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France TST 2000-07, 2012-20 45p (-01)

 geometry problems from French Team Selection Tests (TST) with aops links in the names


(only those not in IMO Shortlist)

collected inside aops here

2000, 2002-07, 2012-20 
2001, 2008-11 missing 


Points $P,Q,R,S$ lie on a circle and $\angle PSR$ is right. $H,K$ are the projections of $Q$ on lines $PR,PS$. Prove that $HK$ bisects segment $ QS$.

$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.

2001 missing

In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$.
a) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$.
b) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.

2002 France TST 2.2 (CHKMO 1999)
Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB+AC$.

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

2004 France TST 1.2 (Bundeswettbewerb 2003)
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.
Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively.
Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle). Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.

Let $ABC$ be a triangle such that $BC=AC+\frac{1}{2}AB$. Let $P$ be a point of $AB$ such that $AP=3PB$. Show that $\widehat{PAC} = 2 \widehat{CPA}.$

Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$. Prove that $(PS)$ and $(QR)$ are perpendicular.

Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel. Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.

2008 missing
2009 missing
2010 missing
2011 missing

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. We denote by $\Delta_1$ and $\Delta_2$ the images of the line $(AB)$ wrt the lines of the interior bisectors of $\angle CAD$  and $\angle CBD$ . Let $P$ be the intersection of $\Delta_1$ with $\Delta_2$ . Prove that $(OP)$ and $(CD)$ are perpendicular.

2012 France TST 1.2 (March Test)
Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.

Let $\Gamma$ and $\Gamma'$ be two circles of the plane which intersect at $A$ and $B$, and let $\Delta$ be a line which passes from $B$ and which intersects $\Gamma$ at $M$ and $\Gamma'$ at $N$. We assume that $B$ lies between $M$ and $N$. Let $I$ be the midpoint of the arc $AM$ of the circle $\Gamma$ which does not contain $B$, and let $J$ be the midpoint of the arc $AN$ of the circle $\Gamma'$ which does not contain $B$. Finally, we denote by $K$ the midpoint of $[MN]$. Prove that the triangle $IKJ$ is right in $K$.

$A, B, C, D, E$ are five points on the same circle, so that $ABCDE$ is convex and we have $AB = BC$ and $CD = DE$. Suppose that the lines $(AD)$ and $(BE)$ intersect at $P$, and that the line $(BD)$ meets line $(CA)$ at $Q$ and line $(CE)$ at $T$. Prove that the triangle $PQT$ is isosceles.

Let $ABC$ be a triangle and $\omega$ a circle which passes through $A$ and $B$ and intersects the sides $[BC]$ and $[AC]$ respectively in $D$ and $E$. The points $K$ and $L$ are respectively the centers of the circles inscribed in triangles $DAC$ and $BEC$ respectively . Let N be the intersection of lines $(EL)$ and $(DK)$. Prove that the triangle $KNL$ is isosceles.

Let $ABC$ be an triangle, isosceles  at $A$. We denote by $O$ the center of its circumscribed circle. Let $D$ be a point of $[BC]$. The line parallel to $(AB)$ passing through $D$ intersects $(AC)$ at $E$. The line parallel to $(AC)$ passing through $D$ intersects $(AB)$ at $F$. Show that $A, E, O, F$ are concyclic.

Let $ABC$ be a triangle. Let $D \in [AC]$ and $E  \in [AB]$ such that $BE = CD$. Let $P$ be the point intersection of $(BD)$ and $(CE)$. The circles circumscribed around $BEP$ and $CDP$ intersect in $Q$. Let $K,L$ be respectively the midpoints of $[BE], [CD]$. Let $R$ be the intersection between the perpendicular to $(QK)$ passing through $K$ and the perpendicular to $(QL)$ passing through $L$. Prove that:
a) $R$ lies on the circle circumscribed to $ABC$
b) $Q$ lies on the bisector of the angle $\angle BAC$.

Let $ABC$ be a triangle whose angles are acute, and such that $AB \ne AC$. We denote by $D$ the foot of the bisector of $\angle BAC$ . Points $E,F$ denotes the feet of the altitude resulting from  $B,C$ respectively.  The circle circumscribed to triangle $DBF$ meets the circle circumscribed to triangle $DCE$ at a point $M$ other than $D$. Prove that $ME = MF$.

Let C,C′ be two external circles, of centers O,O′ respectively. We conctruct two rays parallel, of the same sense, (OM)  and (O′M′) and also  two more parallel rays of the same sense, (OP) and (O′P′) (points M,M′ are in the same half-plane determined by OO′, as well as the points P,P′  are on the same side of OO′  ). Line MM′intersects the circle C′ for  second time at N, and line PP′ intersects the circle C′ for  second time at Q. Prove that the points M,N,P,Q are concyclic.

Let $D$ and $E$ be points belonging respectively to the interiors of sides $[AB]$ and $[AC]$ of a triangle $ABC$, such that $DB = BC = CE$. Let $F$ be the point of intersection of the lines $CD$ and $BE$, $I$ the center of the circle inscribed in triangle $ABC, H$ the orthocenter of triangle $DEF$ and $M$ the midpoint of the arc $BAC$ of the circumscribed circle to the triangle $ABC$. Show that $I, H$ and $M$ are collinear.

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

Let $ABCD$ be a trapezoid such that $(AB) \parallel (CD)$. Two circles $\omega_1$ and $\omega_2$ are located inside the trapezoid so that $\omega_1$ is tangent to $(DA), (AB), (BC)$ and $\omega_2$ is tangent to $(BC), (CD), (DA)$. Let $d_1$ be a line passing through $A$, other than $(AD)$, tangent to $\omega_2$. Let $d_2$ be a line passing through $C$, other than $(CB)$, tangent to $\omega_1$. Show that $d_1 \parallel d_2$.

2015 France RMM TST p2  (Zhautykov 2007 p6)
Let $ABCDEF$ be a convex hexagon and it`s diagonals have one common point $M$. It is known that the circumcenters of triangles $MAB,MBC,MCD,MDE,MEF,MFA$ lie on a circle. Show that the quadrilaterals $ABDE,BCEF,CDFA$ have equal areas.

Let $ABC$ be a triangle, $O$ the center of the circumscribed circle, $A'$ the orthogonal project of  $A$ on $(BC)$, and $X$ a point of the ray $[AA')$. The bisector of the angle $\angle BAC$ intersects the circle circumscribed to $ABC$ at $D$. Let $L$ be the midpoint of $[DX]$. The line passing through $O$ and parallel to (AD) intersects the line $(DX)$ at $N$. Show that $\angle BAM =\angle  CAN$ .

Let $ABC$ be a triangle with all angles acute, and $H$ its orthocenter. The bisectors of $\angle ABH$  and $\angle ACH$   intersect at a point $I$. Show that $I$ is collinear with the midpoints of $[BC]$ and of $[AH]$.

2016 France Training Test January p3 (2012 Swiss IMO TST p11)
Let $ABC$ be a triangle. Let $I$ be the center of the inscribed circle and $AD$ the diameter of the circle circumscribed around $ABC$. Let $E$ and $F$ be points on the rays $BA$ and $CA$ such that $BE = CF =\frac{AB + BC + CA}{2}$. Show that the lines $EF$ and $DI$ are perpendicular.

Let $ABC$ be a non-right triangle such that $AB <AC$. We denote by $H$ the projection of $A$ on $(BC)$, and $E, F$ the projections of  $H$ on $(AB), (AC)$ respectively . Line $(EF)$i ntersects $(BC)$ at point $D$. We consider the semicircle of diameter $[CD]$ located in the same semiplane by $(CD)$ that contains $A$. Let $K$ be the point of this semicircle whose projection on $CD$ is $B$. Show that $(DK)$ is tangent to the circumcircle  of $KEF$ .

Let $ABC$ be a triangle. We denote by $P$ the symmetric of $B$ with respect to $(AC)$ and $Q$ the symmetric of  $C$ wrt to $(AB)$. Let $T$ be the intersection between $(PQ)$ and the tangent at $A$ to the circle circumscribed at $(APQ)$. Show that the symmetric of $T$ wrt to $A$ belongs to $(BC)$.

Let $ABC$ be a triangle right in $C$. Let $D$ be the foot of the altitude in $C$, and $Z$ a point on $[AB]$ such that $AC = AZ$. Angle bisector $ \angle BAC$ intersects $(CB)$ and $(CZ)$ in $X$ and $Y$, respectively. Show that points $B, X, Y, D$ are concyclic.

Let $ABC$ be an acute triangle. The altitudes $[AA_1], [BB_1]$ and $[CC_1]$ intersect at point $H$. Let $A_2$ be the symmetric of $A$ wrt $B_1C_1$ and $O$ be the center of the circle circumscribed to the triangle $ABC$.
a) Prove that points $O, A_2, B_1$ and $C$ are concyclic.
b) Prove that $O, H, A_1$ and $A_2$ are concyclic.

Let $\omega_1, \omega_2$ be two circles tangent to each other at a point $T$, such that $\omega_1$ is the interior of  $\omega_2$. Let $M$ and $N$ be two distinct points on $\omega_1$, different from $T$. Let $[AB]$ and $[CD]$ two chords of the circle $\omega_2$ passing respectively through $M$ and $N$. We suppose that the segments $[BD], [AC]$, and $[MN]$ intersect at a point $K$. Show that $(TK)$ is the bisector of the angle $\angle MTN$ .

Let $ABCD$ be a convex quadrilateral of area $S$. We denote $a = AB, b = BC, c = CD$ and $d = DA$. For any permutation $x, y, z, t$ of $a, b, c, d$, show that $S \le \frac12 (xy + zt)$.

Let $ABC$ be an acute  triangle, and let $P$ be a point inside the triangle $ABC$. Let $D$ be the midpoint of the segment $[PC]$, $E$ the point of intersection of lines $(AP)$ and $(BC)$, and $Q$ the point of intersection of lines $(BP)$ and $(DE)$. Show that, if the angles $\angle PAC$ and  $\angle PCB$  are equal, then  $\sin ( \angle BCQ ) = \sin ( \angle BAP)$.

We have, in the plane, $16$ distinct two by two points, which we denote by $A_{i, j}$ for $i, j \in {1, 2, 3, 4}$. These points verify the following alignment and concyclicity relationships:
$\bullet$ for all $i \in  {1, 2, 3, 4}$, the points $A_{i, 1}, A_{i, 2}, A_{i, 3}$ and $A_{i, 4}$ are collinear,
$\bullet$ for all $j \in  {1, 2, 3, 4}$, the points $A_{1, j}, A_{2, j},, A_{3, j}$ and $A_{4, j}$ are collinear,
$\bullet$ the quadrilaterals $A_{1,1}A_{1,2}A_{2,2}A_{2,1}$,  $A_{2,1}A_{2,2}A_{3,2}A_{3,1}$,  $A_{3,1}A_{3,2}A_{4,2}A_{4,1}$,  $A_{1,2}A_{1,3}A_{2,3}A_{2,2}$, $A_{1,3}A_{1,4}A_{2,4}A_{2,3}$, $ A_{1,1}A_{2,2}A_{2,3}A_{1,4}$ and $A_{1,1}A_{2,2}A_{3,2}A_{4,1}$ are cyclic.
Show that the quadrilateral $A_{4,1}A_{3,2}A_{3,3}A_{4,4}$ is also cyclic.

Let $ABC$ be a triangle, and let $E$ and $F$ be two points belonging to lines $(AB)$ and $(AC)$, distinct from $A, B$ and $C$. Let $\Omega$ also be the circle circumscribed to $ABC$, either $O$ is the center of $\Omega$, and $\Gamma$ is the circle bounded by $AEF$. Finally, let $P$ be the point of intersection of $\Gamma$ and $\Omega$ other than $A$, and let $Q$ be the symmetric of $P$ with respect to the line $(EF)$. Show that $Q$  lies on the line $(BC)$ if and only if $O$  lies onthe circle $\Gamma$.

Let $ABC$ be an isosceles triangle with $AB = AC$ and let $D$ be a point on $(AC)$ such that  $A$ lies between $C$ and $D$, but not be the midpoint of the segment $[CD]$. We denote with $d_1$ and $d_2$ the inner and outer bisector of the angle $ \angle BAC$ ¸ and with $\Delta$ the perpendicular bisector of $[BD]$. Finally, let $E$ and $F$ be the intersection points of $\Delta$ with the lines $d_1$ and $d_2$, respectively. Prove that points $A, D, E$ and $F$ are concyclic.

Let $A, B, C$ and $P$ be four points of the plane such that $ABC$ is an equilateral triangle and that $AP <BP <CP.$ We suppose that using only data of the lengths $AP, BP$ and $CP$ is allows to determine, in a unique way, the length $AB$. Prove that $P$ belongs to the circle circumscribed to $ABC$.

Let $ABC$ be a triangle with all angles acute, with $AB \ne AC$. Let $\Gamma$ be the circle circumscribed to $ABC$, and $D$ the midpoint of the arc $BC$ not containing $A$. Let $E$ and $F$ be points belonging to segments $[AB]$ and $[AC]$ respectively, so that $AE = AF$. Let $P$ be the point of intersection, other than $A$, between $\Gamma$ and the circle circumscribed to $AEF$. Let $G$ and $H$ be the respective points of intersection, other than $P$, between $\Gamma$ and the lines $(PE)$ and $(PF)$. Finally, let $J$ be the point of intersection between lines $(AB)$ and $(DG)$, and let $K$ be the point of intersection between lines $(AC)$ and $(DH)$. Prove that the midpoint of the segment $[BC]$ belongs to the line $(JK)$.

Let $f$ be a function which, with each line $d$ of the plane, associates a point $f (d)$ belonging to $d$. We suppose that, since three lines $d_1, d_2$ and $d_3$ are concurrent in one point $X$, the points $f (d_1), f (d_2), f (d_3)$ and $X$ belong to the same circle.
a) Prove that there exists a unique point $P$ such that $f (d) = P$ for any line $d$ passing through $P$.
b) Show that there exists indeed such a function $f$.

Let $ABC$ be an acute triangle such that $\angle CAB> \angle BCA$  and let $P$ be the point of segment $[BC]$ such that $\angle PAB = \angle BCA$  . Let $Q$ be the point of intersection, other than $A$, between the circle circumscribed to $ABP$ and the line $(AC)$. Let $D$ then be the point of the segment $[AP]$ such that $\angle QDC  = \angle CAP$  , then $E$ the point of $(BD)$, other than $D$, such that $CE = CD$. Finally, let $F$ be the point intersection, other than $C$, between the circle circumscribed to $CQE$ and the line $(CD)$, and let $G$ be point of intersection of lines $(QF)$ and $(BC)$. Prove that the points $B, D, F$ and $G$ are concyclic.


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