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 == | ||
| + | define angle DBT as <math>/alpha</math>, the angle BEM as <math>/betta</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 angle ABT is the auxillary angle of the angle BEM | ||
| + | |||
| + | 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 09:15, 5 May 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
define angle DBT as 
, the angle BEM as 
.
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 angle ABT is the auxillary angle of the angle BEM
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. 
