Olympiad problems

2022 Durer Math Competition (First Round), 4

Tags: geometry , collinear

Let $ABC$ be an acute triangle, and let $F_A$ and $F_B$ be the midpoints of sides $BC$ and $CA$, respectively. Let $E$ and $F$ be the intersection points of the circle centered at $F_A$ and passing through $A$ and the circle centered at $F_B$ and passing through $B$. Prove that if segments $CE$ and $CF$ have midpoints $N$ and $M$, respectively, then the intersection points of the circle centered at $M$ and passing through $E$ and the circle centered at $N$ and passing through $F$ lie on the line $AB$.

2001 Singapore Team Selection Test, 2

Tags: geometry , collinear

Let $P, Q$ be points taken on the side $BC$ of a triangle $ABC$, in the order $B, P, Q, C$. Let the circumcircles of $\vartriangle PAB$, $\vartriangle QAC$ intersect at $M$ ($\ne A$) and those of $\vartriangle PAC, \vartriangle QAB$ at N. Prove that $A, M, N$ are collinear if and only if $P$ and $Q$ are symmetric in the midpoint $A' $ of $BC$.

2010 Bundeswettbewerb Mathematik, 3

Tags: projection , altitude , concyclic , perpendicular , collinear , geometry

Given an acute-angled triangle $ABC$. Let $CB$ be the altitude and $E$ a random point on the line $CD$. Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$. Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

KoMaL A Problems 2017/2018, A. 724

Tags: sphere , 3d geometry , geometry , collinear , tetrahedron

A sphere $S$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $S$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $S$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$. KöMaL A.723. (April 2018), G. Kós

2011 Greece JBMO TST, 4

Tags: circles , circumcircle , geometry , collinear

Let $ABC$ be an acute and scalene triangle with $AB<AC$, inscribed in a circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ intersects side $BC$ at point $E$ and circle $c$ at point $F$. $EF$ intersects for the second time circle $c$ at point $D$ and side $AC$ at point $M$. $AD$ intersects $BC$ at point $K$. Circumcircle of triangle $BKD$ intersects $AB$ at point $L$ . Prove that points $K,L,M$ lie on a line parallel to $BF$.

Kharkiv City MO Seniors - geometry, 2014.11.5

Tags: right angle , perpendicular , geometry , collinear

In the convex quadrilateral of the $ABCD$, the diagonals of $AC$ and $BD$ are mutually perpendicular and intersect at point $E$. On the side of $AD$, a point $P$ is chosen such that $PE = EC$. The circumscribed circle of the triangle $BCD$ intersects the segment $AD$ at the point $Q$. The circle passing through point $A$ and tangent to the line $EP$ at point $P$ intersects the segment $AC$ at point $R$. It turns out that points $B, Q, R$ are collinear. Prove that $\angle BCD = 90^o$.

Champions Tournament Seniors - geometry, 2012.2

Tags: equal angles , geometry , circumcenter , collinear

About the triangle $ABC$ it is known that $AM$ is its median, and $\angle AMC = \angle BAC$. On the ray $AM$ lies the point $K$ such that $\angle ACK = \angle BAC$. Prove that the centers of the circumcircles of the triangles $ABC, ABM$ and $KCM$ lie on the same line.

2021 Dutch IMO TST, 3

Tags: geometry , collinear , circumcircle , orthocenter

Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.

Swiss NMO - geometry, 2005.8

Tags: circles , geometry , orthocenter , collinear

Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ 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.

2022 EGMO, 1

Tags: angle chase , angle chasing , collinear , incenter , geometry

Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.

2016 Costa Rica - Final Round, G3

Tags: collinear , rectangle , geometry

Let the $JHIZ$ be a rectangle and let $A$ and $C$ be points on the sides $ZI$ and $ZJ$, respectively. The perpendicular from $A$ on $CH$ intersects line $HI$ at point $X$ and perpendicular from $C$ on $AH$ intersects line $HJ$ at point $Y$. Show that points $X, Y$, and $Z$ are collinear.

1995 Mexico National Olympiad, 3

Tags: regular polygon , collinear , geometry

$A, B, C, D$ are consecutive vertices of a regular $7$-gon. $AL$ and $AM$ are tangents to the circle center $C$ radius $CB$. $N$ is the intersection point of $AC$ and $BD$. Show that $L, M, N$ are collinear.

2022 SAFEST Olympiad, 5

Tags: tangency , quadrilateral , collinear , geometry

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

Indonesia MO Shortlist - geometry, g5

Tags: geometry , collinear

Given an arbitrary triangle $ABC$, with $\angle A = 60^o$ and $AC < AB$. A circle with diameter $BC$, intersects $AB$ and $AC$ at $F$ and $E$, respectively. Lines $BE$ and $CF$ intersect at $D$. Let $\Gamma$ be the circumcircle of $BCD$, where the center of $\Gamma$ is $O$. Circle $\Gamma$ intersects the line $AB$ and the extension of $AC$ at $M$ and $N$, respectively. $MN$ intersects $BC$ at $P$. Prove that points $A$, $P$, $O$ lie on the same line.

1971 IMO Shortlist, 12

Tags: congruent triangles , geometry , midpoint , collinear

Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.

Ukraine Correspondence MO - geometry, 2013.7

Tags: right triangle , collinear , geometry

An arbitrary point $D$ is marked on the hypotenuse $AB$ of a right triangle $ABC$. The circle circumscribed around the triangle $ACD$ intersects the line $BC$ at the point $E$ for the second time, and the circle circumscribed around the triangle $BCD$ intersects the line $AC$ for the second time at the point $F$. Prove that the line $EF$ passes through the point $D$.

2020 Korea Junior Math Olympiad, 4

Tags: geometry , collinear

In an acute triangle $ABC$ with $\overline{AB} > \overline{AC}$, let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $P$ be an intersection of lines $EF$ and $BC$, and let $Q$ be a point on the segment $BD$ such that $\angle QFD = \angle EPC$. Let $O, H$ denote the circumcenter and the orthocenter of triangle $ABC$, respectively. Suppose that $OH$ is perpendicular to $AQ$. Prove that $P, O, H$ are collinear.

2003 Junior Balkan Team Selection Tests - Romania, 1

Tags: rectangle , collinear , collinearity , geometry

Suppose $ABCD$ and $AEFG$ are rectangles such that the points $B,E,D,G$ are collinear (in this order). Let the lines $BC$ and $GF$ intersect at point $T$ and let the lines $DC$ and $EF$ intersect at point $H$. Prove that points $A, H$ and $T$ are collinear.

2012 Lusophon Mathematical Olympiad, 1

Tags: incenter , geometry , collinear , combinatorial geometry , combinatorics

Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?

2015 Estonia Team Selection Test, 4

Tags: orthocenter , collinear , geometry

Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.

2000 All-Russian Olympiad Regional Round, 11.6

Tags: geometry , collinear

A circle inscribed in triangle $ABC$ has center $O$ and touches side $AC$ at point $K$. A second circle also has center $O$, intersects all sides of triangle $ABC$. Let $E$ and $F$ be the corresponding points of intersection with sides $AB$ and $BC$, closest to vertex $B$; $B_1$ and $B_2$ are the points of its intersection with side $AC$, and $B_1$ is closer to $A$. Prove that points $B$, $K$ and point $P$, the intersections of the segments $B_2E$ and $B_1F$ lie on the same straight line.

Indonesia MO Shortlist - geometry, g2

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

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 $P$.

2010 Saudi Arabia Pre-TST, 1.4

Tags: geometry , centroid , collinear

In triangle $ABC$ with centroid $G$, let $M \in (AB)$ and $N \in (AC)$ be points on two of its sides. Prove that points $M, G, N$ are collinear if and only if $\frac{MB}{MA}+\frac{NC}{NA}=1$.

2019 Costa Rica - Final Round, 6

Tags: right triangle , geometry , isosceles , collinear

Consider the right isosceles $\vartriangle ABC$ at $ A$. Let $L$ be the intersection of the bisector of $\angle ACB$ with $AB$ and $K$ the intersection point of $CL$ with the bisector of $BC$. Let $X$ be the point on line $AK$ such that $\angle KCX = 90^o$ and let $Y$ be the point of intersection of $CX$ with the circumcircle of $\vartriangle ABC$. Let $Y'$ the reflection of point $Y$ wrt $BC$. Prove that $B - K -Y'$. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.