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### European Mathematical Cup 2012-19 (EMC) 16p

geometry problems from European Mathematical Cups (EMC)
with aops links in the names

### 2012 - 2019

Juniors
EMC Junior 2012 P1
Let ABC be a triangle and Q a point on the internal angle bisector of <BAC. Circle ω1 is circumscribed to triangle BAQ and intersects the segment AC in point P ≠ C. Circle ω2 is circumscribed to the triangle CQP. Radius of the circle ω1 is larger than the radius of ω2. Circle centered at Q with radius QA intersects the circle ω1 in points A and A1. Circle centered at Q with radius QC intersects ω1 in points C1 and C2. Prove <A1BC1 = <C2PA.

(Matija Bucic)
EMC Junior 2013 P2
Let P be a point inside a triangle ABC. A line through P parallel to AB meets BC and CA at points L and F, respectively. A line through P parallel to BC meets CA and BA at points M and D respectively, and a line through P parallel to CA meets AB and BC at points N and E respectively. Prove (PDBL) ∙ (PECM) ∙ (PFAN) = 8 (PFM) ∙ (PEL) ∙ (PDN), where (XY Z) and (XY ZW) denote the area of the triangle XY Z and the area of quadrilateral XY ZW.

(Steve Dinh)
EMC Junior 2014 P3
Let ABC be a triangle. The external and internal angle bisectors of <CAB intersect side BC at D and E, respectively. Let F be a point on the segment BC. The circumcircle of triangle ADF intersects AB and AC at I and J, respectively. Let N be the mid-point of IJ and H the foot of E on DN. Prove that E is the incenter of triangle AHF.

(Steve Dinh)
EMC Junior 2015 P4
Let ABC be an acute angled triangle. Let B΄, A΄ be points on the perpendicular bisectors of AC, BC respectively such that B΄A AB and A΄B AB. Let P be a point on the segment AB and O the circumcenter of the triangle ABC. Let D,E be points on BC,AC respectively such that DP BO and EP   AO. Let O΄ be the circumcenter of the triangle CDE. Prove that B΄,A΄ and O΄ are collinear.

(Steve Dinh)
EMC Junior 2016 P2
Two circles C1 and C2 intersect at points A and B. Let P,Q be points on circles C1,C2 respectively, such that |AP| = |AQ|. The segment PQ intersects circles C1 and C2 in points M,N respectively. Let C be the center of the arc BP of C1 which does not contain point A and let D be the center of arc BQ of C2 which does not contain point A. Let E be the intersection of CM and DN. Prove that AE is perpendicular to CD.

(Steve Dinh)
Let $ABC$ be an acute triangle. Denote by $H$ and $M$ the orthocenter of $ABC$ and the midpoint of side $BC,$ respectively. Let $Y$ be a point on $AC$ such that $YH$ is perpendicular to $MH$ and let $Q$ be a point on $BH$ such that $QA$ is perpendicular to $AM.$ Let $J$ be the second point of intersection of $MQ$ and the circle with diameter $MY.$ Prove that $HJ$ is perpendicular to $AM.$

(Steve Dinh)
Let $ABC$ be an acute triangle with $|AB | < |AC |$and orthocenter $H$. The circle with center A and radius$|AC |$ intersects the circumcircle of $\triangle ABC$ at point $D$ and the circle with center $A$ and radius$|AB |$ intersects the segment $\overline{AD}$ at point $K.$ The line through $K$ parallel to $CD$ intersects $BC$ at the point $L.$  If $M$ is the midpoint of $\overline{BC}$  and N is the foot of the perpendicular from $H$ to $AL,$ prove that the line $MN$ bisects the segment $\overline{AH}$.

Let $ABC$ be a triangle with circumcircle ω. Let $l_B$ and $l_C$ be two lines through the points $B$ and $C$, respectively, such that $l_B || l_C$. The second intersections of $l_B$ and $l_C$ with ω are $D$ and $E$, respectively. Assume that $D$ and $E$ are on the same side of $BC$ as $A$. Let $DA$ intersect $l_C$ at $F$ and let $EA$ intersect $l_B$ at $G$. If $O$, $O_1$ and $O_2$ are circumcenters of the triangles $ABC$, $ADG$ and $AEF$, respectively, and $P$ is the circumcenter of the triangle $OO_1O_2$, prove that $l_B || OP || l_C$.

(Stefan Lozanovski)

Seniors
EMC Senior 2012 P2
Let ABC be an acute triangle with orthocenter H. Segments AH and CH intersect segments BC and AB in points A1 and C1 respectively. The segments BH and A1C1 meet at point D. Let P be the midpoint of the segment BH. Let D΄ be the reflection of the point D in AC. Prove that quadrilateral APCD΄ is cyclic.

(Matko Ljulj)
EMC Senior  2013 P4
Given a triangle ABC let D,E, F be orthogonal projections from A,B,C to the opposite sides respectively. Let X, Y,Z denote midpoints of AD,BE,CF respectively. Prove that perpendiculars from D to Y Z, from E to XZ and from F to XY are concurrent.

(Matija Bucic)
EMC Senior 2014 P3
Let ABCD be a cyclic quadrilateral with the intersection of internal angle bisectors of <ABC and <ADC lying on the diagonal AC. Let M be the midpoint of AC. The line parallel to BC that passes through D intersects the line BM in E and the circumcircle of ABCD at F where F≠ D. Prove that BCEF is a parallelogram.

(Steve Dinh)
EMC Senior 2015 P3
Circles k1 and k2 intersect in points A and B, such that k1 passes through the center O of the circle k2. The line p intersects k1 in points K and O and k2 in points L and M, such that the point L is between K and O. The point P is orthogonal projection of the point L to the line AB. Prove that the line KP is parallel to the M-median of the triangle ABM.

(Matko Ljulj)
EMC Senior 2016 P4
Let C1,C2 be circles intersecting in X, Y . Let A,D be points on C1 and B,C on C2 such that A,X,C are collinear and D,X,B are collinear. The tangent to circle C1 at D intersects BC and the tangent to C2 at B in P,R respectively. The tangent to C2  at C intersects AD and tangent to C1 at A, in Q, S respectively. Let W be the intersection of AD with the tangent to C2 at B and Z the intersection of BC with the tangent to C1 at A. Prove that the circumcircles of triangles YWZ,RSY and PQY have two points in common, or are tangent in the same point.

(Misiakos Panagiotis)
Let $ABC$ be a scalene triangle and let its incircle touch sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$ respectively. Let line $AD$ intersect this incircle at point $X$. Point $M$ is chosen on the line $FX$ so that the quadrilateral $AFEM$ is cyclic. Let lines $AM$ and $DE$ intersect at point $L$ and let $Q$ be the midpoint of segment $AE$. Point $T$ is given on the line $LQ$ such that the quadrilateral $ALDT$ is cyclic. Let $S$ be a point such that the quadrilateral $TFSA$ is a parallelogram, and let $N$ be the second point of intersection of the circumcircle of triangle $ASX$ and the line $TS$. Prove that the circumcircles of triangles $TAN$ and $LSA$ are tangent to each other.

Let ABC be a triangle with$|AB|< |AC|.$ Let $k$ be the circumcircle of $\triangle ABC$ and let $O$ be the center of $k$. Point $M$ is the midpoint of the arc $BC$ of $k$ not containing $A$. Let $D$ be the second intersection of the perpendicular line from $M$ to $AB$ with $k$ and $E$ be the second intersection of the perpendicular line from $M$ to $AC$ with $k$. Points $X$and $Y$ are the intersections of $CD$ and $BE$ with $OM$ respectively. Denote by $k_b$ and $k_c$  circumcircles of triangles $BDX$ and $CEY$ respectively. Let $G$ and $H$ be the second intersections of $k_b$ and $k_c$ with $AB$ and $AC$ respectively. Denote by ka the circumcircle of triangle $AGH.$
Prove that $O$ is the circumcenter of $\triangle O_aO_bO_c,$where $O_a, O_b, O_c$ are the centers of $k_a, k_b, k_c$  respectively.

In an acute triangle $ABC$ with $|AB| \not= |AC|$, let $I$ be the incenter and $O$ the circumcenter. The incircle is tangent to $\overline{BC}, \overline{CA}$ and $\overline{AB}$ in $D,E$ and $F$ respectively. Prove that if the line parallel to $EF$ passing through $I$, the line parallel to $AO$ passing through $D$ and the altitude from $A$ are concurrent, then the point of concurrence is the orthocenter of the triangle $ABC$.

(Petar Nizié-Nikola )

source: emc.mnm.hr