Olympiad problems

2014 Czech-Polish-Slovak Junior Match, 4

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.

1996 Singapore Team Selection Test, 1

Tags: geometry , square , perpendicular , equal segments

Let $P$ be a point on the side $AB$ of a square $ABCD$ and $Q$ a point on the side $BC$. Let $H$ be the foot of the perpendicular from $B$ to $PC$. Suppose that $BP = BQ$. Prove that $QH$ is perpendicular to $HD$.

2011 District Olympiad, 2

Tags: diagonal , trapezoid , perpendicular , geometry

The isosceles trapezoid $ABCD$ has perpendicular diagonals. The parallel to the bases through the intersection point of the diagonals intersects the non-parallel sides $[BC]$ and $[AD]$ in the points $P$, respectively $R$. The point $Q$ is symmetric of the point $P$ with respect to the midpoint of the segment $[BC]$. Prove that: a) $QR = AD$, b) $QR \perp AD$.

Croatia MO (HMO) - geometry, 2022.7

Tags: geometry , perpendicular

In the triangle $ABC$ holds $|AB| = |AC|$ and the inscribed circle touches the sides $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ at the points $D$, $E$ and $F$ respectively . The perpendicular from the point $D$ to the line $EF$ intersects the side $\overline{AB}$ at the point $G$, and the circles circumscribed around the triangles $AEF$ and $ABC$ intersect at the points $A $and $T$. Prove that the lines $T G$ and $T F$ are perpendicular.

2021 Bangladeshi National Mathematical Olympiad, 4

Tags: geometry , trapezium , perpendicular , egmo book

$ABCD$ is an isosceles trapezium such that $AD=BC$, $AB=5$ and $CD=10$. A point $E$ on the plane is such that $AE\perp{EC}$ and $BC=EC$. The length of $AE$ can be expressed as $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by any square number other than $1$. Find the value of $(a+b)$.

1952 Moscow Mathematical Olympiad, 225

Tags: tangent , geometry , perpendicular

From a point $C$, tangents $CA$ and $CB$ are drawn to a circle $O$. From an arbitrary point $N$ on the circle, perpendiculars $ND, NE, NF$ are drawn on $AB, CA$ and $CB$, respectively. Prove that the length of $ND$ is the mean proportional of the lengths of $NE$ and $NF$.

Kyiv City MO Seniors Round2 2010+ geometry, 2018.10.3.1

Tags: circumcircle , perpendicular , geometry

The point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AC$ intersects the circumscribed circle $\Delta ABO$ for second time at the point $X$. Prove that $XO \bot BC$.

2023 Yasinsky Geometry Olympiad, 1

Tags: perpendicular , geometry

Let $BD$ and $CE$ be the altitudes of triangle $ABC$ that intersect at point $H$. Let $F$ be a point on side $AC$ such that $FH\perp CE$. The segment $FE$ intersects the circumcircle of triangle $CDE$ at the point $K$. Prove that $HK\perp EF$ . (Matthew Kurskyi)

May Olympiad L2 - geometry, 2015.3

Tags: geometry , perpendicular , regular polygon

Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.

1972 All Soviet Union Mathematical Olympiad, 165

Tags: geometry , perpendicular

Let $O$ be the intersection point of the diagonals of the convex quadrangle $ABCD$ . Prove that the line drawn through the points of intersection of the medians of triangles $AOB$ and $COD$ is orthogonal to the line drawn through the points of intersection of the heights of triangles $BOC$ and $AOD$ .

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 $AE // MN$.

2017 Singapore Senior Math Olympiad, 2

Tags: geometry , right angle , perpendicular , cyclic quadrilateral , cyclic

In the cyclic quadrilateral $ABCD$, the sides $AB, DC$ meet at $Q$, the sides $AD,BC$ meet at $P, M$ is the midpoint of $BD$, If $\angle APQ=90^o$, prove that $PM$ is perpendicular to $AB$.

2006 MOP Homework, 2

Tags: geometry , equal angles , orthocenter , perpendicular , right angle , projection

Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.

2019 Azerbaijan BMO TST, 2

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

2015 Indonesia MO Shortlist, G6

Tags: geometry , equal segments , circumcircle , perpendicular

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

2009 Oral Moscow Geometry Olympiad, 3

Tags: geometry , incenter , perpendicular , incircle

Altitudes $AA_1$ and $BB_1$ are drawn in the acute-angled triangle $ABC$. Prove that the perpendicular drawn from the touchpoint of the inscribed circle with the side $BC$, on the line $AC$ passes through the center of the inscribed circle of the triangle $A_1CB_1$. (V. Protasov)

2017 Saudi Arabia JBMO TST, 3

Tags: tangent , geometry , fixed point , perpendicular , orthocenter

Let $(O)$ be a circle, and $BC$ be a chord of $(O)$ such that $BC$ is not a diameter. Let $A$ be a point on the larger arc $BC$ of $(O)$, and let $E, F$ be the feet of the perpendiculars from $B$ and $C$ to $AC$ and $AB$, respectively. 1. Prove that the tangents to $(AEF)$ at $E$ and $F$ intersect at a fixed point $M$ when $A$ moves on the larger arc $BC$ of $(O)$. 2. Let $T$ be the intersection of $EF$ and $BC$, and let $H$ be the orthocenter of $ABC$. Prove that $TH$ is perpendicular to $AM$.

2017 Romania National Olympiad, 2

Tags: perpendicular , geometry , angle

Consider the triangle $ABC$, with $\angle A= 90^o, \angle B = 30^o$, and $D$ is the foot of the altitude from $A$. Let the point $E \in (AD)$ such that $DE = 3AE$ and $F$ the foot of the perpendicular from $D$ to the line $BE$. a) Prove that $AF \perp FC$. b) Determine the measure of the angle $AFB$.

2009 All-Russian Olympiad Regional Round, 9.3

Tags: geometry , right angle , perpendicular

In an acute triangle $ABC$ the altitudes $AA_1$, $BB_1$, $CC_1$ are drawn. A line perpendicular to side $AC$ and passing through a point $A_1$, intersects the line $B_1C_1$ at point $D$. Prove that angle $ADC$ is right.

2018 Balkan MO Shortlist, G2

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

2017 Yasinsky Geometry Olympiad, 2

Tags: 3-dimensional geometry , 3d geometry , tetrahedron , geometry , equal angles , perpendicular

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

2011 Oral Moscow Geometry Olympiad, 5

Tags: angle chasing , perpendicular , angle , geometry

In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.

2021-IMOC, G5

Tags: geometry , radical axis , tangential quadrilateral , perpendicular

The incircle of a cyclic quadrilateral $ABCD$ tangents the four sides at $E$, $F$, $G$, $H$ in counterclockwise order. Let $I$ be the incenter and $O$ be the circumcenter of $ABCD$. Show that the line connecting the centers of $\odot(OEG)$ and $\odot(OFH)$ is perpendicular to $OI$.

1982 Polish MO Finals, 2

Tags: cyclic , parallel , perpendicular

In a cyclic quadrilateral $ABCD$ the line passing through the midpoint of $AB$ and the intersection point of the diagonals is perpendicular to $CD$. Prove that either the sides $AB$ and $CD$ are parallel or the diagonals are perpendicular.

2021 Regional Olympiad of Mexico West, 5

Tags: perpendicular , geometry

Let $ABC$ be a triangle such that $AC$ is its shortest side. A point $P$ is inside it and satisfies that $BP = AC$. Let $R$ be the midpoint of $BC$ and let $M$ be the midpoint of $AP$. Let $E$ be the intersection of $BP$ and $AC$. Prove that the bisector of angle $\angle BE A$ is perpendicular to segment $MR$.