Francophone 2020-22 (OFM) 6p

geometry problems from Francophone Mathematical Olympiad (OFM) with aops links

Olympiade Francophone de Mathématiques 

it started in 2020 and participated France, Belgium, Luxembourg, Switzerland, Morocco and Ivory Coast

Francophone MO 2020-21 (OFM) with solutions

 (in French )

collected inside aops: Juniors and Seniors

2020-22

Juniors

2020 Francophone MO Juniors p1

Let $ABC$ be a triangle such that $AB <AC$, $\omega$ its inscribed circle and $\Gamma$ its circumscribed circle. Let also $\omega_b$ be the excircle relative to vertex $B$, then $B'$ is the point of tangency between $\omega_b$ and $(AC)$. Similarly, let the circle $\omega_c$ be the excircle exinscribed relative to vertex $C$, then $C'$ is the point of tangency between $\omega_c$ and $(AB)$. Finally, let $I$ be the center of $\omega$  and $X$ the point of $\Gamma$  such that $\angle XAI$ is a right angle. Prove that the triangles $XBC'$ and $XCB'$ are congruent.

2021 Francophone MO Juniors p3

Every point in the plane was colored in red or blue.

Prove that one the two following statements is true:

$\bullet$ there exist two red points at distance $1$ from each other;

$\bullet$ there exist four blue points $B_1$, $B_2$, $B_3$, $B_4$ such that the points $B_i$ and $B_j$ are at distance $|i-j|$ from each other, for all integers $i$ and $j$ such as $1 \le i \le 4$ and $1 \le  j  \le  4$.

2022 Francophone MO Juniors p3

Let $\triangle ABC$ a triangle, and $D$ the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. the line parallel to $AC$ passing by the point $B$, intersect the line $AD$ at $X$. the line parallel to $CX$ passing by the point $B$, intersect $AC$ at $Y$. $E = (AYB) \cap BX$ . prove that $C$ , $D$ and $E$ collinear.

Seniors

2020 Francophone MO Seniors p1

Let $ABC$ be an acute triangle with $AC>AB$, Let $DEF$ be the intouch triangle with $D \in (BC)$,$E \in (AC)$,$F \in (AB)$,, let $G$ be the intersecttion of the perpendicular from $D$ to $EF$ with $AB$, and $X=(ABC)\cap (AEF)$. Prove that $B,D,G$ and $X$ are concylic .

2022 Francophone MO Seniors p3

Let $ABCD$ be a square with incircle $\Gamma$. Let $M$ be the midpoint of the segment $[CD]$. Let $P \neq B$ be a point on the segment $[AB]$. Let $E \neq M$ be the point on $\Gamma$ such that $(DP)$ and $(EM)$ are parallel. The lines $(CP)$ and $(AD)$ meet each other at $F$. Prove that the line $(EF)$ is tangent to $\Gamma$

2022 Francophone MO Seniors p3

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Denote $\Delta$ the tangent at $A$ to the circle $\Gamma$. $\Gamma_1$ is a circle tangent to the lines $\Delta$, $(AB)$ and $(BC)$, and $E$ its touchpoint with the line $(AB)$. Let $\Gamma_2$ be a circle tangent to the lines $\Delta$, $(AC)$ and $(BC)$, and $F$ its touchpoint with the line $(AC)$. We suppose that $E$ and $F$ belong respectively to the segments $[AB]$ and $[AC]$, and that the two circles $\Gamma_1$ and $\Gamma_2$ lie outside triangle $ABC$. Show that the lines $(BC)$ and $(EF)$ are parallel.

source: http://igm.univ-mlv.fr/~juge/ofm/