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
(in French)
collected inside aops:here
2020-22
Juniors
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.
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 $1 \le j \le 4$.
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
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 .
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 $\Gamma$
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.
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