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European Girls 2012-20 (EGMO) 17p

geometry problems from European Girls' Mathematical Olympiad  (EGMO)
with aops links in the names 

2012 - 2020

EGMO  2012 / 1
Let ABC be a triangle with circumcentre O. The points D, E and F lie in the interiors of the sides BC, CA and AB respectively, such that DE is perpendicular to CO and DF is perpendicular to BO. Let K be the circumcentre of triangle AFE. Prove that the lines DK and BC are perpendicular.

by Merlijn Staps, Netherlands
EGMO  2012 / 7
Let ABC be an acute-angled triangle with circumcircle Γ and orthocentre H. Let K be a point of Γ on the other side of BC from A. Let L be the reflection of K in the line AB, and let M be the reflection of K in the line BC. Let E be the second point of intersection of Γ with the circumcircle of triangle BLM. Show that the lines KH, EM and BC are concurrent.
by Pierre Haas, Luxembourg
EGMO  2013 / 1
The side BC of the triangle ABC is extended beyond C to D so that CD = BC. The side CA is extended beyond A to E so that AE = 2CA. Prove that, if AD = BE, then the triangle ABC is right-angled.
by David Monk, United Kingdom
EGMO  2013 / 5
Let Ω be the circumcircle of the triangle ABC. The circle ω is tangent to the sides AC and BC, and it is internally tangent to the circle Ω at the point P. A line parallel to AB and intersecting the interior of triangle ABC is tangent to ω at Q. Prove that <ACP = <QCB.
by Waldemar Pompe, Poland
EGMO  2014 / 2
Let D and E be points in the interiors of sides AB and AC, respectively, of a triangle ABC, such that DB = BC = CE. Let the lines CD and BE meet at F. Prove that the incentre I of triangle ABC, the orthocentre H of triangle DEF and the midpoint M of the arc BAC of the circumcircle of triangle ABC are collinear.
by Danylo Khilko, Ukraine
EGMO  2015 / 1
Let ∆ABC be an acute-angled triangle, and let D be the foot of the altitude from C. The angle bisector of <ABC intersects CD at E and meets the circumcircle ω of triangle ∆ADE again at F. If <ADF = 45°, show that CF is tangent to ω.
by Luxembourg
EGMO  2015 / 6
Let H be the orthocentre and G be the centroid of acute-angled triangle ∆ABC with AB ≠ AC. The line AG intersects the circumcircle of ∆ABC at A and P. Let P΄ be the reflection of P in the line BC. Prove that <CAB = 60° if and only if HG = GP΄.
by Ukraine
EGMO  2016 / 2
Let ABCD be a cyclic quadrilateral, and let diagonals AC and BD intersect at X. Let C1, D1 and M be the midpoints of segments CX, DX and CD, respectively. Lines AD1 and BC1 intersect at Y , and line MY intersects diagonals AC and BD at different points E and F, respectively. Prove that line XY is tangent to the circle through E, F and X.

EGMO  2016 / 4
Two circles, ω1and ω2, of equal radius intersect at different points X1 and X2. Consider a circle ω externally tangent to ω1 at a point T1, and internally tangent to ω2 at a point T2. Prove that lines X1T1 and X2T2 intersect at a point lying on ω.
by Cody Johnson & Charles Leytem, Luxembourg
EGMO  2017 / 1
Let ABCD be a convex quadrilateral with <DAB = <BCD = 90° and <ABC  >   <CDA. Let Q and R be points on segments BC and CD, respectively, such that line QR intersects lines AB and AD at points P and S, respectively. It is given that PQ = RS. Let the midpoint of BD be M and the midpoint of QR be N. Prove that the points M, N, A and C lie on a circle.
by Mark Mordechai Etkind, Israel
EGMO  2017 / 6 
Let ABC be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid G and the circumcentre O of ABC in its sides BC,CA,AB are denoted by G1,G2,G3, and O1,O2,O3, respectively. Show that the circumcircles of the triangles G1G2C, G1G3B, G2G3A, O1O2C, O1O3B, O2O3A and ABC have a common point.
by Charles Leytem, Luxembourg

EGMO  2018 / 1
Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$. Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.

EGMO  2018 /5
Let $\Gamma $ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\Gamma$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle BCA$ intersects $\Omega$ at two different points $P$ and $Q$.  Prove that $\angle ABP = \angle QBC$.

Let $ABC$ be a triangle such that $\angle CAB > \angle ABC$, and let $I$ be its incentre. Let $D$ be the point on segment $BC$ such that $\angle CAD = \angle ABC$. Let $\omega$ be the circle tangent to $AC$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the circumcircle of $ABC$. Prove that the angle bisectors of $\angle DAB$ and $\angle CXB$ intersect at a point on line $BC$.

Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$

Let $ABCDEF$ be a convex hexagon such that $\angle A = \angle C = \angle E$ and $\angle B = \angle D = \angle F$ and the (interior) angle bisectors of $\angle A, ~\angle C,$ and $\angle E$ are concurrent. Prove that the (interior) angle bisectors of $\angle B, ~\angle D, $ and $\angle F$ must also be concurrent.

Note that $\angle A = \angle FAB$.  The other interior angles of the hexagon are similarly described.

Consider the triangle $ABC$ with $\angle BCA > 90^{\circ}$.  The circumcircle $\Gamma$ of $ABC$ has radius $R$.  There is a point $P$ in the interior of the line segment $AB$ such that $PB = PC$ and the length of $PA$ is $R$.  The perpendicular bisector of $PB$ intersects $\Gamma$ at the points $D$ and $E$. Prove $P$ is the incentre of triangle $CDE$.

source: www.egmo.org/

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