Olympiad problems

Ukraine Correspondence MO - geometry, 2011.7

Let $ABCD$ be a trapezoid in which $AB \parallel CD$ and $AB = 2CD$. A line $\ell$ perpendicular to $CD$ was drawn through point $C$. A circle with center at point $D$ and radius $DA$ intersects line $\ell$ at points $P$ and $Q$. Prove that $AP \perp BQ$.

2006 Kazakhstan National Olympiad, 6

Tags: tetrahedron , perpendicular , geometry , 3d geometry , plane

In the tetrahedron $ ABCD $ from the vertex $ A $, the perpendiculars $ AB '$, $ AC' $ are drawn, $ AD '$ on planes dividing dihedral angles at edges $ CD $, $ BD $, $ BC $ in half. Prove that the plane $ (B'C'D ') $ is parallel to the plane $ (BCD) $.

1990 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: perpendicular , trapezoid , geometry , collinear

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.

2014 Sharygin Geometry Olympiad, 5

Tags: perpendicular , geometry , circles

In triangle $ABC$ $\angle B = 60^o, O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$. The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$. (D. Shvetsov)

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 $AFB$.

2006 Tournament of Towns, 2

Tags: geometry , perpendicular , incenter

The incircle of the quadrilateral $ABCD$ touches $AB,BC, CD$ and $DA$ at $E, F,G$ and $H$ respectively. Prove that the line joining the incentres of triangles $HAE$ and $FCG$ is perpendicular to the line joining the incentres of triangles $EBF$ and $GDH$. (4)

2017 Singapore Senior Math Olympiad, 2

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

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 $AB$.

VMEO III 2006 Shortlist, G1

Tags: perpendicular , geometry

Given a circle $(O)$ and a point $P$ outside that circle. $M$ is a point running on the circle $(O)$. The circle with center $I$ and diameter $PM$ intersects circle $(O)$ again at $N$. The tangent of $(I)$ at $P$ intersects $MN$ at $Q$. The line through $Q$ perpendicular to $PO$ intersects $PM$ at $ A$. $AN$ intersects $(O)$ further at $ B$. $BM$ intersects $PO$ at $C$. Prove that $AC$ is perpendicular to $OQ$.

2014 China Northern MO, 1

Tags: geometry , perpendicular , isosceles

As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles. [img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]

2016 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: perpendicular , orthocenter , midpoint , circumcenter , geometry

Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.

2015 Germany Team Selection Test, 2

Tags: perpendicular , geometry , circumcircle , orthogonal , incenter

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]

1976 Polish MO Finals, 4

Tags: perpendicular , geometry

The diagonals of some quadrilateral with sides $a,b,c,d$ are perpendicular. Prove that the diagonals of any other quadrilateral with sides $a,b,c,d $ also are perpendicular

2021 Czech-Polish-Slovak Junior Match, 2

Tags: geometry , perpendicular , angle bisector

An acute triangle $ABC$ is given. Let us denote by $D$ and $E$ the orthogonal projections, respectively of points $ B$ and $C$ on the bisector of the external angle $BAC$. Let $F$ be the point of intersection of the lines $BE$ and $CD$. Show that the lines $AF$ and $DE$ are perpendicular.

2011 Sharygin Geometry Olympiad, 4

Tags: perpendicular , incenter , circumcircle , tangent circle , geometry

Point $D$ lies on the side $AB$ of triangle $ABC$. The circle inscribed in angle $ADC$ touches internally the circumcircle of triangle $ACD$. Another circle inscribed in angle $BDC$ touches internally the circumcircle of triangle $BCD$. These two circles touch segment $CD$ in the same point $X$. Prove that the perpendicular from $X$ to $AB$ passes through the incenter of triangle $ABC$

2015 Regional Olympiad of Mexico Center Zone, 5

Tags: perpendicular , geometry

In the triangle $ABC$, we have that $M$ and $N$ are points on $AB$ and $AC$, respectively, such that $BC$ is parallel to $MN$. A point $D$ is chosen inside the triangle $AMN$. Let $E$ and $F$ be the points of intersection of $MN$ with $BD$ and $CD$, respectively. Show that the line joining the centers of the circles circumscribed to the triangles $DEN$ and $DFM$ is perpendicular to $AD$.

1999 Kazakhstan National Olympiad, 7

Tags: perpendicular , sphere , 3d geometry , area of triangle , fixed point , geometry

On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $

2014 Contests, 2

Tags: perpendicular , circumcircle , geometry

The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.

2003 District Olympiad, 2

Tags: equal segments , geometry , perpendicular , parallel , right triangle

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) $AH \perp BC$.

Russian TST 2015, P2

Tags: perpendicular , geometry , concyclic , altitude

Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.

2017 Ecuador Juniors, 3

Tags: perpendicular , isosceles , geometry

Given an isosceles triangle $ABC$ with $AB = AC$. Let $O$ be the circumcenter of $ABC$, $D$ the midpoint of $AB$ and $E$ the centroid of $ACD$. Prove that $CD \perp EO$.

2015 Irish Math Olympiad, 8

Tags: perpendicular , circumcircle , geometry

In triangle $\triangle ABC$, the angle $\angle BAC$ is less than $90^o$. The perpendiculars from $C$ on $AB$ and from $B$ on $AC$ intersect the circumcircle of $\triangle ABC$ again at $D$ and $E$ respectively. If $|DE| =|BC|$, find the measure of the angle $\angle BAC$.

2012 Indonesia MO, 3

Tags: geometry , circumcircle , angle bisector , perpendicular

Given an acute triangle $ABC$ with $AB>AC$ that has circumcenter $O$. Line $BO$ and $CO$ meet the bisector of $\angle BAC$ at $P$ and $Q$, respectively. Moreover, line $BQ$ and $CP$ meet at $R$. Show that $AR$ is perpendicular to $BC$. [i]Proposer: Soewono and Fajar Yuliawan[/i]

2017 Balkan MO Shortlist, G6

Tags: perpendicular , geometry , concurrency , isosceles

Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B, ABB_A , ACA_C,ACC_A ,BCB_C$ and $BCC_B$, so that $$AB = AB_A = BA_B, AC = AC_A=CA_C, BC = BC_B = CB_C$$ and $$\angle BAB_A = \angle ABA_B =\angle CAC_A=\angle ACA_C= \angle BCB_C =\angle CBC_B = a < 90^o$$. Prove that the perpendiculars from $A$ to $B_AC_A$, from $B$ to $A_BC_B$ and from $C$ to $A_CB_C$ are concurrent

2015 Czech-Polish-Slovak Junior Match, 3

Tags: equal segments , geometry , perpendicular , geometric inequality

Different points $A$ and $D$ are on the same side of the line $BC$, with $|AB| = | BC|= |CD|$ and lines $AD$ and $BC$ are perpendicular. Let $E$ be the intersection point of lines $AD$ and $BC$. Prove that $||BE| - |CE|| < |AD| \sqrt3$

2009 Regional Olympiad of Mexico Center Zone, 5

Tags: geometry , perpendicular

Let $ABC$ be a triangle and let $D$ be the foot of the altitude from $A$. Let points $E$ and $F$ on a line through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is perpendicular to $CF$, where $E$ and $F$ are points other than the point $D$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.