Olympiad problems

2019 Azerbaijan BMO TST, 2

Tags: perpendicular , tangent , perpendicularity , orthocenter , perpendicular bisector , geometry , circumcircle

Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular. by Michael Sarantis, Greece

Ukraine Correspondence MO - geometry, 2019.11

Tags: arc midpoint , geometry , perpendicular

Let $O$ be the center of the circle circumscribed around the acute triangle $ABC$, and let $N$ be the midpoint of the arc $ABC$ of this circle. On the sides $AB$ and $BC$ mark points $D$ and $E$ respectively, such that the point $O$ lies on the segment $DE$. The lines $DN$ and $BC$ intersect at the point $P$, and the lines $EN$ and $AB$ intersect at the point $Q$. Prove that $PQ \perp AC$.

2011 Indonesia TST, 3

Tags: maximization , triangle , perpendicular , geometry , trigonometry

Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define \[ p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. \] Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

2014 Romania National Olympiad, 2

Tags: rhombus , right triangle , geometry , perpendicular , square , isosceles

Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

1999 Ukraine Team Selection Test, 1

Tags: geometry , locus , cyclic , perpendicular

A triangle $ABC$ is given. Points $E,F,G$ are arbitrarily selected on the sides $AB,BC,CA$, respectively, such that $AF\perp EG$ and the quadrilateral $AEFG$ is cyclic. Find the locus of the intersection point of $AF$ and $EG$.

Estonia Open Senior - geometry, 2016.2.5

Tags: perpendicular , circumcircle , geometry

The circumcentre of an acute triangle $ABC$ is $O$. Line $AC$ intersects the circumcircle of $AOB$ at a point $X$, in addition to the vertex $A$. Prove that the line $XO$ is perpendicular to the line $BC$.

2024 Regional Olympiad of Mexico Southeast, 2

Tags: acute triangle , circumradius , perpendicular , geometry

Let $ABC$ be an acute triangle with circumradius $R$. Let $D$ be the midpoint of $BC$ and $F$ the midpoint of $AB$. The perpendicular to $AC$ through $F$ and the perpendicular to $BC$ through $B$ intersect at $N$. Prove that $ND = R$.

Novosibirsk Oral Geo Oly IX, 2020.3

Tags: geometry , perpendicular , equal segments

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

Kyiv City MO Juniors 2003+ geometry, 2019.9.2

Tags: median , geometry , perpendicular

In a right triangle $ABC$, the lengths of the legs satisfy the condition: $BC =\sqrt2 AC$. Prove that the medians $AN$ and $CM$ are perpendicular. (Hilko Danilo)

2018 JBMO Shortlist, G5

Tags: area of a triangle , similar triangles , rectangle , geometry , perpendicular

Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is $$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$ Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.

2016 Sharygin Geometry Olympiad, 2

Tags: arc midpoint , perpendicular , circumcircle , parallelogram , geometry

A circumcircle of triangle $ABC$ meets the sides $AD$ and $CD$ of a parallelogram $ABCD$ at points $K$ and $L$ respectively. Let $M$ be the midpoint of arc $KL$ not containing $B$. Prove that $DM \perp AC$. by E.Bakaev

2022 Indonesia TST, G

Tags: acute , collinear , extension , triangle , orthocenter , perpendicular , geometry

Given an acute triangle $ABC$. with $H$ as its orthocenter, lines $\ell_1$ and $\ell_2$ go through $H$ and are perpendicular to each other. Line $\ell_1$ cuts $BC$ and the extension of $AB$ on $D$ and $Z$ respectively. Whereas line $\ell_2$ cuts $BC$ and the extension of $AC$ on $E$ and $X$ respectively. If the line through $D$ and parallel to $AC$ and the line through $E$ parallel to $AB$ intersects at $Y$, prove that $X,Y,Z$ are collinear.

2004 Junior Tuymaada Olympiad, 3

Tags: perpendicular , arc , collinear , geometry

Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.

Brazil L2 Finals (OBM) - geometry, 2005.2

Tags: right triangle , area , geometry , perpendicular

In the right triangle $ABC$, the perpendicular sides $AB$ and $BC$ have lengths $3$ cm and $4$ cm, respectively. Let $M$ be the midpoint of the side $AC$ and let $D$ be a point, distinct from $A$, such that $BM = MD$ and $AB = BD$. a) Prove that $BM$ is perpendicular to $AD$. b) Calculate the area of the quadrilateral $ABDC$.

2014 Ukraine Team Selection Test, 8

Tags: collinear , perpendicular , cyclic quadrilateral , geometry

The quadrilateral $ABCD$ is inscribed in the circle $\omega$ with the center $O$. Suppose that the angles $B$ and $C$ are obtuse and lines $AD$ and $BC$ are not parallel. Lines $AB$ and $CD$ intersect at point $E$. Let $P$ and $R$ be the feet of the perpendiculars from the point $E$ on the lines $BC$ and $AD$ respectively. $Q$ is the intersection point of $EP$ and $AD, S$ is the intersection point of $ER$ and $BC$. Let K be the midpoint of the segment $QS$ . Prove that the points $E, K$, and $O$ are collinear.

2000 Singapore MO Open, 1

Tags: concyclic , circumcircle , perpendicular , geometry

Triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ and $E$ be points on the respective sides $AB$ and $AC$ so that $DE$ is perpendicular to $AO$. Show that the four points $B,D,E$ and $C$ lie on a circle.

1991 Tournament Of Towns, (286) 2

Tags: perpendicular , pentagon , geometry

The pentagon $ABCDE$ has an inscribed circle and the diagonals $AD$ and $CE$ intersect in its centre $O$. Prove that the segment $BO$ and the side $DE$ are perpendicular. (Folklore)

1981 All Soviet Union Mathematical Olympiad, 305

Tags: cyclic , perpendicular , parallel

Given points $A,B,M,N$ on the circumference. Two chords $[MA_1]$ and $[MA_2]$ are orthogonal to lines $(NA)$ and $(NB)$ respectively. Prove that $(AA_1)$ and $(BB_1)$ lines are parallel.

2018 Dutch IMO TST, 2

Tags: geometry , perpendicular , right triangle , tangent , circumcircle

Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.

2015 Grand Duchy of Lithuania, 2

Tags: perpendicular , circles , geometry

Let $\omega_1$ and $\omega_2$ be two circles , with respective centres $O_1$ and $O_2$ , that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\omega_2$ in $A$ and $C$ and the line $O_2A$ inetersects $\omega_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\omega_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

2004 BAMO, 2

Tags: geometry , straightedge , perpendicular , construction

A given line passes through the center $O$ of a circle. The line intersects the circle at points $A$ and $B$. Point $P$ lies in the exterior of the circle and does not lie on the line $AB$. Using only an unmarked straightedge, construct a line through $P$, perpendicular to the line $AB$. Give complete instructions for the construction and prove that it works.

2018 Yasinsky Geometry Olympiad, 6

Tags: perpendicular , isosceles triangle , geometry , perpendicularity , circumcircle , incenter , isosceles

Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular. (Vyacheslav Yasinsky)

2022 Oral Moscow Geometry Olympiad, 4

Tags: perpendicularity , perpendicular , geometry

In triangle $ABC$, angle $C$ is equal to $60^o$. Bisectors $AA'$ and $BB'$ intersect at point $I$. Point $K$ is symmetric to $I$ with respect to line $AB$. Prove that lines $CK$ and $A'B'$ are perpendicular. (D. Shvetsov, A. Zaslavsky)

2001 Switzerland Team Selection Test, 7

Tags: circumcenter , perpendicular , geometry

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circle $S$ through $A,B$, and $O$ intersects $AC$ and $BC$ again at points $P$ and $Q$ respectively. Prove that $CO \perp PQ$.

2012 Oral Moscow Geometry Olympiad, 4

Tags: excenter , perpendicular , incenter , geometry , circumcircle

In triangle $ABC$, point $I$ is the center of the inscribed circle points, points $I_A$ and $I_C$ are the centers of the excircles, tangent to sides $BC$ and $AB$, respectively. Point $O$ is the center of the circumscribed circle of triangle $II_AI_C$. Prove that $OI \perp AC$