geometry problems from French Fall Animath Cup (a pre TST competition) with aops links
collected inside aops here
2011 - 2020
Let $ABC$ be an isosceles triangle in which $AB = AC$. On the circle circumscribed to this triangle we take a point $D$ belonging to the smallest arc joining $A$ to $B$. Finally we take a point $E$ belonging to the line $(AD)$, outside the segment $[AD]$ and such that the points $A$ and $E$ are in the same half-plane limited by the line $(BC)$. The circle circumscribed to the triangle $BDE$ intersects the line $(AB)$ at $F$. Prove that the lines $(EF)$ and $(BC)$ are parallel.
Let $ABC$ be a right triangle at A with $AB <AC$, $M$ the midpoint of $[BC]$, $D$ the intersection of $(AC)$ with the line perpendicular to $(BC)$ passing through $M$ and $E$ the point of intersection of the line parallel to $(AC)$ passing through $M$ with the line perpendicular to $(BD)$ passing through $B$. Show that the triangles $AEM$ and $MCA$ are similar if, and only if, $\angle ABC = 60^o$
Let $ABC$ be a triangle and $I$ the center of its inscribed circle. We suppose that $AI = BC$ and that $\angle ICA = 2\angle IAC$. What is the value of $\angle ABC$ ?
Four circles $C_1, C_2, C_3, C_4$ of the same radii $r$ are internally tangent to a circle of radius $R$. We set $C_5 = C_1$. We suppose that for all $i = 1, 2, 3, 4$, the circles $C_i$ and $C_{i + 1}$ are tangent. Determine the value of the ratio $r / R$ .
Let $A, B, C$ and $D$ be four distinct points in the plane. It turns out that any circle that goes through $A$ and $B$ meets any circle that goes through $C$ and $D$. Prove that $A, B, C$ and $D$ are all on the same line or that they are all on the same circle.
On a circle of perimeter $p$, we mark the three vertices of an equilateral triangle as well as the four vertices of a square. These seven points divide the circle into $7$ arcs. Prove that one of these arcs is no longer than $p/24$.
Let $ABC$ be a triangle. Let $M, N$ be two points on $[BC]$ such that the $\angle BAM = \angle NAC$ . Let $O_1$ be the center of the circle circumscribed to triangle $ABC$ and $O_2$ the center of circle circumscribed to triangle $AMN$. Show that the points $O_1, O_2$ and $A$ are collinear.
Each side of a unit square is divided into $3$ equal segments. We draw the figure below from this share. What is the area of the gray polygon? Justify your answer.
Two circles $C_1$ and $C_2$ are tangent externally at a point $X$. A tangent common to the two circles meets $C_1$ in $Y$ and $C_2$ in $Z$ (with $Y \ne Z$). Let $T$ be such that $[YT]$ is a diameter of $C_1$. Show that $T, X, Z$ are collinear.
We construct the following figure: we draw an triangle $ABC$ isosceles at $A$, then the line $(d)$ perpendicular to $(BC)$ passing through $C$. We choose a point $D$ on $(d)$. We place $E$ so that $AEDB$ or a parallelogram. Finally, $M$ is the point of intersection of $(AE)$ and $(d)$. Prove that $M$ is the midpoint of $[AE]$.
Let $ABCD$ be a rectangle such that $AB> BC$. Let $E$ be the orthogonal project of $B$ on $(AC), \Gamma$ the circle passing through $A$ and $E$ and whose center is on $(AD)$. Let $F$ be the point of intersection of $\Gamma$ and $[CD]$. Prove that $(BF)$ is the bisector of $AFC$.
Let $ABCD$ be a convex quadrilateral, with $\angle ABC= 90^o$, $\angle BAD=\angle ADC = 80^o$. Let $M$ and $N$ be points of $[AD]$ and $[BC]$ such that $\angle CDN = \angle ABM = 20^o$. Suppose finally $MD = AB$. Calculate $\angle MNB$ .
Let $ABCD$ be a rectangle of area $4$. Let $I$ be the midpoint of $[AD]$ and let $J$ be the midpoint of $[BC]$. Let $X$ be the point of intersection of $(AJ)$ and $(BI)$, and let $Y$ be the point of intersection of $(DJ)$ and $(CI)$. What is the area of the quadrilateral $IXJY$ ?
Let $ABC$ be a triangle. Let $E$ be the foot of the altitude of $ABC$ from $B$, and $F$ the foot of the altitude of $ABC$ from $C$. We also denote by $H$ the point of intersection of lines $(BE)$ and $(FC)$, and $O$ the center of the circle circumscribed to $ABC$. Show that, if $AF = FC$, then the quadrilateral $EHFO$ is a parallelogram.
Let $ABCD$ be a square and $E$ the point of the segment $[BD]$ such that $EB = AB$. We define the point $F$ as the point of intersection of lines $(CE)$ and $(AD)$. Find the value of the angle $\angle FEA$.
Let $ABCD$ be a parallelogram of area $1$ and $M$ a point belonging to the segment $[BD]$ such that $MD = 3MB$. We denote by $N$ the point of intersection of lines $(AM)$ and $(CB)$. Calculate the area of triangle $MND$ .
source: https://maths-olympiques.fr/
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