2024 USAMO Problems/Problem 5

The following problem is from both the 2024 USAMO/5 and 2024 USAJMO/6, so both problems redirect to this page.

Problem

Point $D$ is selected inside acute triangle $ABC$ so that $\angle DAC=\angle ACB$ and $\angle BDC=90^\circ+\angle BAC$ . Point $E$ is chosen on ray $BD$ so that $AE=EC$ . Let $M$ be the midpoint of $BC$ . Show that line $AB$ is tangent to the circumcircle of triangle $BEM$ .

Solution 1

define angle DBT as $/alpha$ , the angle BEM as $/betta$ . 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 as $/betta &= 180-B$ (Error compiling LaTeX. Unknown error_msg)

2024 USAMO (Problems • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6
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2024 USAMO Problems/Problem 5

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Problem

Solution 1

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