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Geometry Problems from Middle European Mathematical Olympiads Geometry Problems from IMOs blogspot page http //imogeometry blogspot gr/ Romantics of Geometry facebook group https //web facebook com/gr[.]
Geometry Problems from Middle European Mathematical Olympiads 2007-2017 [with aops links] MEMO 2007 Individual Let k be a circle and k1, k2, k3 and k4 four smaller circles with their centres O1, O2,O3 and O4 respectively on k For i = 1, 2, 3, and k5 = k1 the circles ki and ki+1 meet at Ai and Bi such that Ai lies on k The points O1, A1, O2, A2, O3, A3, O4, A4, lie in that order on k and are pairwise different Prove that B1B2B3B4 is a rectangle (Switzerland) MEMO 2008 Individual Let ABC be an isosceles triangle with AC = BC It’s incircle touches AB and BC at D and E, respectively A line (different from AE) passes through A and intersects the incircle at F and G The lines EF and EG intersect the line AB at K and L, respectively Prove that DK = DL (Ηungary) MEMO 2008 Team Given an acute-angled triangle ABC, let E be a point situated on the different side of the line AC than B, and let D be an interior point of the line segment AE Suppose that ADB = CDE, BAD = ECD and ACB = EBA Prove that B, C and E are collinear (Slovenia) MEMO 2009 Individual Let ABCD be a convex quadrilateral such that AB and CD are not parallel and AB = CD The midpoints of the diagonals AC and BD are E and F The line EF meets segments AB and CD at G and H, respectively Show that AGH = DHG (Hungary) MEMO 2009 Team Let ABCD be a parallelogram with BAD = 60o and denote by E the intersection of its diagonals The circumcircle of the triangle ACD meets the line BA at K ≠ A, the line BD at P ≠D and the line BC at L ≠ C The line EP intersects the circumcircle of the triangle CEL at points E and M Prove that the triangles KLM and CAP are congruent (Slovenia) MEMO 2009 Team Suppose that ABCD is a cyclic quadrilateral and CD = DA Points E and F belong to the segments AB and BC respectively, and ADC = EDF Segments DK and DM are height and median of the triangle DEF, respectively L is the point symmetric to K with respect to M Prove that the lines DM and BL are parallel (Poland) Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group: http://imogeometry.blogspot.gr/ https://web.facebook.com/groups/parmenides52/ Geometry Problems from Middle European Mathematical Olympiads MEMO 2010 Individual We are given a cyclic quadrilateral ABCD with a point E on the diagonal AC such that AD=AE and CB = CE Let M be the center of the circumcircle k of the triangle BDE The circle k intersects the line AC in the points E and F Prove that the lines FM, AD, and BC meet at one point (Switzerland) MEMO 2010 Team The incircle of the triangle ABC touches the sides BC, CA, and AB in the points D, E, and F, respectively Let K be the point symmetric to D with respect to the incenter The lines DE and FK intersect at S Prove that AS is parallel to BC (Poland) MEMO 2010 Team Let A, B, C, D, E be points such that ABCD is a cyclic quadrilateral and ABDE is a parallelogram The diagonals AC and BD intersect at S and the rays AB and DC intersect at F Prove that AFS = ECD (Croatia) MEMO 2011 Individual In a plane the circles K1 and K2 with centers I1 and I2, respectively, intersect in two points A and B Assume that I1AI2 is obtuse The tangent to K1 in A intersects K2 again in C and the tangent to K2 in A intersects K1 again in D Let K3 be the circumcircle of the triangle BCD Let E be the midpoint of that arc CD of K3 that contains B The lines AC and AD intersect K3 again in K and L, respectively Prove that the line AE is perpendicular to KL (Nik Stopar, Slovenia) MEMO 2011 Team Let ABCDE be a convex pentagon with all five sides equal in length The diagonals AD and EC meet in S with ASE = 60o Prove that ABCDE has a pair of parallel sides (Michal Szabados, Slovakia) MEMO 2011 Team Let ABC be an acute triangle Denote by B0 and C0 the feet of the altitudes from vertices B and C, respectively Let X be a point inside the triangle ABC such that the line BX is tangent to the circumcircle of the triangle AXC0 and the line CX is tangent to the circumcircle of the triangle AXB0 Show that the line AX is perpendicular to BC (Michal Rolinek, Josef Tkadlec, Czech Republic) Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group: http://imogeometry.blogspot.gr/ https://web.facebook.com/groups/parmenides52/ Geometry Problems from Middle European Mathematical Olympiads MEMO 2012 Individual In a given trapezium ABCD with AB parallel to CD and AB > CD, the line BD bisects the angle ADC The line through C parallel to AD meets the segments BD and AB in E and F, respectively Let O be the circumcentre of the triangle BEF Suppose that ACO = 60ο Prove the equality CF = AF + FO (Croatia) MEMO 2012 Team Let K be the midpoint of the side AB of a given triangle ABC Let L and M be points on the sides AC and BC, respectively, such that CLK = KMC Prove that the perpendiculars to the sides AB, AC, and BC passing through K, L, and M, respectively, are concurrent (Poland) MEMO 2012 Team Let ABCD be a convex quadrilateral with no pair of parallel sides, such that ABC =CDA Assume that the intersections of the pairs of neighbouring angle bisectors of ABCD form a convex quadrilateral EFGH Let K be the intersection of the diagonals of EFGH Prove that the lines AB and CD intersect on the circumcircle of the triangle BKD (Croatia) MEMO 2013 Individual Let ABC be an isosceles triangle with AC = BC Let N be a point inside the triangle such that 2ANB = 180ο + ACB Let D be the intersection of the line BN and the line parallel to AN that passes through C Let P be the intersection of the angle bisectors of the angles CAN and ABN Show that the lines DP and AN are perpendicular (Matija Basic, Croatia) MEMO 2013 Team Let ABC be an acute triangle Construct a triangle PQR such that AB = 2PQ, BC = 2QR, CA = 2RP, and the lines PQ, QR, and RP pass through the points A, B, and C, respectively (All six points A, B, C, P, Q, and R are distinct.) (Gerd Baron, Austria) MEMO 2013 Team Let K be a point inside an acute triangle ABC, such that BC is a common tangent of the circumcircles of AKB and AKC Let D be the intersection of the lines CK and AB, and let E be the intersection of the lines BK and AC Let F be the intersection of the line BC and the perpendicular bisector of the segment DE The circumcircle of ABC and the circle k with centre F and radius FD intersect at points P and Q Prove that the segment PQ is a diameter of k (Patrik Bak, Slovakia) Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group: http://imogeometry.blogspot.gr/ https://web.facebook.com/groups/parmenides52/ Geometry Problems from Middle European Mathematical Olympiads MEMO 2014 Individual Let ABC be a triangle with AB < AC and incentre I Let E be the point on the side AC such that AE = AB Let G be the point on the line EI such that IBG = CBA and such that E and G lie on opposite sides of I Prove that the line AI, the perpendicular to AE at E, and the bisector of the angle BGI are concurrent (Croatia) MEMO 2014 Team Let ABC be a triangle with AB < AC Its incircle with centre I touches the sides BC, CA, and AB in the points D, E, and F respectively The angle bisector AI intersects the lines DE and DF in the points X and Y respectively Let Z be the foot of the altitude through A with respect to BC Prove that D is the incentre of the triangle XY Z (Germany) MEMO 2014 Team Let the incircle k of the triangle ABC touch its side BC at D Let the line AD intersect k at ≠ D and denote the excentre of ABC opposite to A by K Let M and N be the midpoints of BC and KM respectively Prove that the points B, C, N, and L are concyclic (Patrik Bak, Slovakia) MEMO 2015 Individual Let ABCD be a cyclic quadrilateral Let E be the intersection of lines parallel to AC and BD passing through points B and A, respectively The lines EC and ED intersect the circumcircle of AEB again at F and G, respectively Prove that points C, D, F, and G lie on a circle (Patrik Bak, Slovakia) MEMO 2015 Team Let ABC be an acute triangle with AB ą AC Prove that there exists a point D with the following property: whenever two distinct points X and Y lie in the interior of ABC such that the points B, C, X, and Y lie on a circle and AXB - ACB = CY A - CBA holds, the line XY passes through D (Patrik Bak, Slovakia) MEMO 2015 Team Let I be the incentre of triangle ABC with AB ą AC and let the line AI intersect the side BC at D Suppose that point P lies on the segment BC and satisfies PI = PD Further, let J be the point obtained by reflecting I over the perpendicular bisector of BC, and let Q be the other intersection of the circumcircles of the triangles ABC and APD Prove that =BAQ = =CAJ (Patrik Bak, Slovakia) Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group: http://imogeometry.blogspot.gr/ https://web.facebook.com/groups/parmenides52/ Geometry Problems from Middle European Mathematical Olympiads MEMO 2016 Individual Let ABC be an acute-angled triangle with BAC > 45ο and with circumcentre O The point P lies in its interior such that the points A, P, O, B lie on a circle and BP is perpendicular to CP The point Q lies on the segment BP such that AQ is parallel to PO Prove that QCB=PCO (Patrik Bak, Slovakia) MEMO 2016 Team Let ABC be an acute-angled triangle with AB ≠ AC, and let O be its circumcentre The line AO intersects the circumcircle ω of ABC a second time in point D, and the line BC in point E The circumcircle of CDE intersects the line CA a second time in point P The line PE intersects the line AB in point Q The line through O parallel to PE intersects the altitude of the triangle ABC that passes through A in point F Prove that FP = FQ (Croatia) MEMO 2016 Team Let ABC be a triangle with AB ≠AC The points K, L, M are the midpoints of the sides BC, CA, AB, respectively The inscribed circle of ABC with centre I touches the side BC at point D The line g, which passes through the midpoint of segment ID and is perpendicular to IK, intersects the line LM at point P Prove that PIA = 90 ο (Poland) MEMO 2017 Individual Let ABCDE be a convex pentagon Let P be the intersection of the lines CE and BD Assume that PAD =ACB and CAP = EDA Prove that the circumcentres of the triangles ABC and ADE are collinear with P (Patrik Bak, Slovakia) MEMO 2017 Team Let ABC be an acute-angled triangle with AB >AC and circumcircle Γ Let M be the midpoint of the shorter arc BC of Γ, and let D be the intersection of the rays AC and BM Let E ≠ C be the intersection of the internal bisector of the angle ACB and the circumcircle of the triangle BDC Let us assume that E is inside the triangle ABC and there is an intersection N of the line DE and the circle Γ such that E is the midpoint of the segment DN Show that N is the midpoint of the segment IBIC, where IB and IC are the excentres of ABC opposite to B and C, respectively (Croatia) MEMO 2017 Team Let ABC be an acute-angled triangle with AB ≠ AC, circumcentre O and circumcircle Γ Let the tangents to Γ through B and C meet each other at D, and let the line AO intersect BC at E Denote the midpoint of BC by M and let AM meet Γ again at N ≠ A Finally, let F ≠ A be a point on Γ such that A, M, E and F are concyclic Prove that FN bisects the segment MD (Patrik Bak, Slovakia) Geometry Problems from IMOs blogspot page: Romantics of Geometry facebook group: http://imogeometry.blogspot.gr/ https://web.facebook.com/groups/parmenides52/ ... lie in the interior of ABC such that the points B, C, X, and Y lie on a circle and AXB - ACB = CY A - CBA holds, the line XY passes through D (Patrik Bak, Slovakia) MEMO 2015 Team Let I... https://web.facebook.com/groups/parmenides52/ Geometry Problems from Middle European Mathematical Olympiads MEMO 20 16 Individual Let ABC be an acute-angled triangle with BAC > 45ο and with circumcentre O The point P lies in its... such that AQ is parallel to PO Prove that QCB=PCO (Patrik Bak, Slovakia) MEMO 20 16 Team Let ABC be an acute-angled triangle with AB ≠ AC, and let O be its circumcentre The line AO intersects