Middle European 2007-18 (MEMO) 33p

geometry problems from Middle European Mathematical Olympiad (MEMO)
with aops links in the names

To be more exact, the Slovak pdf collects all the problems and all solutions that are not contained in the English pdf.

2007 - 2018

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)
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)
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 ÐADB = ÐCDE, ÐBAD = ÐECD and ÐACB = ÐEBA. Prove that B, C and E are collinear.

(Slovenia)
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 ÐAGH = ÐDHG.

(Hungary)
Let ABCD be a parallelogram with ÐBAD = 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)
Suppose that ABCD is a cyclic quadrilateral and CD = DA. Points E and F belong to the segments AB and BC respectively, and ÐADC = 2 ÐEDF. 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)
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)
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)
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 ÐAFS = ÐECD.

(Croatia)
In a plane the circles K1 and K2 with centers I1 and I2, respectively, intersect in two points A and B. Assume that ÐI1AI2 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)
Let ABCDE be a convex pentagon with all five sides equal in length. The diagonals AD and EC meet in S with ÐASE = 60o. Prove that ABCDE has a pair of parallel sides.

(Michal Szabados, Slovakia)
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)
In a given trapezium ABCD with AB parallel to CD and AB > CD, the line BD bisects the angle ÐADC. 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 ÐACO = 60ο. Prove the equality CF = AF + FO.

(Croatia)
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 ÐCLK = ÐKMC. Prove that the perpendiculars to the sides AB, AC, and BC passing through K, L, and M, respectively, are concurrent.

(Poland)
Let ABCD be a convex quadrilateral with no pair of parallel sides, such that ÐABC =ÐCDA. 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)
Let ABC be an isosceles triangle with AC = BC. Let N be a point inside the triangle such that 2ÐANB = 180ο + ÐACB. 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)
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)
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)
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 ÐIBG = ÐCBA 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 ÐBGI are concurrent.

(Croatia)
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)
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)
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)
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 ÐAXB  - ÐACB = ÐCY A  - Ð CBA holds, the line XY passes through D.

(Patrik Bak, Slovakia)
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)
Let ABC be an acute-angled triangle with ÐBAC > 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 ÐQCB=ÐPCO.

(Patrik Bak, Slovakia)
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)
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 ÐPIA = 90 ο.

(Poland)
Let ABCDE be a convex pentagon. Let P be the intersection of the lines CE and BD. Assume that ÐPAD =ÐACB and ÐCAP = ÐEDA. Prove that the circumcentres of the triangles ABC and ADE are collinear with P.

(Patrik Bak, Slovakia)
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)

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)

MEMO 2018 Individual 3
Let $ABC$ be an acute-angled  triangle with $AB<AC,$  and let $D$ be the foot of its altitude from$A.$ Let $R$ and $Q$ be the centroids of  triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and points $P$ $Q$ $R$ and $D$ are concyclic .Prove that the lines $AP$ $BQ$ and $CR$  are concurrent



Let $ABC$ be an acute-angled  triangle with $AB<AC,$  and let $D$ be the foot of its altitude from$A,$ points $B'$ and $C'$ lie on the rays $AB$ and $AC,$ respectively , so that points $B',$ $C'$ and $D$ are collinear and points $B,$ $C,$ $B'$ and $C'$ lie on one circle with center $O.$ Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of  $ABC,$  then $DHMO$ is a parallelogram.


Let $ABC$ be a triangle . The internal bisector of $ABC$ intersects the side $AC$ at $ L$ and the circumcircle of $ABC$ again at $W \neq B.$ Let $K$ be the perpendicular projection of $L$ onto $AW.$ the circumcircle of $BLC$ intersects  line $CK$ again at $P \neq C.$ Lines $BP$ and $AW$ meet at point $T.$ Prove that $$AW=WT.$$


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