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

continue

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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