Balkan BMO SHL 2014-18 24p

geometry problems from Balkan Shortlists (BMO Shortlists)
[and Team Selection Tests for Balkan Mathematical Olympiads (BMO TST) in the future]
with aops links in the names


                                                          under construction

BMO Shortlist 2014


(complete)
2014 BMO Shortlist G1
Let $ABC$ be an isosceles triangle, in which $AB=AC$ , and let $M$ and $N$ be two points on the sides $BC$ and $AC$, respectively such that $\angle BAM = \angle MNC$. Suppose that the lines $MN$ and $AB$ intersects at $P$. Prove that the bisectors of the angles $\angle BAM$ and $\angle BPM$ intersects at a point lying on the line $BC$

2014 BMO Shortlist G2
Triangle $ABC$ is said to be perpendicular to triangle $DEF$ if the perpendiculars from $A$ to $EF$,from $B$ to $FD$ and from $C$ to $DE$ are concurrent. Prove that if $ABC$ is perpendicular to $DEF$,then $DEF$ is perpendicular to $ABC$

2014 BMO Shortlist G3
Let $\triangle ABC$ be an isosceles.$(AB=AC)$.Let $D$ and $E$ be two points on the side $BC$ such that $D\in BE$,$E\in DC$ and $2\angle DAE = \angle BAC$.Prove that we can construct a triangle $XYZ$ such that $XY=BD$,$YZ=DE$ and $ZX=EC$.Find $\angle BAC + \angle YXZ$.

2014 BMO Shortlist G4
Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$. We consider the excenters $A_1$,$B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$,of the triangle $A_1B_1C_1$. By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.

2014 BMO Shortlist G5 (problem 3)
Let $ABCD$ be a trapezium inscribed in a circle $k$ with diameter $AB$. A circle with center $B$ and radius $BE$,where $E$ is the intersection point of the diagonals $AC$ and $BD$ meets $k$ at points $K$ and $L$. If the line ,perpendicular to $BD$ at $E$,intersects $CD$ at $M$,prove that $KM\perp DL$.

by Silouanos Brazitikos, Greece
2014 BMO Shortlist G6
In $\triangle ABC$ with $AB=AC$,$M$ is the midpoint of $BC$,$H$ is the projection of $M$ onto $AB$ and $D$ is arbitrary point on the side $AC$.Let $E$ be the intersection point of the parallel line through $B$ to $HD$ with the parallel line through $C$ to $AB$.Prove that $DM$ is the bisector of $\angle ADE$.

2014 BMO Shortlist G7
Let $I$ be the incenter of $\triangle ABC$ and let $H_a$, $H_b$, and $H_c$ be the orthocenters of $\triangle BIC$ , $\triangle CIA$, and $\triangle AIB$, respectively. The lines $H_aH_b$ meets $AB$ at $X$ and the line $H_aH_c$ meets $AC$ at $Y$. If the midpoint $T$ of the median $AM$ of $\triangle ABC$ lies on $XY$, prove that the line $H_aT$ is perpendicular to $BC$


BMO Shortlist 2015


(incomplete)
In an acute angled triangle $ABC$ , let $BB' $ and  $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle  $\widehat{ABB''}$. Similarly are defined the angles $\alpha_B$ and $\alpha_C$. Prove that

$$\displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}$$

Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ του $\omega$ and is different than $B,C$ and the midpoint of arc $BC$.  Tangent of $\Gamma$ on $D$ intersects lines $BC,CA,AB$ at $A',B',C'$, respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA'$ intersects again the circle  $\omega$ at $F$. Prove that points $D,E,F$ are collinear.

A set of points of the plane is called [i] obtuse-angled[/i] if every three of it's points are not collinear and every triangle with vertices inside the set has one angle $ >91^o$. Is it correct that every finite [i] acute-angled[/i] set can be extended to an infinite  [i]obtuse-angled[/i] set?

2015 BMO Shortlist G4 (problem 2)
Let ABC be a scalene triangle with incentre I and circumcircle (ω).The lines AI,BI,CI intersect (ω) for the second time at the points D,E, F, respectively. The lines through I parallel to the sides BC,AC,AB intersect the lines EF,DF,DE at the points K, L,M, respectively. Prove that  the points K, L,M are collinear.
by Theoklitos Paragyiou, Cyprus
2015 BMO Shortlist G5
2015 BMO Shortlist G6
2015 BMO Shortlist G7


BMO Shortlist 2018
(complete)
Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$.

Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular.
by Michael Sarantis, Greece
Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of
$$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$.

A quadrilateral $ABCD$ is inscribed in a circle $k$ where $AB$ $>$ $CD$,and $AB$ is not paralel to $CD$.Point $M$ is the intersection of diagonals $AC$ and $BD$, and the perpendicular from $M$ to $AB$ intersects the segment $AB$ at a point $E$.If $EM$ bisects the angle $CED$ prove that $AB$ is diameter of $k$.

Let $ABC$ be an acute triangle with $AB<AC<BC$  and let $D$ be a point on it's extension towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$  respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $  respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide.
by Evangelos Psychas, Greece
In a triangle $ABC$ with $AB=AC$, $\omega$ is the circumcircle and $O$ its center. Let $D$ be a point on the extension of $BA$ beyond $A$. The circumcircle $\omega_{1}$ of triangle $OAD$ intersects the line $AC$ and the circle $\omega$ again at points $E$ and $G$, respectively. Point $H$ is such that $DAEH$ is a parallelogram. Line $EH$ meets circle $\omega_{1}$ again at point $J$. The line through $G$ perpendicular to $GB$ meets $\omega_{1}$ again at point $N$ and the line through $G$ perpendicular to $GJ$ meets $\omega$ again at point $L$. Prove that the points $L, N, H, G$ lie on a circle.


by Theoklitos Paragyiou, Cyprus

BMO Shortlist before 2004 
random geometry problems
mentioned in aops by enescu here


BMO Shortlist before 2004 1
A line passing through the center $O$ of an equilateral triangle $ABC$ intersects the circumcircles of the triangles $OAB,OBC,$ and $OCA$ at $K,L,$ and $M.$ Prove that $OK^{2}+OL^{2}+OM^{2}=2AB^{2}.$

BMO Shortlist before 2004 2
Let $M$ be a point inside the equilateral triangle $ABC$ and let $% A^{\prime },$ $B^{\prime },$ and $C^{\prime }$ be its projections on the sides $BC,$ $CA,$ and $AB,$ respectively. Denote by $% r_{1},r_{2},r_{3},r_{1}^{\prime },r_{2}^{\prime },$ and $r_{3}^{\prime }$ the inradii of the triangles $MAC^{\prime },$ $MBA^{\prime },$ $MCB^{\prime },$ $MAB^{\prime },$ $MBC^{\prime },$ and $MCA^{\prime }.$ Prove that
$r_{1}+r_{2}+r_{3}=r_{1}^{\prime }+r_{2}^{\prime }+r_{3}^{\prime }.$

BMO Shortlist before 2004 3
Let $ABCD$ be a convex quadrilateral and let $M,N$ be points on the sides $AB$ and $BC.$ The line segments $DM$ and $DN$ intersect $AC$ at $K$ and $L$ and the lines $BK$ and $BL$ intersect the sides $AD$ and $CD$ at $R$ and $S,$ respectively. Suppose that
$AK=KL=LC$ and $\left[ ADM\right] =\left[ CDN\right] =\frac{1}{4}\left[ABCD\right] .$
Prove that $ \left[ ABR\right] =\left[ BCS\right] =\frac{1}{4}\left[ ABCD\right] .$

BMO Shortlist before 2004 4
On each side of the triangle $ABC$ a regular $n-$gon is constructed in the exterior on the triangle and sharing the side with the triangle. Find the values of $n$ for which the centers of the three $n-$gons are the vertices of an equilateral triangle.

BMO Shortlist before 2004 5
Let $O$ be a point in the interior of the triangle $ABC.$ The lines $%OA,OB,$ and $OC$ intersect the sides $BC,CA,$ and $AB$ at thepoints $D,E,$ and $F,$ respectively. The same lines intersect$EF,FD,$ and $DE$ at the points $K,L,$ and $M.$ Prove that $\frac{KA}{KD}\cdot \frac{LB}{LE}\cdot \frac{MC}{MF}\geq 1.$

BMO Shortlist before 2004 6
Find the maximum number of points that can be chosen in the interior of a regular hexagon with side length $1$ such that all mutual distances between the points are at least $\sqrt{2}.$

BMO Shortlist before 2004 7
The side lengths of the obtuse triangle $ABC$ are three consecutive odd integers and $11\cos \angle B=13\cos \angle C.$ Prove that $OI=\frac{2}{3}\sqrt{ab}$ ($O$ is the circumcenter and $I$ is the incenter of the triangle).

BMO Shortlist before 2004 8
Let $A_{1}A_{2}\ldots A_{n}$ be a regular polygon. Find all points $P$ in the polygon's plane with the property: the squares of distances from $P$ to the polygon's vertices are consecutive
terms of an arithmetic sequence.

Let $ABC$ be a triangle and let $P$ be a variable point. We denote by $A',B',C'$ the symmetric points of $P$ across the triangle's sides and by $P^*$ the centroid of triangle $A'B'C'$. Find $P$ such that $P=P^*$.

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