Geometry Problems from Balkan MOs
1984 – 2017
[with aops links]
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 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
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Geometry Problems from Balkan MOs
1987 BMO Problem 4 (BUL) Circles Κ1 (O1 ,1) and Κ2 (O2 , 2 ) with O1O2 = 2 intersect at A and B. Find the length of the chord AC of circle Κ2 whose midpoint lies on Κ1. 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 1 3 △ABC are equal to and 1 , respectively, determine the angles of △ABC. 4 2 1988 BMO Problem 4 (GRE) Show that every tetrahedron A1A2A3A4 can be placed between two parallel planes which are at 1 p the distance at most , where P 122 132 124 2 32 2 42 342 2 3 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 4 that S BB1GC1 SCC1GB1 S ABC 9 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
P ≤ 3/2 H.
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 2
4S DEF EF vertices). If the quadrilateral AFDE is cyclic, prove that . S ABC AD Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:
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Geometry Problems from Balkan MOs
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 r1, intersect at A and B so that
Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:
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Geometry Problems from Balkan MOs
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 4 S MNP 1 AB,BC,CA, respectively. Prove that 27 S ABC 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.
Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:
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Geometry Problems from Balkan MOs
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 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
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Geometry Problems from Balkan MOs
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 90o. 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
Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:
http://imogeometry.blogspot.gr/ https://web.facebook.com/groups/parmenides52/
Geometry Problems from Balkan MOs
2017 BMO Problem 2 (GRE) Consider an acute-angled triangle ABC with AB
Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group:
http://imogeometry.blogspot.gr/ https://web.facebook.com/groups/parmenides52/
