European Mathematical Cup 2012-22 (EMC) 20p

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

EMC Geometry Problems 2012-16 EN in pdf with aops links

EMC all 2012-21 EN in pdf with solutions

collected inside aops here

2012 - 2022   

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)

EMC Junior 2017 P3

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)

EMC Junior 2018 P3

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 $\overline{AH}$. 

EMC Junior 2019 P3

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)

EMC Junior 2020 P1

Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be the midpoints of sides $\overline{AB}$ and $\overline{AC}$ respectively. Let $F$ be the point such that $D$ is the midpoint of $\overline{EF}$. Let $\Gamma$ be the circumcircle of triangle $FDB$. Let $G$ be a point on the segment $\overline{CD}$ such that the midpoint of $\overline{BG}$ lies on $\Gamma$. Let $H$ be the second intersection of $\Gamma$ and $FC$. Show that the quadrilateral $BHGC$ is cyclic.

(Art Waeterschoot)

EMC Junior 2021 P2

Let $ABC$ be an acute-angled triangle such that $|AB|<|AC|$. Let $X$ and $Y$ be points on the minor arc ${BC}$ of the circumcircle of $ABC$ such that $|BX|=|XY|=|YC|$. Suppose that there exists a point $N$ on the segment $\overline{AY}$ such that $|AB|=|AN|=|NC|$. Prove that the line $NC$ passes through the midpoint of the segment $\overline{AX}$.

(Ivan Novak)

EMC Junior 2022 P3

Let $ABC$ be an acute-angled triangle with $AC > BC$, with incircle $\tau$ centered at $I$ which touches $BC$ and $AC$ at points $D$ and $E$, respectively. The point $M$ on $\tau$ is such that $BM \parallel DE$ and $M$ and $B$ lie on the same halfplane with respect to the angle bisector of $\angle ACB$. Let $F$ and $H$ be the intersections of $\tau$ with $BM$ and $CM$ different from $M$, respectively. Let $J$ be a point on the line $AC$ such that $JM \parallel EH$. Let $K$ be the intersection of $JF$ and $\tau$ different from $F$. Prove that $ME \parallel KH$.

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)

EMC Senior 2017 P3

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.

EMC Senior 2018 P2

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. 

EMC Senior 2019 P3

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 )

EMC Senior 2020 P1

Let $ABCD$ be a parallelogram such that $|AB| > |BC|$. Let $O$ be a point on the line $CD$ such

that $|OB| = |OD|$. Let $\omega$ be a circle with center $O$ and radius $|OC|$. If $T$ is the second

intersection of ω and $CD$, prove that $AT, BO$ and $\omega$ are concurrent.

(Ivan Novak)

EMC Senior 2021 P2

Let $ABC$ be a triangle and let $D, E$ and $F$ be the midpoints of sides $BC, CA$ and $AB$, respectively. Let $X\ne A$ be the intersection of $AD$ with the circumcircle of $ABC$. Let $\Omega$ be the circle through $D and X$, tangent to the circumcircle of $ABC$. Let $Y$ and $Z$ be the intersections of the tangent to $Ω\Omega$ at $D$ with the perpendicular bisectors of segments $DE$ and $DF$, respectively. Let $P$ be the intersection of $YE$ and $ZF$ and let $G$ be the centroid of $ABC$. Show that the tangents at $B$ and $C$ to the circumcircle of $ABC$ and the line $PG$ are concurrent.

EMC Senior 2022 P4

Five points $A$, $B$, $C$, $D$ and $E$ lie on a circle $\tau$ clockwise in that order such that $AB \parallel CE$ and $\angle ABC > 90^{\circ}$. Let $k$ be a circle tangent to $AD$, $CE$ and $\tau$ such that $k$ and $\tau$ touch on the arc $\widehat{DE}$ not containing $A$, $B$ and $C$. Let $F \neq A$ be the intersection of $\tau$ and the tangent line to $k$ passing through $A$ different from $AD$. Prove that there exists a circle tangent to $BD$, $BF$, $CE$ and $\tau$.

source: emc.mnm.hr