Olympiad problems

2016 Czech-Polish-Slovak Junior Match, 1

Let $AB$ be a given segment and $M$ be its midpoint. We consider the set of right-angled triangles $ABC$ with hypotenuses $AB$. Denote by $D$ the foot of the altitude from $C$. Let $K$ and $L$ be feet of perpendiculars from $D$ to the legs $BC$ and $AC$, respectively. Determine the largest possible area of the quadrilateral $MKCL$. Czech Republic

2003 Junior Tuymaada Olympiad, 7

Tags: tangent , geometry , angle bisector , perpendicular

Through the point $ K $ lying outside the circle $ \omega $, the tangents are drawn $ KB $ and $ KD $ to this circle ($ B $ and $ D $ are tangency points) and a line intersecting a circle at points $ A $ and $ C $. The bisector of angle $ ABC $ intersects the segment $ AC $ at the point $ E $ and circle $ \omega $ at $ F $. Prove that $ \angle FDE = 90^\circ $.

2008 District Olympiad, 3

Tags: geometry , 3d geometry , collinear , perpendicular , symmetric

Let $ABCDA' B' C' D '$ be a cube , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.

2022 Costa Rica - Final Round, 1

Tags: geometry , rhombus , perpendicular , perpendicularity

Let $\Gamma$ be a circle with center $O$. Consider the points $A$, $B$, $C$, $D$, $E$ and $F$ in $\Gamma$, with $D$ and $E$ in the (minor) arc $BC$ and $C$ in the (minor) arc $EF$, such that $DEFO$ is a rhombus and $\vartriangle ABC$ It is equilateral. Show that $\overleftrightarrow{BD}$ and $\overleftrightarrow{CE}$ are perpendicular.

Brazil L2 Finals (OBM) - geometry, 2013.3

Tags: midpoint , projection , geometry , circumcircle , perpendicular

Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.

2015 Irish Math Olympiad, 8

Tags: geometry , perpendicular , circumcircle

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 $\angle BAC$.

1999 All-Russian Olympiad Regional Round, 9.8

Tags: perpendicular , geometry

In triangle $ABC$ ($AB > BC$), $K$ and $M$ are the midpoints of sides $AB$ and $AC$, $O$ is the point of intersection of the angle bisectors. Let $P$ be the intersection point of lines $KM$ and $CO$, and the point $Q$ is such that $QP \perp KM$ and $QM \parallel BO$. Prove that $QO \perp AC$.

2001 BAMO, 2

Tags: collinear , geometry , perpendicular

Let $JHIZ$ be a rectangle, and let $A$ and $C$ be points on sides $ZI$ and $ZJ,$ respectively. The perpendicular from $A$ to $CH$ intersects line $HI$ in $X$ and the perpendicular from $C$ to $AH$ intersects line $HJ$ in $Y.$ Prove that $X,$ $Y,$ and $Z$ are collinear (lie on the same line).

1987 Tournament Of Towns, (160) 4

Tags: inradius , altitude , perpendicular , geometry

From point $M$ in triangle $ABC$ perpendiculars are dropped to each altitude. It can be shown that each of the line segments of altitudes, measured between the vertex and the foot of the perpendicular drawn to it, are of equal length. Prove that these lengths are each equal to the diameter of the circle inscribed in the triangle.

2011 Indonesia TST, 3

Tags: geometry , trigonometry , maximization , triangle , perpendicular

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?

2017 Bundeswettbewerb Mathematik, 3

Tags: incenter , centroid , perpendicular , geometry

Given is a triangle with side lengths $a,b$ and $c$, incenter $I$ and centroid $S$. Prove: If $a+b=3c$, then $S \neq I$ and line $SI$ is perpendicular to one of the sides of the triangle.

2018 Czech-Polish-Slovak Junior Match, 5

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

An acute triangle $ABC$ is given in which $AB <AC$. Point $E$ lies on the $AC$ side of the triangle, with $AB = AE$. The segment $AD$ is the diameter of the circumcircle of the triangle $ABC$, and point $S$ is the center of this arc $BC$ of this circle to which point $A$ does not belong. Point $F$ is symmetric of point $D$ wrt $S$. Prove that lines $F E$ and $AC$ are perpendicular.

Champions Tournament Seniors - geometry, 2019.2

Tags: geometry , perpendicular , equal segments , cyclic quadrilateral , arc midpoint

The quadrilateral $ABCD$ is inscribed in the circle and the lengths of the sides $BC$ and $DC$ are equal, and the length of the side $AB$ is equal to the length of the diagonal $AC$. Let the point $P$ be the midpoint of the arc $CD$, which does not contain point $A$, and $Q$ is the point of intersection of diagonals $AC$ and $BD$. Prove that the lines $PQ$ and $AB$ are perpendicular.

2020-21 IOQM India, 22

Tags: geometry , area , angle bisector , perpendicular

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 $ABC$?

2001 IMO Shortlist, 7

Tags: geometry , circumcircle , perimeter , perpendicular

Let $O$ be an interior point of acute triangle $ABC$. Let $A_1$ lie on $BC$ with $OA_1$ perpendicular to $BC$. Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly. Prove that $O$ is the circumcenter of $ABC$ if and only if the perimeter of $A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1, BC_1A_1$, and $CA_1B_1$.

2020 Yasinsky Geometry Olympiad, 2

Tags: perpendicular , square , geometry

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)

May Olympiad L2 - geometry, 2003.2

Tags: geometry , rectangle , perpendicular

Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$.

2017 Saudi Arabia BMO TST, 3

Tags: geometry , similar triangles , cyclic quadrilateral , 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 $GA+ FB = GB + FA$

2011 Sharygin Geometry Olympiad, 3

Tags: geometry , tetrahedron , perpendicular , solid geometry

Given two tetrahedrons $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$. Consider six pairs of edges $A_iA_j$ and $B_kB_l$, where ($i, j, k, l$) is a transposition of numbers ($1, 2, 3, 4$) (for example $A_1A_2$ and $B_3B_4$). It is known that for all but one such pairs the edges are perpendicular. Prove that the edges in the remaining pair also are perpendicular.

2018 Hanoi Open Mathematics Competitions, 9

Tags: geometry , perpendicular , circumcenter , similar triangles

Let $ABC$ be acute, non-isosceles triangle, inscribed in the circle $(O)$. Let $D$ be perpendicular projection of $A$ onto $BC$, and $E, F$ be perpendicular projections of $D$ onto $CA,AB$ respectively. (a) Prove that $AO \perp EF$. (b) The line $AO$ intersects $DE,DF$ at $I,J$ respectively. Prove that $\vartriangle DIJ$ and $\vartriangle ABC$ are similar. (c) Prove that circumcenter of $\vartriangle DIJ$ is equidistant from $B$ and $C$

2014 Contests, 2

Tags: geometry , circumcircle , perpendicular

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 $PQ$.

2018 Dutch IMO TST, 2

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

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 $AB$.

1998 Poland - Second Round, 6

Tags: perpendicular , tetrahedron , 3d geometry , geometry

Prove that the edges $AB$ and $CD$ of a tetrahedron $ABCD$ are perpendicular if and only if there exists a parallelogram $CDPQ$ such that $PA = PB = PD$ and $QA = QB = QC$.

2013 Oral Moscow Geometry Olympiad, 5

Tags: geometry , perpendicular , circumcenter

In the acute-angled triangle $ABC$, let $AP$ and $BQ$ be the altitudes, $CM$ be the median . Point $R$ is the midpoint of $CM$. Line $PQ$ intersects line $AB$ at $T$. Prove that $OR \perp TC$, where $O$ is the center of the circumscribed circle of triangle $ABC$.

1986 Polish MO Finals, 6

Tags: geometry , concyclic , cyclic quadrilateral , perpendicular , altitude

$ABC$ is a triangle. The feet of the perpendiculars from $B$ and $C$ to the angle bisector at $A$ are $K, L$ respectively. $N$ is the midpoint of $BC$, and $AM$ is an altitude. Show that $K,L,N,M$ are concyclic.