Olympiad problems

1952 Moscow Mathematical Olympiad, 220

A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.

2017 Saudi Arabia IMO TST, 2

Tags: cyclic quadrilateral , parallel , geometry , perpendicular

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 $IQ$.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.2

Tags: rectangle , angle , perpendicular , geometry

Given a rectangle $ABCD$ with $AB> BC$. On the side $CD$, take a point $L$ such that $BL$ and $AC$ are perpendicular. Let $K$ be the intersection point of segments $BL$ and $AC$. It is known that segments $AL$. and $DK$ are perpendicular. Find $\angle ACB.$

2016 Saudi Arabia IMO TST, 2

Tags: circles , perpendicular , circumcircle , geometry

Let $ABC$ be a triangle inscribed in the circle $(O)$ and $P$ is a point inside the triangle $ABC$. Let $D$ be a point on $(O)$ such that $AD \perp AP$. The line $CD$ cuts the perpendicular bisector of $BC$ at $M$. The line $AD$ cuts the line passing through $B$ and is perpendicular to $BP$ at $Q$. Let $N$ be the reflection of $Q$ through $M$. Prove that $CN \perp CP$.

2020-21 IOQM India, 22

Tags: angle bisector , geometry , perpendicular , area

In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. If $PQ = 7, QR = 6$ and $RS = 8$, what is the area of triangle $ABC$?

2017 Federal Competition For Advanced Students, P2, 5

Tags: midpoint , perpendicular , geometry

Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$, respectively. Let the common point of $DF$ and the altitude through $B$ be $P$. The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$. Furthermore, $EQ$ intersects the altitude through $A$ in $N$. Prove that $N$ is the midpoint of $AH$. Proposed by Karl Czakler

1998 ITAMO, 4

Tags: trapezoid , geometry , perpendicular

Let $ABCD$ be a trapezoid with the longer base $AB$ such that its diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of the triangle $ABC$ and $E$ be the intersection of the lines $OB$ and $CD$. Prove that $BC^2 = CD \cdot CE$.

2016 Indonesia MO, 1

Tags: angle bisector , projection , perpendicular , geometry

Let $ABCD$ be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point $O$. Let $E,F,G,H$ be the orthogonal projections of $O$ on sides $AB,BC,CD,DA$ respectively. a. Prove that $\angle EFG + \angle GHE = 180^o$ b. Prove that $OE$ bisects angle $\angle FEH$ .

2010 Dutch IMO TST, 1

Tags: equal segments , geometry , perpendicular

Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

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 $AOD$ .

Swiss NMO - geometry, 2007.4

Tags: geometry , perpendicular , orthocenter

Let $ABC$ be an acute-angled triangle with $AB> AC$ and orthocenter $H$. Let $D$ the projection of $A$ on $BC$. Let $E$ be the reflection of $C$ wrt $D$. The lines $AE$ and $BH$ intersect at point $S$. Let $N$ be the midpoint of $AE$ and let $M$ be the midpoint of $BH$. Prove that $MN$ is perpendicular to $DS$.

Croatia MO (HMO) - geometry, 2020.7

Tags: geometry , perpendicular

A circle of diameter $AB$ is given. There are points $C$ and $ D$ on this circle, on different sides of the diameter such that holds $AC <BC$ or $AC<AD$. The point $P$ lies on the segment $BC$ and $\angle CAP = \angle ABC$. The perpendicular from the point $C$ to the line $AB$ intersects the direction $BD$ at the point $Q$. Lines $PQ$ and $AD$ intersect at point $R$, and the lines $PQ$ and $CD$ intersect at point $T$. If $AR=RQ$, prove that the lines $AT$ and $PQ$ are perpendicular.

2020 Ukrainian Geometry Olympiad - April, 1

Tags: geometry , isosceles , angle bisector , perpendicular

In triangle $ABC$, bisectors are drawn $AA_1$ and $CC_1$. Prove that if the length of the perpendiculars drawn from the vertex $B$ on lines $AA1$ and $CC_1$ are equal, then $\vartriangle ABC$ is isosceles.

2021 OMpD, 2

Tags: geometry , perpendicular , arc midpoint

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $D$ the midpoint of the arc $AC$ of $\Gamma$ that does not contain $B$. If $O$ is the center of $\Gamma$ and I is the incenter of $ABC$, prove that $OI$ is perpendicular to $BD$ if and only if $AB + BC = 2AC$.

Kyiv City MO Seniors 2003+ geometry, 2015.11.4.1

Tags: angle bisector , perpendicular , equal segments , geometry

On the bisector of the angle $ BAC $ of the triangle $ ABC $ we choose the points $ {{B} _ {1}}, \, \, {{C} _ {1}} $ for which $ B {{B} _ {1 }}\perp AB $, $ C {{C} _ {1}} \perp AC $. The point $ M $ is the midpoint of the segment $ {{B} _ {1}} {{C} _ {1}} $. Prove that $ MB = MC $.

2006 Sharygin Geometry Olympiad, 19

Tags: geometry , incenter , perpendicular , circumcenter , orthocenter , midpoint

Through the midpoints of the sides of the triangle $T$, straight lines are drawn perpendicular to the bisectors of the opposite angles of the triangle. These lines formed a triangle $T_1$. Prove that the center of the circle circumscribed about $T_1$ is in the midpoint of the segment formed by the center of the inscribed circle and the intersection point of the heights of triangle $T$.

Kyiv City MO Juniors 2003+ geometry, 2018.7.41

Tags: geometry , angle , perpendicular , midpoint

In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.

2014 Belarus Team Selection Test, 1

Tags: geometry , incircle , perpendicular

Given triangle $ABC$ with $\angle A = a$. Let $AL$ be the bisector of the triangle $ABC$. Let the incircle of $\vartriangle ABC$ touch the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $X$ be the intersection point of the lines $AQ$ and $LP$. Prove that the lines $BX$ and $AL$ are perpendicular. (V. Karamzin)

2015 Dutch IMO TST, 4

Tags: perpendicular , parallel , circles , geometry

Let $\Gamma_1$ and $\Gamma_2$ be circles - with respective centres $O_1$ and $O_2$ - that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ in $A$ and $C$ and the line $O_2A$ intersects $\Gamma_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\Gamma_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

2000 Switzerland Team Selection Test, 1

Tags: geometry , arc midpoint , perpendicular

A convex quadrilateral $ABCD$ is inscribed in a circle. Show that the line connecting the midpoints of the arcs $AB$ and $CD$ and the line connecting the midpoints of the arcs $BC$ and $DA$ are perpendicular.

2005 Oral Moscow Geometry Olympiad, 4

Tags: geometry , hexagon , perpendicular , equal segments , right angle

Given a hexagon $ABCDEF$, in which $AB = BC, CD = DE, EF = FA$, and angles $A$ and $C$ are right. Prove that lines $FD$ and $BE$ are perpendicular. (B. Kukushkin)

1957 Moscow Mathematical Olympiad, 359

Tags: geometry , locus , perpendicular

Straight lines $OA$ and $OB$ are perpendicular. Find the locus of endpoints $M$ of all broken lines $OM$ of length $\ell$ which intersect each line parallel to $OA$ or $OB$ at not more than one point.

2010 Dutch IMO TST, 1

Tags: geometry , perpendicular , equal segments

Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

1996 Argentina National Olympiad, 4

Tags: geometry , parallelogram , perpendicular , equal angles , parallel

Let $ABCD$ be a parallelogram with center $O$ such that $\angle BAD <90^o$ and $\angle AOB> 90^o$. Consider points $A_1$ and $B_1$ on the rays $OA$ and $OB$ respectively, such that $A_1B_1$ is parallel to $AB$ and $\angle A_1B_1C = \frac12 \angle ABC$. Prove that $A_1D$ is perpendicular to $B_1C$.

2016 Saudi Arabia BMO TST, 2

Tags: perpendicular , tangent , geometry

Let $A$ be a point outside the circle $\omega$. Two points $B, C$ lie on $\omega$ such that $AB, AC$ are tangent to $\omega$. Let $D$ be any point on $\omega$ ($D$ is neither $B$ nor $C$) and $M$ the foot of perpendicular from $B$ to $CD$. The line through $D$ and the midpoint of $BM$ meets $\omega$ again at $P$. Prove that $AP \perp CP$