Balkan BMO Shortlist 2014- 37p

geometry problems from Balkan Shortlists (BMO Shortlists)
with aops links in the names


2014-2018

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
Quadrilateral $ABCD$ is given with $AD \nparallel BC$. The midpoints of $AD$ and $BC$ are denoted by $M$ and $N$, respectively. The line $MN$ intersects the diagonals $AC$ and $BD$ in points $K$ and $L$, respectively. Prove that the circumcircles of the triangles $AKM$ and $BNL$ have common point on the line $AB$.
by Emil Stoyanov, Bulgaria
Let $AB$ be a diameter of a circle $(\omega)$  with centre $O$. From an arbitrary point $M$ on $AB$ such that $MA < MB$ we draw the circles $(\omega_1)$ and $(\omega_2)$ with diameters $AM$ and $BM$ respectively. Let $CD$ be an exterior common tangent of $(\omega_1), (\omega_2)$ such that $C$ belongs to $(\omega_1)$ and $D$ belongs to $(\omega_2)$. The point $E$ is diametrically opposite to $C$ with respect to $(\omega_1)$ and the tangent to $(\omega_1)$ at the point $E$ intersects $(\omega_2)$ at the points $F, G$. If the line of the common chord of the circumcircles of the triangles $CED$ and $CFG$ intersects the circle $(\omega)$ at the points $K, L$ and the circle $(\omega_2)$ at the point $N$ (with $N$ closer to $L$), then prove that $KC = NL$.

2015 BMO Shortlist G7
Let scalene triangle $ABC$ have orthocentre $H$ and circumcircle $\Gamma$. $AH$ meets $\Gamma$ at $D$ distinct from $A$. $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$ respectively, and $EF$ meets $BC$ at $P$. The tangents to $\Gamma$ at $B$ and $C$ meet at $T$. Show that $AP$ and $DT$ are concurrent on the circumcircle of $AFE$.

BMO Shortlist 2016
(complete)


BMO Shortlist 2016 G1 (Bulgaria 2018)
Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.

2016 BMO Shortlist G2 (problem 2, Greece)
Let ABCD be a cyclic quadrilateral with AB < CD. The diagonals intersect at the point F and lines AD and BC intersect at the point E. Let K and L be the orthogonal projections of F onto lines AD and BC respectively, and let M, S and T be the midpoints of EF, CF and DF respectively. Prove that the second intersection point of the circumcircles of triangles MKT and MLS lies on the segment CD.


by Silouanos Brazitikos
Given that $ABC$ is a triangle where $AB < AC$. On the half-lines $BA$ and $CA$ we take points $F$ and $E$ respectively such that $BF = CE = BC$. Let $M,N$ and $H$ be the mid-points of the segments $BF,CE$ and $BC$ respectively and $K$ and $O$ be the circumcenters of the triangles $ABC$ and $MNH$ respectively. We assume that $OK$ cuts $BE$ and $HN$ at the points $A_1$ and $B_1$ respectively and that $C_1$ is the point of intersection of $HN$ and FE. If the parallel line from $A_1$ to $OC_1$ cuts the line $FE$ at $D$ and the perpendicular from $A_1$ to the line $DB_1$ cuts $FE$ at the point $M_1$, prove that $E$ is the orthocenter of the triangle $A_1OM_1$.

BMO Shortlist 2017
(complete)
Let $ABC$ be an acute triangle. Variable points $E$ and $F$ are on sides $AC$ and $AB$ respectively such that $BC^2 = BA\cdot  BF + CE \cdot CA$ . As $E$ and $F$ vary prove that the circumcircle of $AEF$ passes through a fixed point other than $A$ .

Let $ABC$ be an acute triangle and $D$ a variable point on side $AC$ . Point $E$ is on $BD$ such that $BE =\frac{BC^2-CD\cdot CA}{BD}$ . As $D$ varies on side $AC$ prove that the circumcircle of $ADE$ passes through a fixed point other than $A$ .

2017 BMO Shortlist G3 (problem 2, Greece)
Consider an acute-angled triangle ABC with AB<AC and let ω be its circumscribed circle. Let tB and tC be the tangents to the circle ω at points B and C, respectively, and let L be their intersection. The straight line passing through the point B and parallel to AC intersects tC in point D. The straight line passing through the point C and parallel to AB intersects tB in point E. The circumcircle of the triangle BDC intersects AC in T, where T is located between A and C. The circumcircle of the triangle BEC intersects the line AB (or its extension) in S, where B is located between S and A. Prove that ST, AL, and BC are concurrent.


by Evangelos Psychas and Silouanos Brazitikos
The acuteangled triangle $ABC$ with circumcenter $O$ is given. The midpoints of the sides $BC, CA$ and $AB$ are $D, E$ and $F$ respectively. An arbitrary point $M$ on the side $BC$, different of $D$, is choosen. The straight lines $AM$ and $EF$ intersects at the point $N$ and the straight line $ON$ cut again the circumscribed circle of the triangle $ODM$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $DP$ belongs on the nine points circle of the triangle $ABC$.


Let $ABC$ be an acute angled triangle with orthocenter $H$. centroid $G$ and circumcircle $\omega$. Let $D$ and  $M$ respectively be the intersection of lines $AH$ and $AG$ with side $BC$. Rays $MH$ and $DG$ interect $ \omega$ again at $P$ and $Q$ respectively. Prove that $PD$ and $QM$ intersect on  $\omega$.


Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B, ABB_A , ACA_C,ACC_A ,BCB_C$ and $BCC_B$, so that $$AB = AB_A = BA_B, AC = AC_A=CA_C, BC = BC_B = CB_C$$ and $$\angle BAB_A = \angle ABA_B =\angle CAC_A=\angle ACA_C= \angle BCB_C =\angle CBC_B = a < 90^o$$.
Prove that the perpendiculars from $A$ to $B_AC_A$, from $B$ to $A_BC_B$ and from $C$ to $A_CB_C$ are concurrent.

Let $ABC$ be an acute triangle with $AB\ne AC$ and circumcircle $\omega$. The angle bisector of $BAC$ intersects $BC$ and $\omega$ at $D$ and $E$ respectively. Circle with diameter $DE$ intersects $\omega$ again at $F \ne E$. Point $P$ is on $AF$ such that $PB = PC$ and $X$ and $Y$ are feet of perpendiculars from $P$ to $AB$ and $AC$ respectively. Let $H$ and $H'$ be the orthocenters of $ABC$ and $AXY$ respectively. $AH$ meets $\omega$ again at $Q$ . If $AH'$ and $HH'$ intersect the circle with diameter $AH$ again at points $S$ and $T$, respectively, prove that the lines $AT , HS$ and $FQ$ are concurrent.

Given an acute triangle  $ABC$ ($AC\ne AB$) and let $(C)$ be its circumcircle. The excircle $(C_1)$ corresponding to the vertex $A$, of center $I_a$, tangents to the side $BC$ at the point $D$ and to the extensions of the sides $AB,AC$ at the points $E,Z$ respectively. Let $I$ and $L$ are the intersection points of the circles $(C)$ and $(C_1)$, $H$  the orthocenter of the triangle $EDZ$ and $N$ the midpoint of segment $EZ$. The parallel line through the point $l_a$ to the line $HL$ meets the line $HI$ at the point $G$. Prove that the perpendicular line $(e)$ through the point $N$ to the line $BC$ and the parallel line $(\delta)$ through the point $G$ to the line $IL$ meet each other on the line $HI_a$.

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 $BF=CG$.

by Petru Braica, Romania
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 $P$.

by Elton Bojaxhiu, Albania
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 $k$.

by Emil Stoyanov, Bulgaria
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 $P=P^*$.

old BMO Shortlist
mentioned in aops

Let $ABO$ be an equilateral triangle with center $S$ and let $A^{\prime}B^{\prime}O$ be another equilateral triangle with the same orientation and $S \neq A, S \neq B$. Consider $M$ and $N$, the midpoints of segments $A^{\prime}B$ and $AB^{\prime}$. Prove that $\triangle SB^{\prime}M \sim \triangle SA^{\prime}N$.

Two circles $\Gamma_1$ and $\Gamma_2$ with radii $r_1$ and $r_2$ ($r_2$$>$$r_1$), respectively are externally tangent. The straight line $t_1$ is tangent to the circles $\Gamma_1$ and $\Gamma_2$ at points $A$ and $D$, respectively.The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\Gamma_1$ and intersects $\Gamma_2$ at points $E$ and $F$. The line $t_3$ through $D$ intersects the line $t_2$ and the circle $\Gamma_2$ at points $B$ and $C$, respectively, different from $E$ and $F$. Prove that the circumcircle of triangle $ABC$ is tangent to the line $t_1$.

BMO Shortlist 2007 (Romanian TST4 2007)
Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that
$ [A_{1}A_{2}A_{3}] = [A_{2}A_{3}A_{4}]=[A_{3}A_{4}A_{5}] =[A_{4}A_{5}A_{1}] =[A_{5}A_{1}A_{2}].$
Prove that there exists a point  $ M$ in the plane of the pentagon such that
$[A_{1}MA_{2}] =[A_{2}MA_{3}] =[A_{3}MA_{4}] =[A_{4}MA_{5}] = [A_{5}MA_{1}].$
Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

Let $ I$ be the incenter of the triangle  $ ABC$ . A circle $ K$  passes through $ I$ and touches both side $ BC$ and the circumcircle of $ \triangle ABC$. Prove that $ AI$ is a tangent to the circle $ K$.

BMO Shortlist 2011 Saudi Arabia
Given a triangle $ABC$, let $D$ be the midpoint of the side $AC$ and let $M$ be the point that divides the segment $BD$ in the ratio $1/2$; that is, $MB/MD=1/2$. The rays $AM$ and $CM$ meet the sides $BC$ and $AB$ at points $E$ and $F$, respectively. Assume the two rays perpendicular: $AM\perp CM$. Show that the quadrangle $AFED$ is cyclic if and only if the median from $A$ in triangle $ABC$ meets the line $EF$ at a point situated on the circle $ABC$.

Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.

The incircle of a triangle $ABC$ touches its sides $BC$,$CA$,$AB$ at the points $A_1$,$B_1$,$C_1$.Let the projections of the orthocenter $H_1$ of the triangle $A_{1}B_{1}C_{1}$ to the lines $AA_1$ and $BC$ be $P$ and $Q$,respectively. Show that $PQ$ bisects the line segment $B_{1}C_{1}$

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[and Team Selection Tests for Balkan Mathematical Olympiads (BMO TST) in the future]
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