Found problems: 412
2011 Tournament of Towns, 4
Four perpendiculars are drawn from four vertices of a convex pentagon to the opposite sides. If these four lines pass through the same point, prove that the perpendicular from the fifth vertex to the opposite side also passes through this point.
2015 Bosnia and Herzegovina Junior BMO TST, 3
Let $AD$ be an altitude of triangle $ABC$, and let $M$, $N$ and $P$ be midpoints of $AB$, $AD$ and $BC$, respectively. Furthermore let $K$ be a foot of perpendicular from point $D$ to line $AC$, and let $T$ be point on extension of line $KD$ (over point $D$) such that $\mid DT \mid = \mid MN \mid + \mid DK \mid$. If $\mid MP \mid = 2 \cdot \mid KN \mid$, prove that $\mid AT \mid = \mid MC \mid$.
2015 IMAR Test, 3
Let $ABC$ be a triangle, let $A_1, B_1, C_1$ be the antipodes of the vertices $A, B, C$, respectively, in the circle $ABC$, and let $X$ be a point in the plane $ABC$, collinear with no two vertices of the triangle $ABC$. The line through $B$, perpendicular to the line $XB$, and the line through $C$, perpendicular to the line $XC$, meet at $A_2$, the points $B_2$ and $C_2$ are defined similarly. Show that the lines $A_1A_2, B_1B_2$ and $C_1C_2$ are concurrent.
2013 Oral Moscow Geometry Olympiad, 5
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 $ABC$.
2018 Bundeswettbewerb Mathematik, 3
Let $H$ be the orthocenter of the acute triangle $ABC$. Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$. Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$.
Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.
2018 Dutch IMO TST, 2
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 $AB$.
the 7th XMO, 1
As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]
2011 Sharygin Geometry Olympiad, 3
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.
2003 Oral Moscow Geometry Olympiad, 4
In triangle $ABC$, $M$ is the point of intersection of the medians, $O$ is the center of the inscribed circle, $A', B', C'$ are the touchpoints with the sides $BC, CA, AB$, respectively. Prove that if $CA'= AB$, then $OM$ and $AB$ are perpendicular.
PS. There is a a typo
1997 Abels Math Contest (Norwegian MO), 2a
Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.
2009 Junior Balkan Team Selection Tests - Romania, 2
Let $ABCD$ be a quadrilateral. The diagonals $AC$ and $BD$ are perpendicular at point $O$. The perpendiculars from $O$ on the sides of the quadrilateral meet $AB, BC, CD, DA$ at $M, N, P, Q$, respectively, and meet again $CD, DA, AB, BC$ at $M', N', P', Q'$, respectively. Prove that points $M, N, P, Q, M', N', P', Q'$ are concyclic.
Cosmin Pohoata
2000 Argentina National Olympiad, 2
Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.
Estonia Open Junior - geometry, 1998.1.3
Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$ at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.
1983 Austrian-Polish Competition, 6
Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$.
2004 Estonia National Olympiad, 5
Let $n$ and $c$ be coprime positive integers. For any integer $i$, denote by $i' $ the remainder of division of product $ci$ by $n$. Let $A_o.A_1,A_2,...,A_{n-1}$ be a regular $n$-gon. Prove that
a) if $A_iA_j \parallel A_kA_i$ then $A_{i'}A_{j'} \parallel A_{k'}A_{i'}$
b) if $A_iA_j \perp A_kA_l$ then $A_{i'}A_{j'} \perp A_{k'}A_{l'}$
2011 Bosnia and Herzegovina Junior BMO TST, 3
In isosceles triangle $ABC$ ($AC=BC$), angle bisector $\angle BAC$ and altitude $CD$ from point $C$ intersect at point $O$, such that $CO=3 \cdot OD$. In which ratio does altitude from point $A$ on side $BC$ divide altitude $CD$ of triangle $ABC$
Brazil L2 Finals (OBM) - geometry, 2003.3
The triangle $ABC$ is inscribed in the circle $S$ and $AB <AC$. The line containing $A$ and is perpendicular to $BC$ meets $S$ in $P$ ($P \ne A$). Point $X$ is on the segment $AC$ and the line $BX$ intersects $S$ in $Q$ ($Q \ne B$). Show that $BX = CX$ if, and only if, $PQ$ is a diameter of $S$.
2018 Danube Mathematical Competition, 2
Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $ \angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.
2004 Singapore MO Open, 3
Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$. A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$. Let $E$ be a point on the segment $AD$, different from $A$ and $D$. The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$. The line $BE$ intersects $\Gamma_2$ at $M$ and $N$.
(i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$.
(ii) If the centre of $\Gamma_3$ is $O$, prove that $OD$ is perpendicular to $BC$.
2017 Saudi Arabia JBMO TST, 3
Let $(O)$ be a circle, and $BC$ be a chord of $(O)$ such that $BC$ is not a diameter. Let $A$ be a point on the larger arc $BC$ of $(O)$, and let $E, F$ be the feet of the perpendiculars from $B$ and $C$ to $AC$ and $AB$, respectively.
1. Prove that the tangents to $(AEF)$ at $E$ and $F$ intersect at a fixed point $M$ when $A$ moves on the larger arc $BC$ of $(O)$.
2. Let $T$ be the intersection of $EF$ and $BC$, and let $H$ be the orthocenter of $ABC$. Prove that $TH$ is perpendicular to $AM$.
2018 Saudi Arabia BMO TST, 4
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 $H$.
2019 Greece JBMO TST, 1
Consider an acute triangle $ABC$ with $AB>AC$ inscribed in a circle of center $O$. From the midpoint $D$ of side $BC$ we draw line $(\ell)$ perpendicular to side $AB$ that intersects it at point $E$. If line $AO$ intersects line $(\ell)$ at point $Z$, prove that points $A,Z,D,C$ are concyclic.
2020 Dutch IMO TST, 4
Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.
1977 IMO Longlists, 22
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.
2019 Canadian Mathematical Olympiad Qualification, 6
Pentagon $ABCDE$ is given in the plane. Let the perpendicular from $A$ to line $CD$ be $F$, the perpendicular from $B$ to $DE$ be $G$, from $C$ to $EA$ be $H$, from $D$ to $AB$ be $I$,and from $E$ to $BC$ be $J$. Given that lines $AF,BG,CH$, and $DI$ concur, show that they also concur with line $EJ$.