start page, in English

This start page in Greek here / Η παρούσα αρχική σελίδα στα Ελληνικά εδώ
Welcome.
Here are gonna be collected all problems proposed already in this blog.
Do not post links, just solutions, if you find any problem interesting.
All links of these problems within the blog are on the right.
Below are JBMO Shortlists.


all problems posted in this blog collected
Balkan Mathematical Olympiad  (BMO)
1984 - 2017

1984 BMO Problem 2 (ROM)                                              
Let ABCD be a cyclic quadrilateral and let HA, HB, HC, HD  be the orthocenters of the triangles BCD, CDA, DAB and ABC respectively. Show that the quadrilaterals ABCD and HAHBHCHD  are congruent.

1985 BMO Problem 1 (BUL)                                             
Let O be the circumcircle of a triangle ABC, D be the midpoint of AB, and E be the centroid of triangle ACD. Prove that CD is perpendiculat to OE if and only if AB = AC.
 by Ivan Tonov
1985 BMO Shortlist 1 (GRE)                                               
Let e1, e2 be two lines perpendicular to the same plane. Find the locus of the points of the space , that we can draw 3 lines, perpendicular in pairs, who intersect e1 or e2 .
by Theodoros Bolis
1985 BMO Shortlist 2 (GRE)                                              

Let ABC be a triangle with <A=120­­­o.  Let AD, CE be the angle bisectors of angles A,C respectively and I be the intersection point of AD, CE. If Z is the intersection point of BI and DE, calculate angle <DAZ .


by Dimitris Kontogiannis
1986 BMO Problem 1 (GRE)                                         
A line through the incenter I of a triangle ABC intersects its circumcircle at F and G, and its incircle at D and E, where D is between I and F. Prove that DF EG ≥ r­­­­2 , where r is the inradius. When does equality occur? 

1986 BMO Problem 2 (BUL)                                      
Let E,F,G,H,K,L respectively be points on the edges AB,BC,CA,DA, DB,DC  of a tetrahedron ABCD. If AE BE = BF CF =CGAG = DH AH = DK BK = DL CL,  prove that the points E,F,G,H,K,L lie on a sphere.

1986 BMO Problem 4 (ROM)                                        
A triangle ABC and a point T are given in the plane so that the triangles TAB, TBC, TCA have the same area and perimeter. Prove that:
(a) If T is inside △ABC, then △ABC is equilateral;
(b) If T is not inside △ABC, then △ABC is right-angled.


1987 BMO Problem 4 (BUL)    
Circles k1 (O1 ,1) and k2 (O2 , 2) with O1O2 = 2 intersect at A and B. Find the length of the chord AC of circle k2 whose midpoint lies on k­1.

1988 BMO Problem 1 (BUL)   
Let CH,CL,CM be the altitude, angle bisector, and median of a triangle ABC, respectively, where H,L,M are on AB. Given that the ratios of the areas of HMC and LMC to the area of ABC are equal to  $$\frac{1}{4}$$ και $$1-\frac{\sqrt{3}}{2}$$, respectively, determine the angles of ABC.

1988 BMO Problem 4 (GRE)  
Show that every tetrahedron A­­12A­­3A­­4 can be placed between two parallel planes which are at the distance at most $\frac{1}{2}\sqrt{\frac{p}{3}}$, where $$P=A_{1}A_{2}^{2}+A_{1}A_{3}^{2}+A_{1}A_{4}^{2}+{A}_{2}A_{3}^{2}+A_{2}A_{4}^{2}+A_{3}A_{4}^{2}$$

1989 BMO Problem 3 (GRE)  
A line l intersects the sides AB and AC of a triangle ABC at points B1 and C1, respectively, so that the vertex A and the centroid G of ABC lie in the same half-plane determined by l. Prove that $${{S}_{B{{B}_{1}}G{{C}_{1}}}}+{{S}_{C{{C}_{1}}G{{B}_{1}}}}\ge \frac{4}{9}{{S}_{ABC}}$$
by Dimitris Kontogiannis
1990 BMO Problem 3 (YUG)            
The feet of the altitudes of a non-equilateral triangle ABC are A1,B1,C1. If A2,B2,C2 are the tangency points of the incircle of the triangle A1B1C1 with its sides, prove that the Euler lines of the triangles ABC and A2B2C2 coincide.

1991 BMO Problem 1 (GRE)      
Let M be a point on the arc AB not containing C of the circumcircle of an acuteangled triangle ABC, and let O be the circumcenter. The perpendicular from M to OA intersects AB at K and AC at L. The perpendicular from M to OB intersects AB at N and BC at P. If KL = MN, express <MLP in terms of the angles of ABC.

1991 BMO Problem 3 (BUL)
A regular hexagon of area H is inscribed in a convex polygon of area P. Prove that P ≤ 3/2 H. When does equality occur?

1992 BMO Problem 3 (GRE)   
Let D,E,F be points on the sides BC,CA,AB respectively of a triangle ABC (distinct from the vertices). If the quadrilateral AFDE is cyclic, prove that $$\frac{4{{S}_{DEF}}}{{{S}_{ABC}}}\le {{\left( \frac{EF}{AD} \right)}^{2}}$$
1993 BMO Problem 3 (GRE)    
Circles C­1 and C­2 with centers O­1 and O­2, respectively, are externally tangent at point G. A circle C with center O touches C­1 at A and C­2 at B so that the centers O­1,O2 lie inside C. The common tangent to C­1 and C­2 at G intersects the circle C at K and L. If D is the midpoint of the segment KL, show that <O­1OO­2 = <ADB.

1994 BMO Problem 3 (GRE) 
An acute angle XAY and a point P inside it are given. Construct (by a ruler and a compass) a line that passes through P and intersects the rays AX and AY at B and C such that the area of the triangle ABC equals AP2.

1995 BMO Problem 2 (GRE)
Circles c1 (O1, r1) and c2 (O2, r2), r2> r1, intersect at A and B so that <O1 AO2 = 90◦. The line O1 O2 meets c1 at C and D, and c2 at E and F (in the order CEDF). The line BE meets c1 at K and AC at M, and the line BD meets c2 at L and AF at N. Prove that $$\frac{{{r}_{2}}}{{{r}_{1}}}=\frac{KE}{KM}\cdot \frac{LN}{LD}$$.
1996 BMO Problem 1 (GRE)  
Let O be the circumcenter and G be the centroid of a triangle ABC. If R and r are the circumradius and inradius of the triangle, respectively, prove that  

1996 BMO Problem 3 (YUG) 
In a convex pentagon ABCDE, M,N,P,Q,R are the midpoints of the sides AB, BC, CD, DE, EA, respectively. If the segments AP, BQ, CR, DM pass through a single point, prove that EN contains that point as well. 

1997 BMO Problem 1 (YUG)
Suppose that O is a point inside a convex quadrilateral ABCD such that OA2+OB2+OC2+OD2 = 2SABCD , where SABCD  denotes the area of ABCD. Prove that ABCD is a square and O its center.

1997 BMO Problem 3 (GRE)  
Circles C1 and C2 touch each other externally at D, and touch a circle G internally at B and C, respectively. Let A be an intersection point of G and the common tangent to C1 and C2 at D. Lines AB and AC meet C1 and C2 again at K and L, respectively, and the line BC meets C1 again at M and C2 again at N. Show that the lines AD, KM, LN are concurrent.
 
1998 BMO Problem 3 (YUG)   
Let L denote the set of points inside or on the border of a triangle ABC, without a fixed point T inside the triangle. Show that L can be partitioned into disjoint closed segments.

1999 BMO Problem 1 (TUR)  
Let D be the midpoint of the shorter arc BC of the circumcircle of an acuteangled triangle ABC. The points symmetric to D with respect to BC and the circumcenter are denoted by E and F, respectively. Let K be the midpoint of EA.
(a) Prove that the circle passing through the midpoints of the sides of ABC also passes through K.
(b) The line through K and the midpoint of BC is perpendicular to AF.

1999 BMO Problem 3  (ALB)  
Let M,N,P be the orthogonal projections of the centroid G of an acute-angled triangle ABC onto AB,BC,CA, respectively. Prove that $$\frac{4}{27}<\frac{{{S}_{MNP}}}{{{S}_{ABC}}}\le \frac{1}{4}$$

2000 BMO Problem 2  (FYROM) 
Let ABC be a scalene triangle and E be a point on the median AD. Point F is the orthogonal projection of E onto BC. Let M be a point on the segment EF, and N,P be the orthogonal projections of M onto AC and AB respectively. Prove that the bisectors of the angles PMN and PEN are parallel.

2001 BMO Problem 2 (MOL) 
Prove that a convex pentagon that satisfies the following two conditions must be regular:
(i) All its interior angles are equal.
(ii) The lengths of all its sides are rational numbers.

2002 BMO Problem 3  (ROM)
Two circles with different radii intersect at A and B. Their common tangents MN and ST touch the first circle at M and S and the second circle at N and T. Show  that the orthocenters of triangles AMN, AST, BMN, and BST are the vertices of a rectangle.

2003 BMO Problem 2 (ROM)   
Let ABC be a triangle with AB AC. The tangent at A to the circumcircle of the triangle ABC meets the line BC at D. Let E and F be the points on the perpendicular bisectors of the segments AB and AC respectively, such that BE and CF are both perpendicular to BC. Prove that the points D,E, and F are collinear.
by Valentin Vornicu
2004 BMO Problem 3 (ROM)   
Let O be an interior point of an acute-angled triangle ABC. The circles centered at the midpoints of the sides of the triangle ABC and passing through point O, meet in points K,L,M different from O. Prove that O is the incenter of the triangle KLM if and only if O is the circumcenter of the triangle ABC.

2005 BMO Problem 1 (BUL)    
The incircle of an acute-angled triangle ABC touches AB at D and AC at E. Let the bisectors of the angles <ACB and <ABC intersect the line DE at X and Y respectively, and let Z be the midpoint of BC. Prove that the triangle XYZ is equilateral if and only if <A = 60o.

2006 BMO Problem 2 (GRE)    
A line m intersects the sides AB, AC and the extension of BC beyond C of the triangle ABC at points D,F,E, respectively. The lines through points A,B,C which are parallel to m meet the circumcircle of triangle ABC again at points A1,B1,C1, respectively. Show that the lines A1E, B1F, C1D are concurrent.
by Dimitris Kontogiannis
2007 BMO Problem 1 (ALB)      
In a convex quadrilateral ABCD with AB = BC =CD, the diagonals AC and BD are of different length and intersect at point E. Prove that AE = DE if and only if <BAD + <ADC = 120 o.

2008 BMO Problem 1  (CYP)    
An acute-angled scalene triangle ABC with AB > BC is given. Let O be its circumcenter, H its orthocenter, and F the foot of the altitude from C. Let P be the point (other than A) on the line AB such that AF = PF, and M be a point on AC. We denote the intersection of PH and BC by X, the intersection of OM and FX by Y, and the intersection of OF and AC by Z. Prove that the points F, M, Y, and Z are concyclic.
by Theoklitos Paragyiou
2009 BMO Problem 1 (MOL)  
Let MN be a line parallel to the side BC of triangle ABC, with M on the side AB and N on the side AC. The lines BN and CM meet at point P. The circumcircles of triangles BMP and CNP meet at two distinct points P and Q. Prove that <BAQ = <CAP.
by Liubomir Chiriac
2010 BMO Problem 2 (KSA) 
Let ABC be an acute triangle with orthocentre H, and let M be the midpoint of AC. The point C1 on AB is such that CC1 is an altitude of the triangle ABC. Let H1 be the reection of H in AB. The orthogonal projections of C1 onto the lines AH1, AC and BC are P, Q and R, respectively. Let M1 be the point such that the circumcentre of triangle PQR is the midpoint of the segment MM1. Prove that M1 lies on the segment BH1.

2011 BMO Problem 1 (GBR)  
Let ABCD be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at E. The midpoints of AB and CD are F and G respectively, and l is the line through G parallel to AB. The feet of the perpendiculars from E onto the lines l and CD are H and K, respectively. Prove that the lines EF and HK are perpendicular.

2011 BMO Problem 4 (BUL)  
Let ABCDEF be a convex hexagon of area 1, whose opposite sides are parallel. The lines AB, CD and EF meet in pairs to determine the vertices of a triangle. Similarly, the lines BC, DE and FA meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least 3 / 2.

2012 BMO Problem 1  (ROM)
Let A, B and C be points lying on a circle Γ with centre  O. Assume that <ABC > 90. Let D be the point of intersection of the line AB with the line perpendicular to AC at C. Let l be the line through D which is perpendicular to AO. Let E be the point of intersection of l with the line AC, and let F be the point of intersection of Γ with l that lies between D and E. Prove that the circumcircles of triangles BFE and CFD are tangent at F.

2013 BMO Problem 1 (BUL)  
In a triangle ABC, the excircle ωa opposite A touches AB at P and AC at Q, and the excircle ωb opposite B touches BA at M and BC at N. Let K be the projection of C onto MN, and let L be the projection of C onto PQ. Show that the quadrilateral MKLP is cyclic.

2014 BMO Problem 3 (GRE)
Let ABCD be a trapezium inscribed in a circle Γ with diameter AB. Let E be the intersection point of the diagonals AC and BD . The circle with center B and radius BE meets Γ at the points K and L (where K is on the same side of AB as C). The line perpendicular to BD at E intersects CD at M. Prove that KM is perpendicular to DL.
by Silouanos Brazitikos  
2015 BMO Problem 2 (CYP)  
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

2016 BMO Problem 2 (GRE)  
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

2017 BMO Problem 2 (GRE)
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

Junior Balkan Mathematical Olympiad Shortlist
(JBMO Shortlist)
2009  - 2016

2009 JBMO Shortlist G1
Parallelogram ${ABCD}$ is given with ${AC>BD}$, and ${O}$ point of intersection of ${AC}$ and ${BD}$. Circle with center at ${O}$and radius ${OA}$ intersects extensions of ${AD}$and ${AB}$at points ${G}$ and ${L}$, respectively. Let ${Z}$ be intersection point of lines ${BD}$and ${GL}$. Prove that $\angle ZCA={{90}^{{}^\circ }}$.



2009 JBMO Shortlist G2
In right trapezoid ${ABCD \left(AB\parallel CD\right)}$ the angle at vertex B measures ${{75}^{{}^\circ }}$. Point ${H}$is the foot of the perpendicular from point ${A}$ to the line ${BC}$. If ${BH=DC}$ and${AD+AH=8}$, find the area of ${ABCD}$.




2009 JBMO Shortlist G3
Parallelogram ${ABCD}$with obtuse angle $\angle ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$, we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear. Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects the straight line ${BD}$at point ${P}$. Prove that ${PC}$is the bisector of the angle $\angle BP{D}'$.



2009 JBMO Shortlist G4 problem 1
Let ${ABCDE}$be convex pentagon such that ${AB+CD=BC+DE}$ and ${k}$ half circle with center on side ${AE}$ that touches sides ${AB, BC, CD}$ and ${DE}$of pentagon, respectively, at points ${P, Q, R}$ and ${S}$ (different from vertices of pentagon). Prove that $PS\parallel AE$.



2009 JBMO Shortlist G5
Let ${A, B, C}$ and ${O}$ be four points in plane, such that $\angle ABC>{{90}^{{}^\circ }}$ and ${OA=OB=OC}$.Define the point ${D\in AB}$ and the line ${l}$ such that ${D\in l, AC\perp DC}$ and ${l\perp AO}$. Line ${l}$ cuts ${AC}$at ${E}$ and circumcircle of ${ABC}$ at ${F}$. Prove that the circumcircles of triangles ${BEF}$and ${CFD}$are tangent at ${F}$.



2010 JBMO Shortlist G1
Consider a triangle ${ABC}$ with${\angle ACB=90^{\circ}.}$ Let ${F}$ be the foot of the altitude from ${C}$. Circle ${\omega}$ touches the line segment ${FB}$at point ${P,}$  the altitude ${CF}$at point ${Q}$ and the circumcircle of ${ABC}$at point ${R.}$ Prove that points ${A,Q,R}$ are collinear and ${AP=AC.}$



2010 JBMO Shortlist G2
Consider a triangle ${ABC}$ and let ${M}$ be the midpoint of the side ${BC.}$ Suppose ${\angle MAC=\angle ABC}$ and ${\angle BAM=105^{\circ}.}$ Find the measure of ${\angle ABC.}$



2010 JBMO Shortlist G3 
Let ${ABC}$be an acute-angled triangle. A circle ${\omega_1(O_1,R_1)}$ passes through points ${B}$ and ${C}$ and meets the sides ${AB}$and ${AC}$at points ${D}$ and ${E,}$ respectively. Let ${\omega_2(O_2,R_2)}$be the circumcircle of the triangle ${ADE.}$. Prove that ${O_1O_2}$is equal to the circumradius of the triangle ${ABC.}$



2010 JBMO Shortlist G4 problem 3
Let ${AL}$and ${BK}$be angle bisectors in the non-isosceles triangle ${ABC}$ ($L\in BC,$$K\in AC$). The perpendicular bisector of ${BK}$ intersects the line ${AL}$ at point ${M}$. Point ${N}$ lies on the line ${BK}$such that $LN\parallel MK$. Prove that ${LN=NA}$.



2011 JBMO Shortlist G1
Let $ABC$ be an isosceles triangle with $AB=AC$. On the extension of the side ${CA}$ we consider the point ${D}$ such that ${AD<AC}$. The perpendicular bisector of the segment ${BD}$ meets the internal and the external bisectors of the angle $\angle BAC$ at the points ${E}$and ${Z}$, respectively. Prove that the points ${A, E, D, Z}$ are concyclic.

2011 JBMO Shortlist G2
Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.



2011 JBMO Shortlist G3
Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$, ${AH}$ is an altitude of$\vartriangle ABC$ $\left( H\in BC \right)$ and ${M}$is the midpoint of the side ${AB}$. It is known  that the midpoints of the segments ${BL}$ and ${MH}$ coincides. Determine the internal angles of triangle $\vartriangle ABC$.



2011 JBMO Shortlist G4
Point ${D}$ lies on the side ${BC}$ of $\vartriangle ABC$. The circumcenters of $\vartriangle ADC$ and $\vartriangle BAD$ are ${O_1}$ and ${O_2}$, respectively and ${O_1O_2\parallel AB}$. The orthocenter of $\vartriangle ADC$is ${H}$ and  ${AH=O_1O_2}.$ Find the angles of $\vartriangle ABC$ if $2m\left( \angle C \right)=3m\left( \angle B \right).$



2011 JBMO Shortlist G5
Inside the square ${ABCD}$, the equilateral triangle △$\vartriangle ABE$is constructed. Let ${M}$ be an interior point of the triangle $\vartriangle ABE$ such that ${MB=\sqrt{2}, MC=\sqrt{6}, MD=\sqrt{5}}$ and ${ME=\sqrt{3}}$. Find the area of the square ${ABCD}$.



2011 JBMO Shortlist G6 problem 4
Let ${ABCD}$ be a convex quadrilateral, $E$ and $F$ points on the sides $AB$ and ${CD}$,  respectively, such that ${AB:AE=CD:DF=n}$. Denote by ${S}$ the area of the quadrilateral${AEFD}$. Prove that $${S\leq\frac{AB\cdot CD+n(n-1)AD^2+nDA\cdot BC}{2n^2}}$$


2012 JBMO Shortlist G1
Let $ABC$ be an equlateral triangle and  ${P}$ a point on the circumcircle of the triangle $ABC$, and distinct from ${A, B}$ and ${C}$. If the lines through ${P}$ and parallel to $BC,CA,AB$ intersect the lines $CA,AB,BC$ at $M,N,Q$ respectively, prove that ${M, N}$ and ${Q}$ are collinear.



2012 JBMO Shortlist G2
Let $ABC$ be an isosceles triangle with $AB=AC$. Let also ${c\left(K, KC\right)}$ be a circle tangent to the line ${AC}$ at point${C}$ which it intersects the segment ${BC}$ again at an interior point ${H}$. Prove that ${HK\perp AB}$.



2012 JBMO Shortlist G3
Let $AB$ and $CD$ be chords in a circle of center ${O}$ with $A,B,C,D$ distinct, and let the  lines $AB$ and $CD$ meet at a right angle at point ${E}$. Let also $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. If ${MN\perp OE}$, prove that ${AD\parallel BC}$.



2012 JBMO Shortlist G4
Let  $ABC$ be an acute-angled triangle with circumcircle $\Gamma $,  and let ${O, H}$ be the triangle’s circumcenter and orthocenter respectively. Let also ${A'}$ be the point where the angle bisector of angle ${\angle BAC}$ meets $\Gamma $.  If ${A'H=AH}$, find the measure of the angle${\angle BAC}$.



2012 JBMO Shortlist G5 problem 2 
Let the circles ${{k}_{1}}$ and ${{k}_{2}}$ intersect at two distinct points ${A}$ and ${B}$ , and let $t$t be a common tangent of ${{k}_{1}}$ and ${{k}_{2}}$, that touches ${{k}_{1}}$ and ${{k}_{2}}$ at ${M}$ and ${N}$, respectively. If  $t\bot AM$ and ${MN=2AM}$, evaluate  ${\angle{NMB}}$.



2012 JBMO Shortlist G6
Let ${O_1}$ be a point in the exterior of the circle ${c\left(O, R\right)}$ and let ${O_1N, O_1D}$ be the tangent segments from ${O_1}$ to the circle. On the segment ${O_1N}$ consider the point ${B}$ such that ${BN=R}$. Let the line from ${B}$ parallel to ${ON}$, intersect the segment ${O_1D}$ at ${C}$. If ${A}$ is a point on the segment ${O_1D}$, other than ${C}$ so that ${BC=BA=a}$, and if  ${c'\left(K, r\right)}$ is the incircle of the triangle ${{O}_{1}}AB$  find the area of $ABC$ in terms of $a,R,r$.



2012 JBMO Shortlist G7   (ROM)
Let ${MNPQ}$ be a square of side length 1, and ${A, B, C, D}$ points on the sides $MN,NP,PQ$ and $QM$  respectively such that ${AC\cdot BD=\dfrac{5}{4}}$.  Can the set ${\left\{AB, BC, CD, DA\right\}}$ be partitioned into two subsets ${{S}_{1}}$and ${{S}_{2}}$of two elements each such that both the sum of the elements of ${{S}_{1}}$and the sum of the elements of ${{S}_{2}}$ are positive integers?


by Flavian Georgescu



2013 JBMO Shortlist G1 (ALB)   
Let ${AB}$ be a diameter of a circle  ${\omega}$ and center ${O}$ ,  ${OC}$ a radius of ${\omega}$ perpendicular to $AB$,${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points ${M,O,P,N}$ are cocyclic.

2013 JBMO Shortlist G2 (CYP)   
Circles ${\omega_1}$ , ${\omega_2}$ are externally tangent at point M and tangent internally with circle ${\omega_3}$ at points ${K}$ and $L$ respectively. Let ${A}$ and  ${B}$be the points that their common tangent at point ${M}$ of circles ${\omega_1}$ and ${\omega_2}$ intersect with circle ${\omega_3.}$ Prove that if ${\angle KAB=\angle LAB}$ then the segment ${AB}$ is diameter of circle ${\omega_3.}$
by Theoklitos Paragyiou
2013 JBMO Shortlist G3 problem 2 (FYROM)
Let ${ABC}$ be an acute triangle with ${AB<AC}$ and ${O}$ be the center of its circumcircle $\omega $. Let ${D}$ be a point on the line segment ${BC}$ such that $\angle BAD=\angle CAO$. Let ${E}$ be the second point of intersection of ${\omega}$ and the line$AD$. If ${M, N}$ and ${P}$ are the midpoints of the line segments ${BE, OD}$ and$\left[ AC \right]$ respectively, show that the points ${M, N}$ and ${P}$ are collinear.

by Stefan Lozanovski
2013 JBMO Shortlist G4 (BIH)
Let ${I}$ be the incenter and ${AB}$the shortest side of a triangle${ABC.}$ The circle with center ${I}$and passing through ${C}$ intersects the ray ${AB}$at the point ${P}$and the ray ${BA}$at the point$Q$. Let ${D}$ be the point where the excircle of the triangle ${ABC}$ belonging to angle ${A}$touches the side${BC}$, and let ${E}$be the symmetric of the point ${C}$with respect to $D$. Show that the lines ${PE}$ and ${CQ}$ are perpendicular.

2013 JBMO Shortlist G5 (BUL)
A circle passing through the midpoint ${M}$of side ${BC}$ and the vertex ${A}$ of a triangle ${ABC}$, intersects sides ${AB}$ and ${AC}$ for the second time at points ${P}$ and ${Q,}$ respectively. Prove that if $\angle BAC={{60}^{{}^\circ }}$ then ${AP+AQ+PQ<AB+AC+\frac{1}{2}BC.}$

2013 JBMO Shortlist G6 (CYP) 
Let ${P}$and ${Q}$be the midpoints of the sides ${BC}$and ${CD}$, respectively in a rectangle $ABCD$. Let ${K}$ and ${M}$be the intersections of the line ${PD}$with the lines ${QB}$and $QA$respectively, and let ${N}$ be the intersection of the lines ${PA}$and ${QB.}$. Let $X,Y$and  ${X,Y,Z}$be the midpoints of the segments , $AN,KN$and $AM$ respectively. Let ${l_1}$be the line passing through ${X}$and perpendicular to $MK,\,\,{{l}_{2}}$ be the line passing through ${Y}$and perpendicular to ${AM}$ and  ${l_3}$ the line passing through ${Z}$and perpendicular to ${KN.}$. Prove that the lines ${{l}_{1}},{{l}_{2}}$and  ${l_1,l_2,l_3}$ are concurrent.
by Theoklitos Paragyiou


2014 JBMO Shortlist G1   
Let ${ABC}$ be a triangle with $m\left( \angle B \right)=m\left( \angle C \right)={{40}^{{}^\circ }}$ Line bisector of ${\angle{B}}$ intersects ${AC}$ at point ${D}$. Prove that $BD+DA=BC$.

2014 JBMO Shortlist G2   (GRE)
Acute-angled triangle ${ABC}$ with ${AB<AC<BC}$ and let be ${c(O,R)}$ it’s circumicircle. Diametes ${BD}$ and ${CE}$ are drawn. Circle ${c_1(A,AE)}$ interescts ${AC}$ at ${K}$ Circle ${{c}_{2}(A,AD)}$ intersects ${BA}$ at ${L}$ (${A}$ lies between ${B}$ και ${L}$).Prove that lines ${EK}$ and ${DL}$ intersect at circle .

by Evangelos Psychas
2014 JBMO Shortlist G3 problem 2   
Consider an acute triangle ${ABC}$ with area ${S}$. Let ${CD\perp AB \; (D\in AB), DM\perp AC \;  (M\in AC)}$ and ${DN \perp BC \;  (N\in BC)}$. Denote by ${H_1}$ and ${H_2}$ the orthocenters of the triangles ${MNC}$ and ${MND}$ respectively. Find the area of the quadrilateral ${AH_1BH_2}$ in terms of ${S}$.

2014 JBMO Shortlist G4 
Let ${ABC}$ be a triangle such that ${AB\ne AC}$. Let ${M}$ be the midpoint of ${BC,H}$ be the orthocenter of triangle $ABC$,${O_1}$ be the midpoint of ${AH}$, ${O_2}$ the circumcentre of triangle $BCH$. Prove that ${O_1AMO_2}$ is a parallelogram.
   
2014 JBMO Shortlist G5
Let $ABC$ be a triangle with ${AB\ne BC}$; and let ${BD}$ be the internal bisector of $\angle ABC,\ $, $\left( D\in AC \right)$. Denote by ${M}$ the midpoint of the arc ${AC}$ which contains point ${B}$. The circumscribed circle of the triangle ${\vartriangle BDM}$ intersects the segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of ${A}$ with respect to ${K}$.  If ${DJ\cap AM=\left\{O\right\}}$, prove that the points ${J, B, M, O}$ belong to the same circle.

2014 JBMO Shortlist G6 (ROM)   
Let ${ABCD}$ be a quadrilateral whose diagonals are not perpendicular and whose sides ${AB\nparallel CD}$ and ${AB\nparallel CD}$ are not parallel. Let ${O}$ be the intersection of its diagonals. Denote with ${H_1}$and ${H_2}$ the orthocenters of triangles $\vartriangle OAB$ and $\vartriangle OCD$, respectively. If ${M}$ and ${N}$ are the midpoints of the segment lines $\left[ AB \right]$ and $\left[ CD \right]$, respectively, prove that the lines ${H_1H_2}$ and ${MN}$are parallel if and only if $AC=BD$.

by Flavian Georgescu

2015 JBMO Shortlist G1 (MNE)
Around the triangle $ABC$ the circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to this tangent intersectsthe lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$, respectively. Prove that the points $A,B,D,E$ belong to the same circle.

2015 JBMO Shortlist G2 (MOL)   
The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle  ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel. 
  
2015 JBMO Shortlist G3 (GRE)  
Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$.

by Evangelos Psychas
2015 JBMO Shortlist G4 problem 3 (CYP)  
Let $\vartriangle ABC$ be an acute triangle. Lines ${l_1, l_2}$ are perpendicular to ${AB}$ at the points ${A}$ and ${B}$, respectively. The perpendicular lines from the midpoints ${M}$ of ${AB}$ to the sides of the triangle ${AC}$ , ${BC}$ intersect ${l_1}$ , ${l_2}$ at the points ${E}$ , ${F}$, respectively. If ${D}$ είναι το σημείο τομής των ${EF}$ και ${MC}$, να αποδείξετε ότι $\angle ADB=\angle EMF$.

by Theoklitos Paragyiou
2015 JBMO Shortlist G5 (ROM)   
Let $ABC$ be an acute triangle with ${AB\neq AC}$. The incircle ${\omega}$ of the triangle  κύκλος  touches the sides ${BC, CA}$ and ${AB}$ at ${D, E}$ and ${F}$, respectively. The perpendicular line erected at ${C}$onto ${BC}$ meets ${EF}$at ${M}$, and similarly the perpendicular line erected at ${B}$onto ${BC}$ meets ${EF}$at${N}$. The line ${DM}$ meets ${\omega}$ again in ${P}$, and the line ${DN}$ meets ${\omega}$ again at ${Q}$. Prove that ${DP=DQ}$. 

by Ruben Dario and Leo Giugiuc


2016 JBMO Shortlist G1 (GRE)   
Let ${ABC}$ be an acute angled triangle, let ${O}$be its circumcentre, and let ${D,E,F}$ be points on the sides ${BC,CA,AB}$, respectively. The circle ${(c_1)}$ of radius ${FA}$, centred at ${F}$, crosses the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centred at $D$, crosses the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$. Finally, the circle ${(c_3)}$ of radius $EC$, centred at $E$, crosses the segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$. Prove that the quadrilaterals $BKF{A}',CLD{B}'$ and $AME{C}'$ are all cyclic, and their circumcircles share a common point.

by Evangelos Psychas
2016 JBMO Shortlist G2
Let${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.

2016 JBMO Shortlist G3 problem 1 (BUL) 
A trapezoid ${ABCD}$ (${AB\parallel CD}$,${AB>CD}$) is circumscribed. The incircle of triangle ${ABC}$ touches the lines ${AB}$and ${AC}$ at ${M}$ and ${N}$, respectively. Prove that the incenter of the trapezoid lies on the line ${MN}$.

2016 JBMO Shortlist G4   
Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.

2016 JBMO Shortlist G5  
Let $ABC$ be an acute angled triangle with orthocenter ${H}$ and circumcenter ${O}$. Assume the circumcenter ${X}$ of ${BHC}$lies on the circumcircle of ${ABC}$. Reflect O across ${X}$ to obtain ${O'}$, and let the lines ${XH}$and ${O'A}$ meet at ${K}$. Let $L,M$ and $N$ be the midpoints of $\left[ XB \right],\left[ XC \right]$] and $\left[ BC \right]$, respectively. Prove that the points $K,L,M$ and ${K,L,M,N}$ are cocyclic.

2016 JBMO Shortlist G6 (BUL)   

Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that ${\angle ADB= \angle AEC=90^o}$ and ${\angle BAD= \angle CAE}$. Let ${{A}_{1}}\in BC,{{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and ${K,L}$ be the midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles $AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear.


2016 JBMO Shortlist G7 (CYP)   
Let ${AB}$ be a chord of a circle ${(c)}$ centered at ${O}$, and let ${K}$ be a point on the segment ${AB}$  such that ${AK<BK}$. Two circles through ${K}$, internally tangent to ${(c)}$ at ${A}$ and ${B}$, respectively, meet again at ${L}$. Let ${P}$ be one of the points of intersection of the line ${KL}$and the circle ${(c)}$,  and let the lines ${AB}$and ${LO}$meet at ${M}$. Prove that the line ${MP}$is tangent to the circle ${(c)}$.
by Theoklitos Paragyiou
2017 JBMO Problem 3  
Let ${ABC}$ be an acute triangle such that ${AB \neq AC}$, with circumcircle ${\Gamma}$ and circumcenter ${O}$. Let ${M}$ be the midpoint of ${BC}$and ${D}$ be a point on ${\Gamma}$ such that ${AD \perp BC}$. Let ${T}$ be a point such that ${BDCT}$is a parallelogram and ${Q}$ a point on the same side of ${BC}$as ${A}$ such that ${ \angle BQM = \angle BCA}$ and ${ \angle CQM = \angle CBA}$. Let the line ${AO}$ intersect ${\Gamma}$ at ${E}$, (${E \neq A}$) and let the circumcircle of ${ETQ}$ intersect ${\Gamma}$ at point ${X \neq E}$. Prove that the points ${A,M}$ and ${X}$ are collinear.
 

To be continued ...

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