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Geometry Problems from Middle European Mathematical Olympiads

2007-2017

[with aops links]

MEMO 2007 Individual 3
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, 4 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 3
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 3
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 3
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 5
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 6
Suppose that ABCD is a cyclic quadrilateral and CD = DA. Points E and F belong to the
segments AB and BC respectively, and ADC = 2 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)

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Geometry Problems from Middle European Mathematical Olympiads

MEMO 2010 Individual 3
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 5
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 6
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 3
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 5
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 6
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)

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Geometry Problems from Middle European Mathematical Olympiads

MEMO 2012 Individual 3
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 5
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 6
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 3
Let ABC be an isosceles triangle with AC = BC. Let N be a point inside the triangle such that
2ANB = 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 5
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 6
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)

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Geometry Problems from Middle European Mathematical Olympiads

MEMO 2014 Individual 3
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 5
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 6
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 3
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 5
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 6
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)

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Geometry Problems from Middle European Mathematical Olympiads

MEMO 2016 Individual 3
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 5
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 6
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 3
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 5
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 6
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:

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