Functional Equations Online Olympiad 2020 (FEOO) 4p (aops)

 geometry problems from Functional Equations Online Olympiad (FEOO) + Shortlist with aops links

it took place inside Aops, details here

shortlist pdf

2020

 

2020 Functional Equations Online Olympiad Shortlist G1  

Let $\mathbb{P}$ be the set of all points on a fixed plane. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that for any two different points $A$ and $B$ on the set $\mathbb{P}$, the points $f(A), f(B)$ and $M_{\overline{AB}}$ are collinear ( where $M_{\overline{AB}}$ is the midpoint of segment $\overline{AB}$ ).

by  Dorlir Ahmeti, Kosovo

2020 Functional Equations Online Olympiad Shortlist G2

Let $\mathbb{P}$ be the set of all points on a fixed plane and let $O$ be a fixed point on this set $\mathbb{P}$. Find all fucntions $f:\mathbb{P}\setminus\{O\}\rightarrow\mathbb{P}\setminus\{O\}$ such that for all points $A,B\in\mathbb{P}\setminus\{O\}$ with $AB=AO$, point $f(B)$ lie on the circle with diameter $Of(A)$.

by Victor Dominguez, Mexico, and Papon Tan Lapate, Thailand

2020 Functional Equations Online Olympiad Shortlist G3 p2

Let $\mathbb{L}$ and $\mathbb{P}$ be the sets of all lines and all points on the same fixed plane, respectively. Find all functions $f:\mathbb{L}\rightarrow\mathbb{P}$ such that, for any two non-parallel lines $\ell_1,\ell_2\in\mathbb{L}$, the points $f(\ell_1)$, $f(\ell_2)$ and $\ell_1\cap \ell_2$ are collinear.

by Dorlir Ahmeti, Kosovo

2020 Functional Equations Online Olympiad Shortlist G4

Let $\mathbb{P}$ be the set of all points on a fixed plane. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that for any two different points $A$ and $B$ on the set $\mathbb{P}$, the points $A, B, f(A)$ and $f(B)$ are concyclic or collinear.

by Dorlir Ahmeti, Kosovo, and Demetres Christofides, Cyprus