Olympiad problems

2001 Singapore Team Selection Test, 1

In the acute triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to $BC$, let $E$ be the foot of the perpendicular from $D$ to $AC$, and let $F$ be a point on the line segment $DE$. Prove that $AF$ is perpendicular to $BE$ if and only if $FE/FD = BD/DC$

2010 Oral Moscow Geometry Olympiad, 4

Tags: parallelogram , geometry , perpendicular

From the vertex $A$ of the parallelogram $ABCD$, the perpendiculars $AM,AN$ on sides $BC,CD$ respectively. $P$ is the intersection point of $BN$ and $DM$. Prove that the lines $AP$ and $MN$ are perpendicular.

2014 Czech-Polish-Slovak Match, 3

Tags: geometry , convex quadrilateral , perpendicular

2021 New Zealand MO, 2

Tags: trapezoid , perpendicular , angle bisector , geometry

Let $ABCD$ be a trapezium such that $AB\parallel CD$. Let $E$ be the intersection of diagonals $AC$ and $BD$. Suppose that $AB = BE$ and $AC = DE$. Prove that the internal angle bisector of $\angle BAC$ is perpendicular to $AD$.

2018 Balkan MO Shortlist, G2

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

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 Hanoi Open Mathematics Competitions, 14

Tags: collinear , geometry , trapezoid , collinearity , perpendicular

Given trapezoid $ABCD$ with bases $AB \parallel CD$ ($AB < CD$). Let $O$ be the intersection of $AC$ and $BD$. Two straight lines from $D$ and $C$ are perpendicular to $AC$ and $BD$ intersect at $E$ , i.e. $CE \perp BD$ and $DE \perp AC$ . By analogy, $AF \perp BD$ and $BF \perp AC$ . Are three points $E , O, F$ located on the same line?

2021 Azerbaijan EGMO TST, 4

Tags: circles , orthocenter , perpendicular , geometry , incenter

Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively. a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$. b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.

2000 All-Russian Olympiad Regional Round, 9.7

Tags: perpendicular , geometry

On side $AB$ of triangle $ABC$, point $D$ is selected. Circle circumscribed around triangle $BCD$, intersects side $AC$ at point $M$, and the circumcircle of triangle $ACD$ intersects the side $BC$ at point $ N$ ($M,N \ne C$). Let $O$ be the circumcenter of the triangle $CMN$. Prove that line $OD$ is perpendicular to side $AB$.

1982 IMO Shortlist, 17

Tags: geometry , right triangle , triangle , perpendicular , similar triangles

The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.

2016 Switzerland Team Selection Test, Problem 8

Tags: bisector , geometry , perpendicular

Let $ABC$ be a triangle with $AB \neq AC$ and let $M$ be the middle of $BC$. The bisector of $\angle BAC$ intersects the line $BC$ in $Q$. Let $H$ be the foot of $A$ on $BC$. The perpendicular to $AQ$ passing through $A$ intersects the line $BC$ in $S$. Show that $MH \times QS=AB \times AC$.

2016 Romania National Olympiad, 4

Tags: geometry , right triangle , angle , perpendicular , isosceles

Consider the isosceles right triangle $ABC$, with $\angle A = 90^o$ and the point $M \in (BC)$ such that $\angle AMB = 75^o$. On the inner bisector of the angle $MAC$ take a point $F$ such that $BF = AB$. Prove that: a) the lines $AM$ and $BF$ are perpendicular; b) the triangle $CFM$ is isosceles.

2005 Cuba MO, 5

Tags: geometry , right angle , perpendicular

On the circumcircle of triangle $ABC$, point $P$ is taken in such a way that the perpendicular drawn by the point $P$ to the line $AC$ cuts the circle also at the point $Q$, the perpendicular drawn by the point $Q$ to the line $AB$ cuts the circle also at point R and the perpendicular drawn by point $R$ to the line BC cuts the circle also at the point $P$. Let $O$ be the center of this circle. Prove that $\angle POC = 90^o$ .

2020 Yasinsky Geometry Olympiad, 2

Tags: geometry , perpendicular , square

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$. (Ercole Suppa, Italy)

Champions Tournament Seniors - geometry, 2007.5

Tags: 3d geometry , perpendicular , area , geometry

The polyhedron $PABCDQ$ has the form shown in the figure. It is known that $ABCD$ is parallelogram, the planes of the triangles of the $PAC$ and $PBD$ mutually perpendicular, and also mutually perpendicular are the planes of triangles $QAC$ and $QBC$. Each face of this polyhedron is painted black or white so that the faces that have a common edge are painted in different colors. Prove that the sum of the squares of the areas of the black faces is equal to the sum of the squares of the areas of the white faces. [img]https://1.bp.blogspot.com/-UM5PKEGGWqc/X1V2cXAFmwI/AAAAAAAAMdw/V-Qr94tZmqkj3_q-5mkSICGF1tMu-b_VwCLcBGAsYHQ/s0/2007.5%2Bchampions%2Btourn.png[/img]

2018 Oral Moscow Geometry Olympiad, 2

Tags: right angle , diagonal , perpendicular , trapezoid , geometry

The diagonals of the trapezoid $ABCD$ are perpendicular ($AD//BC, AD>BC$) . Point $M$ is the midpoint of the side of $AB$, the point $N$ is symmetric of the center of the circumscribed circle of the triangle $ABD$ wrt $AD$. Prove that $\angle CMN = 90^o$. (A. Mudgal, India)

2022 OMpD, 4

Tags: geometry , perpendicular , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral and $M,N$ be the midpoints of $AB$, $CD$ respectively. The diagonals $AC$ and $BD$ intersect at $L$. Suppose that the circumcircle of $LMN$, with center $T$, intersects the circumcircle of $ABCD$ at two distinct points $X,Y$. If the line $MN$ intersects the line $XY$ at $S$ and the line $XM$ intersects the line $YN$ at $P$, prove that $PL$ is perpendicular to $ST$.

1992 All Soviet Union Mathematical Olympiad, 569

Tags: circles , geometry , perpendicular

Circles $C$ and $C'$ intersect at $O$ and $X$. A circle center $O$ meets $C$ at $Q$ and $R$ and meets $C'$ at $P$ and $S$. $PR$ and $QS$ meet at $Y$ distinct from $X$. Show that $\angle YXO = 90^o$.

2010 Junior Balkan Team Selection Tests - Romania, 3

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

Let $ABC$ be a triangle inscribed in the circle $(O)$. Let $I$ be the center of the circle inscribed in the triangle and $D$ the point of contact of the circle inscribed with the side $BC$. Let $M$ be the second intersection point of the bisector $AI$ with the circle $(O)$ and let $P$ be the point where the line $DM$ intersects the circle $(O)$ . Show that $PA \perp PI$.

2015 Indonesia MO, 6

Tags: circumcircle , geometry , equal segments , 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 $BP = CP$.

2013 Thailand Mathematical Olympiad, 9

Tags: geometry , perpendicular , ratio , parallel

Let $ABCD$ be a convex quadrilateral, and let $M$ and$ N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$, and point $Q$ is chosen on $AB$ so that $NQ \perp AB$. Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .

2011 Indonesia TST, 3

Tags: perpendicular , trigonometry , maximization , triangle , geometry

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?

2012 China Northern MO, 1

Tags: geometry , right triangle , perpendicular

As shown in figure, given right $\vartriangle ABC$ with $\angle C=90^o$. $I$ is the incenter. The line $BI$ intersects segment $AC$ at the point $D$ . The line passing through $D$ parallel to $AI$ intersects $BC$ at point $E$. The line $EI$ intersects segment $AB$ at point $F$. Prove that $DF \perp AI$. [img]https://cdn.artofproblemsolving.com/attachments/2/4/6fc94adb4ce12c3bf07948b8c57170ca01b256.png[/img]

2009 Postal Coaching, 2

Tags: geometry , polygon , max , cyclic , diagonal , area , perpendicular

Let $n \ge 4$ be an integer. Find the maximum value of the area of a $n$-gon which is inscribed in the circle of radius $1$ and has two perpendicular diagonals.

2017 Saudi Arabia BMO TST, 3

Tags: geometry , cyclic quadrilateral , similar triangles , perpendicular

Let $ABCD$ be a cyclic quadrilateral and triangles $ACD, BCD$ are acute. Suppose that the lines $AB$ and $CD$ meet at $S$. Denote by $E$ the intersection of $AC, BD$. The circles $(ADE)$ and $(BC E)$ meet again at $F$. a) Prove that $SF \perp EF.$ b) The point $G$ is taken out side of the quadrilateral $ABCD$ such that triangle $GAB$ and $FDC$ are similar. Prove that $GA+ FB = GB + FA$

2016 Novosibirsk Oral Olympiad in Geometry, 5

Tags: parallelogram , perpendicular , geometry

In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]