France JBMO TST 2013-22 34p

geometry problems from French Junior Balkan Mathematical Olympiads Team Selection Tests (JBMO TST) with aops links

[not in JBMO Shortlist]

collected inside aops here

 2013 - 2022

2013 France JBMO Training Test
Let $ABC$ be a triangle of area $1$. Determine the maximum area of a parallelogram whose four vertices are inside or on the border of the triangle.

2013 France JBMO TST 1.3
On a circle $\Gamma$ we consider the points $A, B, C$ so that $AC = BC$. Let $P$ be a point on the arc $AB$ of the circle $\Gamma$, which does not contain the point $C$. Perpendicular from $C$ on $PB$ intersects $PB$ at $D$. Prove that $PA + PB = 2PD$.

2014 France JBMO TST 1.1
Let $C, C'$ be two external circles, of centers $O,O'$ respectively.  An external common tangent intersects the inner common tangents at points $M$ and $N$. Prove that $OM \perp O'M$ and  $ON \perp O'N$.

2014 France JBMO TST 1.4 

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 $ME = MF$

2014 France JBMO TST 2.4
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.

2014 France JBMO TST 3.2
Let $P$ be a point outside the circle $\Gamma$. The tangents from $P$ to the circle $\Gamma$ intersect the circle at points $A$ and $B$. A line passing through $P$ intersects the circle at points $Q$ and $R$. Let $S$ in $\Gamma$ be so $BS \parallel QR$. Prove that $SA$ goes through the midpoint of $[QR]$.

2015 France JBMO TST 1.2
Let ABC be an acute triangle and $D \in BC$ such that $AD = AB$ and $D \ne B$. Let $\Gamma$ be the circle circumscribed to $ABC$, $\Delta$ be the tangent to $\Gamma$ at $C$ such that $\{E\} = AD \cap \Delta$ . Prove that $CD^2 = AD \cdot DE - BD \cdot  DC$.

2015 France JBMO TST 2.4
Let $ABC$ be a scalene triangle. Let $\omega$  be the inscribed circle and $I$ its center. Let $M, N, P$ be the points of contact of $\omega$ with the sides $[BC], [CA] , [AB]$ respectively. Let $\{J\} = MN \cap IC$. The line $PJ$ cuts for second time $\omega$  at $K$. Prove that:
a) $CKIP$ is cyclic
b) $(CI$ is the bisector of the angle $\angle PCK$.

2015 France JBMO TST 3.2
Let$ABC$ be a triangle. We denote by $D$ the foot of the bisector of the angle $\angle A$. Let $M, N$ be two points of $[AD]$ such  that $\angle NBA = \angle  CBM$. Let $E$ be the second intersection point of the line $BM$ with the circle circumscribed around the triangle $ACM$. Let $F$ be  the second intersection point of the line $CN$ with the circle circumscribed around the triangle $ABN$. Prove that $A, E, F$ are collinear.

2016 France JBMO TST 1.1
Let $ABC$ be an isosceles triangle with vertex in $A$, with $\angle A \ne 90^o$. Let $D$ be the point on $BC$ such that $AD \perp AB$. Let $E$ be the projection of $D$ on the $AC$. Finally, let $H$ be the middle of $[BC]$. Prove that $AH = HE$.

2016 France JBMO TST 2.3
Let $ABC$ be a triangle and $M$ the middle of $[BC]$. We denote by $I_b$ and $I_c$ the centers of the circles inscribed in the triangles $AMB$ and $AMC$ respectively. Prove that the second point of intersection of the circles circumscribed to the triangles $AI_bB$ and $AI_cC$ lies on line $AM$.

2016 France JBMO TST 3.2
Given is arbitrary point $P$ and circle $K$. The tangents from $P$ to $K$ intersect $K$ in $A$ and $B$. Let $M$ be the midpoint of $BP$. $AM$ intersects $K$ in $C$. $PC$ intersects $K$ in $D$. Prove that $AD$ is parralel to $BP$

2016 France JBMO TST 4.4 (IMO 2015 Shortlist, G1)
Given is triangle $ABC$ with orthocenter $H$. Also, point $G$ is such that $ABGH$ is parallelogram. Point $I$ is on $GH$, such that $AC$ intersects $HI$ in the middle. Let $J$ be the intersection point of $(GCI)$ and $AC$. Prove that $IJ=AH$.

2017 France JBMO TST 1.4
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.

2017 France JBMO TST 2.4
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.

2017 France JBMO TST 3.1 [typo has been corrected]
Let $\Gamma$ be a circle with center $O$ ¸ and $M$ a point outside the circle. Let $A$ be a point of $\Gamma$ so that the line $MA$ is tangent to the circle. Let $B$ and $C$ be two points of the circle $\Gamma$ such that $B$ belongs to the arc $AC$. Let $H$ be the projection of $A$ on $[MO]$ and $K$ the intersection of $[MO]$ with the circle $\Gamma$. Prove that $(BK$ is the bisector angle $\angle HBM$

2017 France JBMO TST 4.3
Let $ABC$ be a triangle. Let $D$ and $E$ be two points of $[AC]$ so that $D$ is lies between $C$ and $E$. Let $F$ be the intersection of the circle circumscribed to $ABD$ with the parallel to $BC$ through $E$  such that $E, F$ lie in different half-planes wrt $AB$. Let $G$ the intersection of the circle circumscribed to triangle $BCD$ with the line parallel  to $AB$ through $E$,  so that $E,G$  lie in different half-planes wrt $BC$. Prove that the points $D, E, F, G$ are concyclic.

2018 France JBMO TST 1.4

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 $S \le \frac12 (xy + zt)$.

2018 France JBMO TST 2.2
Let $ABC$ be a triangle and $L, M, N$ be the midpoints of the sides $[BC], [CA] , [AB]$ respectively. We denote by $d$ the tangent at $A$ of the circle circumscribed to $ABC$. The line $LM$ intersects $d$  at $P$ and the line $LN$ intersects $d$ at $Q$. Show that the lines $CP$ and $BQ$ are parallel.

2018 France JBMO TST 3.2
Let $ABC$ be an isosceles triangle with vertex in $A$ and $\angle BAC= 100^o$. Let $D$ be the foot of the bisector from $B$. Show that $BC = AD + BD$.

2018 France JBMO TST 4.3
Let $ABCD$ be a trapezoid in which $(AB)$ and $(CD)$ are parallel. We assume that $AD< CD$ and that $ABCD$ is inscribed in a circle $\Gamma$. Let $P \in \Gamma$ be such that $(DP)$ is parallel to $(AC)$. The tangent at $D$ at $\Gamma$ intersects for second time $AB$ in $E$, and the chords $[BP]$ and $[CD]$ intersect in $Q$. Show that $EQ = AC$.

2019 France JBMO TST 1.2
Let $\Gamma$ be a circle with center $O$ and radius $r$, and a straight line $\ell$ that does not intersect. We note with $E$ the point of intersection between $\ell$ and the perpendicular from $O$ on $\ell$. Let $M$ be a point of $\ell$ different from $E$. The tangents from $M$ at $\Gamma$ intersect $\Gamma$ in $A$ and $B$. Finally, let $H$ be the point of intersection of the lines $AB$ and $OE$. Prove that $OH =\frac{r^2}{OE}$.

2019 France JBMO TST 2.3
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.

2019 France JBMO TST 3.1
Let $ABCD$ be a trapezoid in which $(AB)$ is parallel to $(CD)$. Let $P$ be a point on $[AC]$, and $Q$ a point on $[BD]$ so that $\angle APD = \angle BQC$. Prove that $\angle AQD = \angle BPC$.

2019 France JBMO TST 4.3
Let $ABC$ be a triangle and let $M$ be the foot of the median of $A$. Also be $\ell_b$ be the bisector of $\angle AMB$   and $\ell_c$ the bisector of $\angle AMC$. Finally, let $B'$ be the projection of $B$ on $\ell_b$, $C'$ be the projection of $C$ on $\ell_c$ and let $A'$ be the point of intersection of the lines $AM$ and $B'C'$. Prove that $A'B' = A'C'$

2020 France JBMO TST 1.2
Let $ABC$ be a triangle and  $K$ be its circumcircle.  Let $P$ be the point of intersection
of $BC$ with tangent in $A$ to $K$. Let $D$ and $E$ be the symmetrical points of $B$ and $A$, respectively,  from  $P$.  Let $K_1$ be  the circumcircle of triangle $DAC$ and let $K_2$
 the circumscribed circle of triangle $APB$. We denote with $F$ the second intersection point of the circles $K_1$ and $K_2$.  Then denote with $G$ the second intersection point of the circle $K_1$ with $BF$.   Show that the lines $BC$ and $EG$ are parallel.

2020 France JBMO TST 2.1
Given are four distinct points $A, B, E, P$ so that $P$ is the middle of $AE$ and $B$ is on the segment $AP$. Let $k_1$ and $k_2$ be two circles passing through $A$ and $B$. Let $t_1$ and $t_2$ be the tangents of $k_1$ and $k_2$, respectively, to $A$.Let $C$ be the intersection point of $t_2$ and $k_1$ and $Q$ be the intersection point of $t_2$ and the circumscribed circle of the triangle $ECB$. Let $D$ be the intersection posit of $t_1$ and $k_2$ and $R$ be the intersection point of $t_1$ and the circumscribed circle of triangle $BDE$. Prove that $P, Q, R$ are collinear.

2021 France JBMO TST 1.4

Let $P_1P_2...P_{2021}$ a convex polygon with$2021$ sides with the property that, for every vertex $P_i$ , the $2018$ diagonals starting from $P_i$ divide the angle $\angle P_i$ in $2019$ equal angles. Prove that $P_1P_2...P_{2021}$ is a regular polygon, i.e. a polygon that has all the angles of the same measure and all the sides of the same length.

2021 France JBMO TST 3.3

Let $ABC$ be a triangle such that $90^o > \angle ABC) > \angle BCA$. Let $D$ be one point on side $[BC]$ such that $\angle DAC = \angle ABC - \angle BCA$. Denote by $E$ the intersection point, different from $A$, between $(AB)$ and the circumscribed circle of the triangle $ACD$, and by $P$ the intersection point between $(AB)$ and the angle bisector of $\angle BDE$. Analogously, we denote by $F$ the intersection point ,different from $A$, between $(AC)$ and the circumscribed circle of the triangle $ABD$, and by $Q$ the intersection point between $(AC)$ ¸and the angle bisector of $\angle CDF$. Prove that $AB$ and $PQ$ are perpendicular.

2021 France JBMO TST 4.1

Let ABCDE be a convex pentagon such that $\angle ABE  = \angle  ACE  =\angle ADE = 90^o$ and $BC = CD$. Finally, let $K$ be a point on the semiline $[AB$ such that $AK = AD$, and let $L$ be a point on the semi-line $[ED$ such that $EL = BE$. Prove that points $B, D, K$ and $L$ lie on a circle with center at $C$.

2022 France JBMO TST 1.3

Let $ABC$ be a triangle and $\Omega$ its circumscribed circle. Let $D$ be the foot of the altitude taken from vertex $A$. The bisector of angle $A$ intersects segment $[BC]$ at point $P$ and cuts the circle $\Omega$ for second time at point $S$. Let $A'$ be the point diametrically opposite to point $A$ in the circle $\Omega$. Prove that the lines $SD$ and $A'P$ intersect on the circle $\Omega$.

2022 France JBMO TST 2.2

Let $ABC$ be an acute-angled triangle and let $D$ be a point located inside the triangle $ABC$. The lines $AD$ and $BD$ intersect the circumscribed circle of triangle $ABC$ again at points $A_1$ and $B_1$, respectively. The circumscribed circle of the triangle $B_1DA$ again intersects the line $AC$ at point $P$. The circumscribed circle of the triangle $A_1BD$ intersects again the line $BC$ at point $Q$. Prove that the quadrilateral $CPDQ$ is a parallelogram.

2022 France JBMO TST 3.1

Let $ABCD$ be a rectangle. Let $\omega$ be the semicircle of diameter $[BC]$ located on the same part of the line$BC$ as the point $A$. The circle of center $B$ and the radius $AB$ intersect again the semicircle $\omega$ at point $E$. The line $AE$ intersects the semicircle $\omega$ again at point $F$. Prove that $AF = BF$

source: https://pregatirematematicaolimpiadejuniori.wordpress.com/