Olympiad problems

1972 Bulgaria National Olympiad, Problem 5

In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral. (a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed. (b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold? [i]H. Lesov[/i]

2009 All-Russian Olympiad Regional Round, 9.3

Tags: right angle , perpendicular , geometry

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.

2011 Oral Moscow Geometry Olympiad, 1

Tags: rectangle , equal segments , perpendicular , geometry

The bisector of angle $B$ and the bisector of external angle $D$ of rectangle $ABCD$ intersect side $AD$ and line $AB$ at points $M$ and $K$, respectively. Prove that the segment $MK$ is equal and perpendicular to the diagonal of the rectangle.

2021 Regional Olympiad of Mexico West, 5

Tags: geometry , perpendicular

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 $MR$.

2010 Greece Junior Math Olympiad, 2

Tags: equilateral , equal segments , perpendicular , geometry , rectangle

Let $ABCD$ be a rectangle with sides $AB=a$ and $BC=b$. Let $O$ be the intersection point of it's diagonals. Extent side $BA$ towards $A$ at a segment $AE=AO$, and diagonal $DB$ towards $B$ at a segment $BZ=BO$. If the triangle $EZC$ is an equilateral, then prove that: i) $b=a\sqrt3$ ii) $AZ=EO$ iii) $EO \perp ZD$

2011 Belarus Team Selection Test, 1

Tags: perpendicular , geometry , orthocenter , incenter

In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$ A. Voidelevich

2015 Hanoi Open Mathematics Competitions, 12

Tags: perpendicular , geometry , isosceles

Give an isosceles triangle $ABC$ at $A$. Draw ray $Cx$ being perpendicular to $CA, BE$ perpendicular to $Cx$ ($E \in Cx$).Let $M$ be the midpoint of $BE$, and $D$ be the intersection point of $AM$ and $Cx$. Prove that $BD \perp BC$.

2001 BAMO, 2

Tags: perpendicular , collinear , geometry

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).

2021 Saudi Arabia Training Tests, 6

Tags: perpendicular , tangent , geometry

Let $A$ be a point lies outside circle $(O)$ and tangent lines $AB$, $AC$ of $(O)$. Consider points $D, E, M$ on $(O)$ such that $MD = ME$. The line $DE$ cuts $MB$, $MC$ at $R, S$. Take $X \in OB$, $Y \in OC$ such that $RX, SY \perp DE$. Prove that $XY \perp AM$.

2006 Cuba MO, 9

Tags: equal angles , perpendicular , geometry

In the cyclic quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $P$. Let $O$ be the center of the circumcircle $ABCD$, and $E$ a point of the extension of $OC$ beyond $C$. A parallel line to $CD$ is drawn through $E$ that cuts the extension of $OD$ beyonf $D$ at $F$. Let $Q$ be a point interior to $ABCD$, such that $\angle AFQ = \angle BEQ$ and $\angle FAQ = \angle EBQ$. Prove that $PQ \perp CD$.

2023 Sharygin Geometry Olympiad, 5

Tags: perpendicular , geometry

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.

2010 Brazil Team Selection Test, 1

Tags: perpendicular , geometry

Let $ABC$ be an acute triangle and $D$ a point on the side $AB$. The circumcircle of triangle $BCD$ cuts the side $AC$ again at $E$ .The circumcircle of triangle $ACD$ cuts the side $BC$ again at $F$. If $O$ is the circumcenter of the triangle $CEF$. Prove that $OD$ is perpendicular to $AB$.

Cono Sur Shortlist - geometry, 2009.G4

Tags: altitude , geometry , perpendicular , incenter

Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.

2018 Thailand TSTST, 3

Tags: fixed , fixed point , geometry , perpendicular

Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$. (a) Prove that $P$ is independent of the choice of $A$. (b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.

May Olympiad L2 - geometry, 2003.2

Tags: perpendicular , rectangle , geometry

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 $MN$.

2000 Belarus Team Selection Test, 8.1

Tags: equal segments , geometry , perpendicular

The diagonals of a convex quadrilateral $ABCD$ with $AB = AC = BD$ intersect at $P$, and $O$ and $I$ are the circumcenter and incenter of $\vartriangle ABP$, respectively. Prove that if $O \ne I$ then $OI$ and $CD$ are perpendicular

1954 Moscow Mathematical Olympiad, 265

Tags: vector , convex polygon , perpendicular

From an arbitrary point $O$ inside a convex $n$-gon, perpendiculars are drawn on (extensions of the) sides of the $n$-gon. Along each perpendicular a vector is constructed, starting from $O$, directed towards the side onto which the perpendicular is drawn, and of length equal to half the length of the corresponding side. Find the sum of these vectors.

2011 Abels Math Contest (Norwegian MO), 2a

Tags: diagonal , geometry , perpendicular , sidelengths

In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular. You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).

2015 Dutch IMO TST, 4

Tags: circles , perpendicular , parallel , 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 $O_2C$.

Champions Tournament Seniors - geometry, 2019.2

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

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.

2018 India PRMO, 10

Tags: perpendicular , median , geometry

In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.

May Olympiad L2 - geometry, 2015.3

Tags: perpendicular , regular polygon , geometry

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

2006 Bosnia and Herzegovina Team Selection Test, 5

Tags: perpendicular , geometry , isosceles

Triangle $ABC$ is inscribed in circle with center $O$. Let $P$ be a point on arc $AB$ which does not contain point $C$. Perpendicular from point $P$ on line $BO$ intersects side $AB$ in point $S$, and side $BC$ in $T$. Perpendicular from point $P$ on line $AO$ intersects side $AB$ in point $Q$, and side $AC$ in $R$. (i) Prove that triangle $PQS$ is isosceles (ii) Prove that $\frac{PQ}{QR}=\frac{ST}{PQ}$

2003 Poland - Second Round, 2

Tags: bisector , perpendicular , geometry , cyclic quadrilateral

The quadrilateral $ABCD$ is inscribed in the circle $o$. Bisectors of angles $DAB$ and $ABC$ intersect at point $P$, and bisectors of angles $BCD$ and $CDA$ intersect in point $Q$. Point $M$ is the center of this arc $BC$ of the circle $o$ which does not contain points $D$ and $A$. Point $N$ is the center of the arc $DA$ of the circle $o$, which does not contain points $B$ and $C$. Prove that the points $P$ and $Q$ lie on the line perpendicular to $MN$.

2015 Bundeswettbewerb Mathematik Germany, 4

Tags: perpendicular , distance , incenter , geometry , circumcircle

Let $ABC$ be a triangle, such that its incenter $I$ and circumcenter $U$ are distinct. For all points $X$ in the interior of the triangle let $d(X)$ be the sum of distances from $X$ to the three (possibly extended) sides of the triangle. Prove: If two distinct points $P,Q$ in the interior of the triangle $ABC$ satisfy $d(P)=d(Q)$, then $PQ$ is perpendicular to $UI$.