Olympiad problems

Kyiv City MO Seniors 2003+ geometry, 2015.10.5.1

The points $X, \, \, Y$are selected on the sides $AB$ and $AD$ of the convex quadrilateral $ABCD$, respectively. Find the ratio $AX \, \,: \, \, BX$ if you know that $CX || DA$, $DX || CB$, $BY || CD$ and $CY || BA$.

Ukraine Correspondence MO - geometry, 2010.7

Tags: equal segments , parallel , geometry

An arbitrary point $D$ was marked on the median $BM$ of the triangle $ABC$. It is known that the point $DE\parallel AB$ and $CE \parallel BM$. Prove that $BE = AD$

Indonesia Regional MO OSP SMA - geometry, 2007.1

Tags: parallel , equal segments , geometry

Let $ABCD$ be a quadrilateral with $AB = BC = CD = DA$. (a) Prove that point A must be outside of triangle $BCD$. (b) Prove that each pair of opposite sides on $ABCD$ is always parallel.

2006 Korea Junior Math Olympiad, 3

Tags: geometry , concyclic , perpendicular , parallel

In a circle $O$, there are six points, $A,B,C,D,E, F$ in a counterclockwise order. $BD \perp CF$, and $CF,BE,AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE // MN$.

1971 Dutch Mathematical Olympiad, 1

Tags: geometry , trapezoid , parallel

Given a trapezoid $ABCD$, where sides $AB$ and $CD$ are parallel; the points $P$ on $AD$ and $Q$ on $BC$ lie such that the lines $AQ$ and $CP$ are parallel. Prove that lines $PB$ and $DQ$ are parallel.

2015 Costa Rica - Final Round, 6

Tags: geometry , trapezoid , parallel

Given the trapezoid $ABCD$ with the $BC\parallel AD$, let $C_1$ and $C_2$ be circles with diameters $AB$ and $CD$ respectively. Let $M$ and $N$ be the intersection points of $C_1$ with $AC$ and $BD$ respectively. Let $K$ and $L$ be the intersection points of $C_2$ with $AC$ and $BD$ respectively. Given $M\ne A$, $N\ne B$, $K\ne C$, $L\ne D$. Prove that $NK \parallel ML$.

2012 Junior Balkan Team Selection Tests - Romania, 5

Tags: geometry , incircle , parallel , incenter

Let $ABC$ be a triangle and $A', B', C'$ the points in which its incircle touches the sides $BC, CA, AB$, respectively. We denote by $I$ the incenter and by $P$ its projection onto $AA' $. Let $M$ be the midpoint of the line segment $[A'B']$ and $N$ be the intersection point of the lines $MP$ and $AC$. Prove that $A'N $is parallel to $B'C'$

2018-IMOC, G4

Tags: geometry , parallel , circumcircle , incenter

Given an acute $\vartriangle ABC$ with incenter $I$. Let $I'$ be the symmetric point $I$ with respect to the midpoint of $B,C$ and $D$ is the foot of $A$. If $DI$ and the circumcircle of vartriangle $BI'C$ intersect at $T$ and $TI' $ intersects the circumcircle of $\vartriangle ATI$ at $X$. Furthermore, $E,F$ are tangent points of the incircle and $AB,AC, P$ is the another intersection of the circumcircles of $\vartriangle ABC, \vartriangle AEF$. Show that $AX \parallel PI$. [img]https://3.bp.blogspot.com/-tj9A8HIR6Vw/XndLEPMRvnI/AAAAAAAALfk/2vw_pZbhpnkTKIc1BcKf4K7SNZ11vu4TACK4BGAYYCw/s1600/2018%2Bimoc%2Bg4.png[/img]

2021 Novosibirsk Oral Olympiad in Geometry, 7

Tags: incenter , parallel , geometry

A circle concentric with the inscribed circle of $ABC$ intersects the sides of the triangle at six points forming a convex hexagon $A_1A_2B_1B_2C_1C_2$ (points $C_1$ and $C_2$ on the $AB$ side, $A_1$ and $A_2$ on $BC$, $B_1$ and $B_2$ on $AC$). Prove that if line $A_1B_1$ is parallel to the bisector of angle $B$, then line $A_2C_2$ is parallel to the bisector of angle $C$.

Croatia MO (HMO) - geometry, 2014.3

Tags: angle bisector , parallel , geometry

Given a triangle $ABC$ in which $|AB|>|AC|$. Let $P$ be the midpoint of the side $BC$, and $S$ the point in which the angle bisector of $\angle BAC$ intersects that side. The parallel with the line $AS$ through the point $P$ intersects lines $AB$ and $AC$ at points $X$ and $Y$ respectively . Let $Z$ be the point be such that $Y$ is the midpoint of the length $XZ$ and let the lines $BY$ and $CZ$ intersect at point $D$. Prove that the angle bisector of $\angle BDC$ is parallel to the lines $AS$.

2015 Bosnia Herzegovina Team Selection Test, 6

Tags: geometry , incircle , perpendicular bisector , parallel , incenter

Let $D$, $E$ and $F$ be points in which incircle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$, respectively, and let $I$ be a center of that circle.Furthermore, let $P$ be a foot of perpendicular from point $I$ to line $AD$, and let $M$ be midpoint of $DE$. If $\{N\}=PM\cap{AC}$, prove that $DN \parallel EF$

2024 Dutch IMO TST, 4

Tags: circumcircle , parallelogram , parallel , geometry , similar triangles

Let $ABC$ be an acute triangle with circumcenter $O$, and let $D$, $E$, and $F$ be the feet of altitudes from $A$, $B$, and $C$ to sides $BC$, $CA$, and $AB$, respectively. Denote by $P$ the intersection of the tangents to the circumcircle of $ABC$ at $B$ and $C$. The line through $P$ perpendicular to $EF$ meets $AD$ at $Q$, and let $R$ be the foot of the perpendicular from $A$ to $EF$. Prove that $DR$ and $OQ$ are parallel.

2014 Iranian Geometry Olympiad (junior), P3

Tags: ratio , polygon , parallel , geometry

Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$. by Morteza Saghafian

2019 Chile National Olympiad, 4

Tags: parallel , equal angles , geometry

In the convex quadrilateral $ABCD$ , $\angle ADC = \angle BCD > 90^o$ . Let $E$ be the intersection of the line $AC$ with the line parallel to $AD$ that passes through $B$. Let $F$ be the intersection of line $BD$ with the line parallel to $BC$ passing through $A$. Prove that $EF$ is parallel to $CD$.

2020 Ukrainian Geometry Olympiad - December, 5

Tags: circumcircle , centroid , circumcenter , parallel , geometry

Let $ABC$ be an acute triangle with $\angle ACB = 45^o$, $G$ is the point of intersection of the medians, and $O$ is the center of the circumscribed circle. If $OG =1$ and $OG \parallel BC$, find the length of $BC$.

2018-IMOC, G5

Tags: geometry , circumcenter , parallel , euler line , orthocenter , incenter

Suppose $I,O,H$ are incenter, circumcenter, orthocenter of $\vartriangle ABC$ respectively. Let $D = AI \cap BC$,$E = BI \cap CA$, $F = CI \cap AB$ and $X$ be the orthocenter of $\vartriangle DEF$. Prove that $IX \parallel OH$.

1949 Moscow Mathematical Olympiad, 163

Tags: parallel , hexagon , geometry

Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.

2022 IMO Shortlist, G1

Tags: congruent triangles , parallel , computer problems , geometry

Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.

Mathley 2014-15, 1

Tags: midpoint , parallel , collinear , geometry

Let $AD, BE, CF$ be segments whose midpoints are on the same line $\ell$. The points $X, Y, Z$ lie on the lines $EF, FD, DE$ respectively such that $AX \parallel BY \parallel CZ \parallel \ell$. Prove that $X, Y, Z$ are collinear. Tran Quang Hung, High School of Natural Sciences, Hanoi National University

2016 German National Olympiad, 3

Tags: orthocenter , parallel , excenter , geometry , circumcircle

Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$. Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.

1983 All Soviet Union Mathematical Olympiad, 358

Tags: tetrahedron , combinatorial geometry , parallel , 3d geometry , projection , plane , geometry

The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of lines $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ .

1988 Tournament Of Towns, (176) 2

Tags: geometry , diagonal , isosceles trapezoid , parallel , trapezoid

Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.

Kyiv City MO Juniors 2003+ geometry, 2018.8.41

Tags: geometry , angle bisector , trapezoid , parallel

In a trapezoid $ABCD$ with bases $AD$ and $BC$, the bisector of the angle $\angle DAB$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $P$ and $S$, respectively, and the bisector of the angle $\angle BCD$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $Q$ and $R$, respectively. Prove that if $PS\parallel RQ$, then $AB = CD$.

2011 Saudi Arabia Pre-TST, 2.4

Tags: rectangle , parallel , geometry

Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

2018 Oral Moscow Geometry Olympiad, 3

Tags: circumcircle , excircle , parallel , geometry

On the extensions of sides $CA$ and $AB$ of triangle $ABC$ beyond points $A$ and $B$, respectively, the segments $AE = BC$ and $BF = AC$ are drawn. A circle is tangent to segment $BF$ at point $N$, side $BC$ and the extension of side $AC$ beyond point $C$. Point $M$ is the midpoint of segment $EF$. Prove that the line $MN$ is parallel to the bisector of angle $A$.