Olympiad problems

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Found problems: 2

2024 EGMO, 2

Let $ABC$ be a triangle with $AC>AB$ , and denote its circumcircle by $\Omega$ and incentre by $I$. Let its incircle meet sides $BC,CA,AB$ at $D,E,F$ respectively. Let $X$ and $Y$ be two points on minor arcs $\widehat{DF}$ and $\widehat{DE}$ of the incircle, respectively, such that $\angle BXD = \angle DYC$. Let line $XY$ meet line $BC$ at $K$. Let $T$ be the point on $\Omega$ such that $KT$ is tangent to $\Omega$ and $T$ is on the same side of line $BC$ as $A$. Prove that lines $TD$ and $AI$ meet on $\Omega$. [right][i]Tommy Walker Mackay, United Kingdom[/i][/right]

2024 EGMO, 5

Tags: EGMO 2024 P5 , function , number theory , EGMO 2024

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$: $(i)$: $x$ and $f(x)$ have the same number of positive divisors. $(ii)$: If $x \nmid y$ and $y \nmid x$, then: $$\gcd(f(x), f(y)) > f(\gcd(x, y))$$