Olympiad problems

Kyiv City MO Juniors 2003+ geometry, 2006.9.4

On the sides $AB$ and $CD$ of the parallelogram $ABCD$ mark points $E$ and $F$, respectively. On the diagonals $AC$ and $BD$ chose the points $M$ and $N$ so that $EM\parallel BD$ and $FN\parallel AC$. Prove that the lines $AF, DE$ and $MN$ intersect at one point. (B. Rublev)

2005 China Team Selection Test, 1

Tags: concurrency , projective geometry , geometry

Point $P$ lies inside triangle $ABC$. Let the projections of $P$ onto sides $BC$,$CA$,$AB$ be $D$, $E$, $F$ respectively. Let the projections from $A$ to the lines $BP$ and $CP$ be $M$ and $N$ respectively. Prove that $ME$, $NF$ and $BC$ are concurrent.

2009 Philippine MO, 5

Tags: concurrency , perpendicular bisector , geometry

Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$. Let $E$ be the foot of the perpendicular from $P$ to the line $CD$. Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.

2008 Korean National Olympiad, 6

Tags: concurrency , geometry

Let $ABCD$ be inscribed in a circle $\omega$. Let the line parallel to the tangent to $\omega$ at $A$ and passing $D$ meet $\omega$ at $E$. $F$ is a point on $\omega$ such that lies on the different side of $E$ wrt $CD$. If $AE \cdot AD \cdot CF = BE \cdot BC \cdot DF$ and $\angle CFD = 2\angle AFB$, Show that the tangent to $\omega$ at $A, B$ and line $EF$ concur at one point. ($A$ and $E$ lies on the same side of $CD$)

Geometry Mathley 2011-12, 15.1

Tags: geometry , incircle , circumcircle , concurrency , concurrent circles , equal segments

Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ of the triangle touches sides $BC,CA,AB$ at $A_0,B_0$, and $C_0$. Points $A_1,B_1$, and $C_1$ are on $BC,CA,AB$ such that $BA1 = CA_0, CB_1 = AB_0, AC_1 = BC_0$. Prove that the circumcircles $(IAA1), (IBB_1), (ICC_1)$ pass all through a common point, distinct from $I$. Nguyễn Minh Hà

2022 Germany Team Selection Test, 2

Tags: geometry , angle chasing , concurrency , cyclic quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Brazil L2 Finals (OBM) - geometry, 2012.3

Tags: concurrency , geometry

Let be a triangle $ ABC $, the midpoint of the $ AC $ and $ N $ side, and the midpoint of the $ AB $ side. Let $ r $ and $ s $ reflect the straight lines $ BM $ and $ CN $ on the straight $ BC $, respectively. Also define $ D $ and $ E $ as the intersection of the lines $ r $ and $ s $ and the line $ MN $, respectively. Let $ X $ and $ Y $ be the intersection points between the circumcircles of the triangles $ BDM $ and $ CEN $, $ Z $ the intersection of the lines $ BE $ and $ CD $ and $ W $ the intersection between the lines $ r $ and $ s $. Prove that $ XY, WZ $, and $ BC $ are concurrents.

1997 Chile National Olympiad, 5

Tags: geometry , circles , concurrency

Let: $ C_1, C_2, C_3 $ three circles , intersecting in pairs, such that the secant line common to two of them (any) passes through the center of the third. Prove that the three lines thus defined are concurrent.

2006 Belarusian National Olympiad, 7

Tags: concurrency , altitude , geometry

Let $AH_A, BH_B, CH_C$ be altitudes and $BM$ be a median of the acute-angled triangle $ABC$ ($AB > BC$). Let $K$ be a point of intersection of $BM$ and $AH_A$, $T$ be a point on $BC$ such that $KT \parallel AC$, $H$ be the orthocenter of $ABC$. Prove that the lines passing through the pairs of the points $(H_c, H_A), (H, T)$ and $(A, C)$ are concurrent. (S. Arkhipov)

Swiss NMO - geometry, 2015.8

Tags: trapezoid , concurrency , geometry

Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.

1961 Poland - Second Round, 2

Tags: tetrahedron , concurrency , 3d geometry , geometry

Prove that all the heights of a tetrahedron intersect at one point if and only if the sums of the squares of the opposite edges are equal.

Estonia Open Junior - geometry, 2005.1.3

Tags: concurrency , angle bisector , geometry

In triangle $ABC$, the midpoints of sides $AB$ and $AC$ are $D$ and $E$, respectively. Prove that the bisectors of the angles $BDE$ and $CED$ intersect at the side $BC$ if the length of side $BC$ is the arithmetic mean of the lengths of sides $AB$ and $AC$.

2017 Balkan MO Shortlist, G5

Tags: projective geometry , concurrency , geometry , circumcircle

Let $ABC$ be an acute angled triangle with orthocenter $H$. centroid $G$ and circumcircle $\omega$. Let $D$ and $M$ respectively be the intersection of lines $AH$ and $AG$ with side $BC$. Rays $MH$ and $DG$ interect $ \omega$ again at $P$ and $Q$ respectively. Prove that $PD$ and $QM$ intersect on $\omega$.

2003 IMO Shortlist, 1

Tags: quadrilateral , concurrency , geometry , angle bisector , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

2020 Greece National Olympiad, 2

Tags: parallelogram , geometry , concurrency , equal angles

Given a line segment $AB$ and a point $C$ lies inside it such that $AB=3 \cdot AC$ . Construct a parallelogram $ACDE$ such that $AC=DE=CE>AR$. Let $Z$ be a point on $AC$ such that $\angle AEZ=\angle ACE =\omega$. Prove that the line passing through point $B$ and perpendicular on side $EC$, and the line passing through point $D$ and perpendicular on side $AB$, intersect on point , let it be $K$, lying on line $EZ$.

1991 All Soviet Union Mathematical Olympiad, 540

Tags: rectangle , concurrency , geometry

$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$.

Geometry Mathley 2011-12, 12.3

Tags: concurrency , geometry , tangent , circumcircle

Points $E,F$ are chosen on the sides $CA,AB$ of triangle $ABC$. Let $(K)$ be the circumcircle of triangle $AEF$. The tangents at $E, F$ of $(K)$ intersect at $T$ . Prove that (a) $T$ is on $BC$ if and only if $BE$ meets $CF$ at a point on the circle $(K)$, (b) $EF, PQ,BC$ are concurrent given that $BE$ meets $FT$ at $M, CF$ meets $ET$ at $N, AM$ and $AN$ intersects $(K)$ at $P,Q$ distinct from $A$. Trần Quang Hùng

2016 Sharygin Geometry Olympiad, P9

Tags: right triangle , incircle , concurrency , geometry

Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$. Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively, $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$. Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$.

2014 Oral Moscow Geometry Olympiad, 3

Tags: concurrency , circumcircle , angle bisector , geometry

The bisectors $AA_1$ and $CC_1$ of triangle $ABC$ intersect at point $I$. The circumscribed circles of triangles $AIC_1$ and $CIA_1$ intersect the arcs $AC$ and $BC$ (not containing points $B$ and $A$ respectively) of the circumscribed circle of triangle $ABC$ at points $C_2$ and $A_2$, respectively. Prove that lines $A_1A_2$ and $C_1C_2$ intersect on the circumscribed circle of triangle $ABC$.

2021 Balkan MO Shortlist, G1

Tags: concurrency , shortlist , geometry

Let $ABC$ be a triangle with $AB < AC < BC$. On the side $BC$ we consider points $D$ and $E$ such that $BA = BD$ and $CE = CA$. Let $K$ be the circumcenter of triangle $ADE$ and let $F$, $G$ be the points of intersection of the lines $AD$, $KC$ and $AE$, $KB$ respectively. Let $\omega_1$ be the circumcircle of triangle $KDE$, $\omega_2$ the circle with center $F$ and radius $FE$, and $\omega_3$ the circle with center $G$ and radius $GD$. Prove that $\omega_1$, $\omega_2$, and $\omega_3$ pass through the same point and that this point of intersection lies on the line $AK$.

2021 IMO Shortlist, G4

Tags: angle chasing , concurrency , geometry , cyclic quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

1961 Polish MO Finals, 5

Tags: concurrency , geometry

Four lines intersecting at six points form four triangles. Prove that the circles circumscribed around out these triangles have a common point.

2015 Switzerland - Final Round, 8

Tags: concurrency , trapezoid , geometry

Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.

1980 IMO Longlists, 17

Tags: concurrency , geometry

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

1935 Moscow Mathematical Olympiad, 015

Tags: concurrency , geometry , 3d geometry , parallel

Triangles $\vartriangle ABC$ and $\vartriangle A_1B_1C_1$ lie on different planes. Line $AB$ intersects line $A_1B_1$, line $BC$ intersects line $B_1C_1$ and line $CA$ intersects line $C_1A_1$. Prove that either the three lines $AA_1, BB_1, CC_1$ meet at one point or that they are all parallel.