Difference between revisions of "2024 USAMO Problems/Problem 5"
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| + | {{duplicate|[[2024 USAMO Problems/Problem 5|2024 USAMO/5]] and [[2024 USAJMO Problems/Problem 6|2024 USAJMO/6]]}} | ||
__TOC__ | __TOC__ | ||
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== Solution 1 == | == Solution 1 == | ||
| + | Let <math>\angle DBT = \alpha</math> and <math>\angle BEM = \beta</math>. | ||
| + | Extend AD intersects BC at point T, then TC = TA, TE is perpendicular to AC | ||
| + | Thus, AB is the tangent of the circle BEM | ||
| + | |||
| + | Then the question is equivalent as the <math>\angle ABT</math> is the auxillary angle of <math>\angle BEM</math>. | ||
| + | |||
| + | continue | ||
==See Also== | ==See Also== | ||
{{USAMO newbox|year=2024|num-b=4|num-a=6}} | {{USAMO newbox|year=2024|num-b=4|num-a=6}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 04:28, 1 November 2024
- The following problem is from both the 2024 USAMO/5 and 2024 USAJMO/6, so both problems redirect to this page.
Contents
Problem
Point 
is selected inside acute triangle 
so that 
and 
. Point 
is chosen on ray 
so that 
. Let 
be the midpoint of 
. Show that line 
is tangent to the circumcircle of triangle 
.
Solution 1
Let 
and 
.
Extend AD intersects BC at point T, then TC = TA, TE is perpendicular to AC
Thus, AB is the tangent of the circle BEM
Then the question is equivalent as the 
is the auxillary angle of 
.
continue
See Also
| 2024 USAMO (Problems • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
