Olympiad problems

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2024 China Girls Math Olympiad, 4

Let $ABC$ be a triangle with $AB<BC<CA$ and let $D$ be a variable point on $BC$. The point $E$ on the circumcircle of $ABC$ is such that $\angle BAD=\angle BED$. The line through $D$ perpendicular to $AB$ meets $AC$ at $F$. Show that the measure of $\angle BEF$ is constant as $D$ varies.

2010 Bosnia and Herzegovina Junior BMO TST, 4

Tags: circle inversion , clockwise , combinatorics

On circle in clockwise order are written positive integers from $1$ to $2010$. Let us cross out number $1$, then number $10$, then number $19$, and so on every $9$th number in that direction. Which number will be first crossed out twice? How many numbers at that moment are not crossed out?

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