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2024 India Iran Friendly Math Competition, 2

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O_1$. The diagonals $AC$ and $BD$ meet at point $P$. Suppose the four incentres of triangles $PAB, PBC, PCD, PDA$ lie on a circle with centre $O_2$. Prove that $P, O_1, O_2$ are collinear. [i]Proposed by Shantanu Nene[/i]

2024 Bulgaria MO Regional Round, 12.1

Tags: geometry , conditional geometry , right angle , Incentres , isosceles

Let $ABC$ be an acute triangle with midpoint $M$ of $AB$. The point $D$ lies on the segment $MB$ and $I_1, I_2$ denote the incenters of $\triangle ADC$ and $\triangle BDC$. Given that $\angle I_1MI_2=90^{\circ}$, show that $CA=CB$.

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