (180^\circ – \angle AB_1B) – (180^\circ – \angle AA_1C) =</cmath> | ||
+ | <cmath>=\angle AB_1B + \angle AA_1C \implies \angle BDC + \angle BC_1C = 180^\circ \implies</cmath> | ||
+ | <math>BDCC_1B_2</math> is cyclic with diameters <math>BC_1</math> and <math>CB_2 \implies \angle CDB_2 = 90^\circ.</math> | ||
+ | Similarly, <math>\angle CDA_1 = 90^\circ \implies</math> points <math>A_1, D,</math> and <math>B_2</math> are collinear. | ||
+ | |||
+ | Similarly, triples of points <math>A_2, D, C_1</math> and <math>C_2, D, B_1</math> are collinear. | ||
+ | |||
+ | (After USAMO 2021 Solution Notes – Evan Chen) | ||
+ | |||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | {{MAA Notice}} |
Revision as of 22:55, 18 October 2022
Rectangles and are erected outside an acute triangle Suppose thatProve that lines and are concurrent.
Solution
Let be the second point of intersection of the circles and Then is cyclic with diameters and Similarly, points and are collinear.
Similarly, triples of points and are collinear.
(After USAMO 2021 Solution Notes – Evan Chen)
vladimir.shelomovskii@gmail.com, vvsss
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