Olympiad problems

Kharkiv City MO Seniors - geometry, 2018.11.4

The line $\ell$ parallel to the side $BC$ of the triangle $ABC$, intersects its sides $AB,AC$ at the points $D,E$, respectively. The circumscribed circle of triangle $ABC$ intersects line $\ell$ at points $F$ and $G$, such that points $F,D,E,G$ lie on line $\ell$ in this order. The circumscribed circles of the triangles $FEB$ and $DGC$ intersect at points $P$ and $Q$. Prove that points $A, P$ and $Q$ are collinear.

Durer Math Competition CD 1st Round - geometry, 2017.D+2

Tags: geometry , trapezoid , collinear

Let the trapezoids $A_iB_iC_iD_i$ ($i = 1, 2, 3$) be similar and have the same clockwise direction. Their angles at $A_i$ and $B_i$ are $60^o$ and the sides $A_1B_1$, $B_2C_2$ and $A_3D_3$ are parallel. The lines $B_iD_{i+1}$ and $C_iA_{i+1}$ intersect at the point $P_i$ (the indices are understood cyclically, i.e. $A_4 = A_1$ and $D_4 = D_1$). Prove that the points $P_1$, $P_2$ and $P_3$ lie on a line.

Croatia MO (HMO) - geometry, 2010.7

Tags: geometry , circumcircle , circumcenter , collinear

Given a non- isosceles triangle $ABC$. Let the points $B'$ and $C'$ be symmetric to the points $B$ and $C$ wrt $AC$ and $AB$ respectively. If the circles circumscribed around triangles $ABB'$ and $ACC'$ intersect at point $P$, prove that the line $AP$ passes through the center of the circumcircle of the triangle $ABC$.

2007 Korea Junior Math Olympiad, 4

Tags: geometry , perpendicular bisector , concyclic , collinear

Let $P$ be a point inside $\triangle ABC$. Let the perpendicular bisectors of $PA,PB,PC$ be $\ell_1,\ell_2,\ell_3$. Let $D =\ell_1 \cap \ell_2$ , $E=\ell_2 \cap \ell_3$, $F=\ell_3 \cap \ell_1$. If $A,B,C,D,E,F$ lie on a circle, prove that $C, P,D$ are collinear.

2015 IFYM, Sozopol, 6

Tags: geometry , inscribed circle , excircle , collinear , concurrency

In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides. a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$. b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.

2010 IMAC Arhimede, 1

Tags: point , combinatorics , combinatorial geometry , collinear

$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.

2017 Czech-Polish-Slovak Match, 1

Tags: geometry , collinear

Let ${ABC}$ be a triangle. Line [i]l[/i] is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$, side ${AC}$ at point ${E}$, and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$, where points ${F,D,E,G}$ lie in this order on [i]l[/i]. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$. Prove that points ${A, P,Q}$ are collinear. (Slovakia)

2013 ELMO Shortlist, 7

Tags: ratio , parallelogram , circumcircle , trigonometry , collinear , complex , geometry

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

Indonesia MO Shortlist - geometry, g2

Tags: geometry , circles , perpendicular , collinear , tangent circle

Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.

1997 Mexico National Olympiad, 2

Tags: ratio , collinear , geometry

In a triangle $ABC, P$ and $P'$ are points on side $BC, Q$ on side $CA$, and $R $ on side $AB$, such that $\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}$ . Let $G$ be the centroid of triangle $ABC$ and $K$ be the intersection point of $AP'$ and $RQ$. Prove that points $P,G,K$ are collinear.

1990 Swedish Mathematical Competition, 2

Tags: sum , geometric inequalities , geometry , collinear

The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

2010 Ukraine Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , collinear , incircle , excircle

Let $ABCD$ be a quadrilateral inscribled in a circle with the center $O, P$ be the point of intersection of the diagonals $AC$ and $BD$, $BC\nparallel AD$. Rays $AB$ and $DC$ intersect at the point $E$. The circle with center $I$ inscribed in the triangle $EBC$ touches $BC$ at point $T_1$. The $E$-excircle with center $J$ in the triangle $EAD$ touches the side $AD$ at the point T$_2$. Line $IT_1$ and $JT_2$ intersect at $Q$. Prove that the points $O, P$, and $Q$ lie on a straight line.

2014 Sharygin Geometry Olympiad, 16

Tags: midpoint , geometry , collinear

Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA, DB, DC$ meet lines $BC, AC, AB$ at points $A_1, B_1, C_1$ respectively. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ are collinear.

2007 Thailand Mathematical Olympiad, 3

Tags: circles , collinear , geometry

Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.

2018 Federal Competition For Advanced Students, P2, 4

Tags: geometry , circumcircle , collinear

Let $ABC$ be a triangle and $P$ a point inside the triangle such that the centers $M_B$ and $M_A$ of the circumcircles $k_B$ and $k_A$ of triangles $ACP$ and $BCP$, respectively, lie outside the triangle $ABC$. In addition, we assume that the three points $A, P$ and $M_A$ are collinear as well as the three points $B, P$ and $M_B$. The line through $P$ parallel to side $AB$ intersects circles $k_A$ and $k_B$ in points $D$ and $E$, respectively, where $D, E \ne P$. Show that $DE = AC + BC$. [i](Proposed by Walther Janous)[/i]

2019 Regional Olympiad of Mexico Center Zone, 4

Tags: geometry , collinear

Let $ABC$ be a triangle with $\angle BAC> 90 ^ \circ$ and $D$ a point on $BC$. Let $E$ and $F$be the reflections of the point $D$ about $AB$ and $AC$, respectively. Suppose that $BE$ and $CF$ intersect at $P$. Show that $AP$ passes through the circumcenter of triangle $ABC$.

1993 All-Russian Olympiad Regional Round, 9.7

Tags: geometry , rhombus , collinear , equal segments

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.

Kharkiv City MO Seniors - geometry, 2021.10.5

Tags: geometry , collinear , incircle

The inscribed circle $\Omega$ of triangle $ABC$ touches the sides $AB$ and $AC$ at points $K$ and $ L$, respectively. The line $BL$ intersects the circle $\Omega$ for the second time at the point $M$. The circle $\omega$ passes through the point $M$ and is tangent to the lines $AB$ and $BC$ at the points $P$ and $Q$, respectively. Let $N$ be the second intersection point of circles $\omega$ and $\Omega$, which is different from $M$. Prove that if $KM \parallel AC$ then the points $P, N$ and $L$ lie on one line.

1997 All-Russian Olympiad Regional Round, 10.2

Tags: rectangle , collinear , geometry

Circles $S_1$ and $S_2$ intersect at points $M$ and $N$. Prove that if vertices $A$ and $ C$ of some rectangle $ABCD$ lie on the circle $S_1$, and the vertices $B$ and $D$ lie on the circle $S_2$, then the point of intersection of its diagonals lies on the line $MN$.

2019 Brazil EGMO TST, 3

Tags: geometry , collinear , concyclic

Let $ABC$ be a triangle and $E$ and $F$ two arbitrary points on sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. The point $D$ is such that $EF$ bisects the segment $MD$ . Finally, $O$ is the circumcenter of triangle $ABC$. Prove that $D$ lies on line $BC$ if and only if $O$ lies on the circumcircle of triangle $AEF$.

2006 Sharygin Geometry Olympiad, 18

Tags: geometry , collinear , parallel , orthocenter , projection

Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ, BZ$ are parallel, respectively with $HX$ and $HY$. Prove that the points $X, Y, Z$ lie on the same line.

Indonesia MO Shortlist - geometry, g1

Tags: collinear , geometry , angle bisector

Given triangle $ABC$, $AL$ bisects angle $\angle BAC$ with $L$ on side $BC$. Lines $LR$ and $LS$ are parallel to $BA$ and $CA$ respectively, $R$ on side $AC$ and$ S$ on side $AB$, respectively. Through point $B$ draw a perpendicular on $AL$, intersecting $LR$ at $M$. If point $D$ is the midpoint of $BC$, prove that that the three points $A, M, D$ lie on a straight line.

2018 Romania National Olympiad, 3

Tags: geometry , parallelogram , collinear , equilateral , rhombus

On the sides $[AB]$ and $[BC]$ of the parallelogram $ABCD$ are constructed the equilateral triangles $ABE$ and $BCF,$ so that the points $D$ and $E$ are on the same side of the line $AB$, and $F$ and $D$ on different sides of the line $BC$. If the points $E,D$ and $F$ are collinear, then prove that $ABCD$ is rhombus.

2011 Junior Balkan Team Selection Tests - Moldova, 7

Tags: rectangle , collinear , circles , geometry

In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.

2004 Tournament Of Towns, 4

Tags: tangent , circles , geometry , collinear

Two circles intersect in points $A$ and $B$. Their common tangent nearer $B$ touches the circles at points $E$ and $F$, and intersects the extension of $AB$ at the point $M$. The point $K$ is chosen on the extention of $AM$ so that $KM = MA$. The line $KE$ intersects the circle containing $E$ again at the point $C$. The line $KF$ intersects the circle containing $F$ again at the point $D$. Prove that the points $A, C$ and $D$ are collinear.