Found problems: 250
2011 Belarus Team Selection Test, 1
$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola.
I.Voronovich
Kyiv City MO Seniors Round2 2010+ geometry, 2012.11.4
The circles ${{w} _ {1}}$ and ${{w} _ {2}}$ intersect at points $P$ and $Q$. Let $AB$ and $CD$ be parallel diameters of circles ${ {w} _ {1}}$ and ${{w} _ {2}} $, respectively. In this case, none of the points $A, B, C, D$ coincides with either $P$ or $Q$, and the points lie on the circles in the following order: $A, B, P, Q$ on the circle ${{w} _ {1} }$ and $C, D, P, Q$ on the circle ${{w} _ {2}} $. The lines $AP$ and $BQ$ intersect at the point $X$, and the lines $CP$ and $DQ$ intersect at the point $Y, X \ne Y$. Prove that all lines $XY$ for different diameters $AB$ and $CD$ pass through the same point or are all parallel.
(Serdyuk Nazar)
2017 Mexico National Olympiad, 3
Let $ABC$ be an acute triangle with orthocenter $H$. The circle through $B, H$, and $C$ intersects lines $AB$ and $AC$ at $D$ and $E$ respectively, and segment $DE$ intersects $HB$ and $HC$ at $P$ and $Q$ respectively. Two points $X$ and $Y$, both different from $A$, are located on lines $AP$ and $AQ$ respectively such that $X, H, A, B$ are concyclic and $Y, H, A, C$ are concyclic. Show that lines $XY$ and $BC$ are parallel.
Mathley 2014-15, 3
Let the incircle $\gamma$ of triangle $ABC$ be tangent to $BA, BC$ at $D, E$, respectively. A tangent $t$ to $\gamma$ , distinct from the sidelines, intersects the line $AB$ at $M$. If lines $CM, DE$ meet at$ K$, prove that lines $AK,BC$ and $t$ are parallel or concurrent.
Michel Bataille , France
Kyiv City MO Seniors Round2 2010+ geometry, 2021.10.4.1
Let $ABCD$ be an isosceles trapezoid, $AD=BC$, $AB \parallel CD$. The diagonals of the trapezoid intersect at the point $O$, and the point $M$ is the midpoint of the side $AD$. The circle circumscribed around the triangle $BCM$ intersects the side $AD$ at the point $K$. Prove that $OK \parallel AB$.
2007 Regional Olympiad of Mexico Northeast, 2
In the isosceles triangle $ABC$, with $AB=AC$, $D$ is a point on the extension of $CA$ such that $DB$ is perpendicular to $BC$, $E$ is a point on the extension of $BC$ such that $CE=2BC$, and $F$ is a point on $ED$ such that $FC$ is parallel to $AB$. Prove that $FA$ is parallel to $BC$.
2008 Oral Moscow Geometry Olympiad, 6
Opposite sides of a convex hexagon are parallel. Let's call the "height" of such a hexagon a segment with ends on straight lines containing opposite sides and perpendicular to them. Prove that a circle can be circumscribed around this hexagon if and only if its "heights" can be parallelly moved so that they form a triangle.
(A. Zaslavsky)
2012 IMAC Arhimede, 2
Circles $k_1,k_2$ intersect at $B,C$ such that $BC$ is diameter of $k_1$.Tangent of $k_1$ at $C$ touches $k_2$ for the second time at $A$.Line $AB$ intersects $k_1$ at $E$ different from $B$, and line $CE$ intersects $k_2$ at F different from $C$. An arbitrary line through $E$ intersects segment $AF$ at $H$ and $k_1$ for the second time at $G$.If $BG$ and $AC$ intersect at $D$, prove $CH//DF$ .
Kyiv City MO Seniors 2003+ geometry, 2006.10.4
A circle $\omega$ is inscribed in the acute-angled triangle $\vartriangle ABC$, which touches the side $BC$ at the point $K$. On the lines $AB$ and $AC$, the points $P$ and $Q$, respectively, are chosen so that $PK \perp AC$ and $QK \perp AB$. Denote by $M$ and $N$ the points of intersection of $KP$ and $KQ$ with the circle $\omega$. Prove that if $MN \parallel PQ$, then $\vartriangle ABC$ is isosceles.
(S. Slobodyanyuk)
1996 North Macedonia National Olympiad, 4
A polygon is called [i]good [/i] if it satisfies the following conditions:
(i) All its angles are in $(0,\pi)$ or in $(\pi ,2\pi)$,
(ii) It is not self-intersecing,
(iii) For any three sides, two are parallel and equal.
Find all $n$ for which there exists a [i]good [/i] $n$-gon.
2011 Sharygin Geometry Olympiad, 10
The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.
2003 District Olympiad, 2
In the right triangle $ABC$ ( $\angle A = 90^o$), $D$ is the intersection of the bisector of the angle $A$ with the side $(BC)$, and $P$ and $Q$ are the projections of the point $D$ on the sides $(AB),(AC)$ respectively . If $BQ \cap DP=\{M\}$, $CP \cap DQ=\{N\}$, $BQ\cap CP=\{H\}$, show that:
a) $PM = DN$
b) $MN \parallel BC$
c) $AH \perp BC$.
1997 Estonia National Olympiad, 3
Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the ratio of the length of each diagonal to the length of the corresponding parallel side is the same, and find this ratio.
2024 Brazil National Olympiad, 6
Let \(ABC\) be an isosceles triangle with \(AB = BC\). Let \(D\) be a point on segment \(AB\), \(E\) be a point on segment \(BC\), and \(P\) be a point on segment \(DE\) such that \(AD = DP\) and \(CE = PE\). Let \(M\) be the midpoint of \(DE\). The line parallel to \(AB\) through \(M\) intersects \(AC\) at \(X\) and the line parallel to \(BC\) through \(M\) intersects \(AC\) at \(Y\). The lines \(DX\) and \(EY\) intersect at \(F\). Prove that \(FP\) is perpendicular to \(DE\).
1984 IMO Longlists, 50
Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.
2006 Sharygin Geometry Olympiad, 17
In two circles intersecting at points $A$ and $B$, parallel chords $A_1B_1$ and $A_2B_2$ are drawn. The lines $AA_1$ and $BB_2$ intersect at the point $X, AA_2$ and $BB_1$ intersect at the point $Y$. Prove that $XY // A_1B_1$.
1997 Israel National Olympiad, 8
Two equal circles are internally tangent to a larger circle at $A$ and $B$. Let $M$ be a point on the larger circle, and let lines $MA$ and $MB$ intersect the corresponding smaller circles at $A'$ and $B'$. Prove that $A'B'$ is parallel to $AB$.
2009 Belarus Team Selection Test, 3
Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$. A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$. Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$.
I. Voronovich
1981 Austrian-Polish Competition, 8
The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?
2015 Junior Balkan Team Selection Tests - Romania, 5
Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel . Denote by $O$ the intersection of the diagonals , $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$ . Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$ . Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$
2010 Flanders Math Olympiad, 3
In a triangle $ABC$, $\angle B= 2\angle A \ne 90^o$ . The inner bisector of $B$ intersects the perpendicular bisector of $[AC]$ at a point $D$. Prove that $AB \parallel CD$.
2019 Ukraine Team Selection Test, 1
In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$.
(Anton Trygub)
2019 Regional Competition For Advanced Students, 2
The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.
2017 Saudi Arabia IMO TST, 2
Let $ABCD$ be a quadrilateral inscribed a circle $(O)$. Assume that $AB$ and $CD$ intersect at $E, AC$ and $BD$ intersect at $K$, and $O$ does not belong to the line $KE$. Let $G$ and $H$ be the midpoints of $AB$ and $CD$ respectively. Let $(I)$ be the circumcircle of the triangle $GKH$. Let $(I)$ and $(O)$ intersect at $M, N$ such that $MGHN$ is convex quadrilateral. Let $P$ be the intersection of $MG$ and $HN,Q$ be the intersection of $MN$ and $GH$.
a) Prove that $IK$ and $OE$ are parallel.
b) Prove that $PK$ is perpendicular to $IQ$.
2021 Indonesia TST, G
The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.