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. | |
+ | |||
+ | ==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>\begin{align*} | ||
+ | \angle ADB &= 180^\circ – \angle AB_1B,&\angle ADC &= 180^\circ – \angle AA_1C\\ | ||
+ | \angle BDC &= 360^\circ – \angle ADB – \angle ADC\\ | ||
+ | &= 360^\circ – (180^\circ – \angle AB_1B) – (180^\circ – \angle AA_1C)\\ | ||
+ | &= \angle AB_1B + \angle AA_1C\\ | ||
+ | \angle BDC + \angle BC_1C &= 180^\circ | ||
+ | \end{align*}</cmath> | ||
+ | Therefore, <math>BDCC_1B_2</math> is cyclic with diameters <math>BC_1</math> and <math>CB_2</math>, and thus <math>\angle CDB_2 = 90^\circ.</math> | ||
+ | Similarly, <math>\angle CDA_1 = 90^\circ</math>, meaning points <math>A_1</math>, <math>D</math>, and <math>B_2</math> are collinear. | ||
+ | |||
+ | Similarly, the 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}} |
Latest revision as of 13:32, 25 December 2023
Rectangles
and
are erected outside an acute triangle
Suppose that
Prove that lines
and
are concurrent.
Solution
Let be the second point of intersection of the circles
and
Then:
Therefore,
is cyclic with diameters
and
, and thus
Similarly,
, meaning points
,
, and
are collinear.
Similarly, the points and
are collinear.
(After USAMO 2021 Solution Notes – Evan Chen)
vladimir.shelomovskii@gmail.com, vvsss
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.