This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 131

2006 MOP Homework, 2
Tags: geometry , equal angles , orthocenter , perpendicular , right angle , projection
Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.

2017 Ukrainian Geometry Olympiad, 4
Tags: geometry , parallelogram , circumcircle , circumcenter , right angle
Let $ABCD$ be a parallelogram and $P$ be an arbitrary point of the circumcircle of $\Delta ABD$, different from the vertices. Line $PA$ intersects the line $CD$ at point $Q$. Let $O$ be the center of the circumcircle $\Delta PCQ$. Prove that $\angle ADO = 90^o$.

2009 All-Russian Olympiad Regional Round, 9.3
Tags: geometry , right angle , perpendicular
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.

VI Soros Olympiad 1999 - 2000 (Russia), 9.9
Tags: geometric inequality , right angle , geometry
The center of a circle, the radius of which is $r$, lies on the bisector of the right angle $A$ at a distance $a$ from its sides ($a > r$). A tangent to the circle intersects the sides of the angle at points $B$ and $C$. Find the smallest possible value of the area of triangle $ABC$.

Brazil L2 Finals (OBM) - geometry, 2004.5
Tags: circumcenter , right triangle , right angle , tangent , geometry
Let $D$ be the midpoint of the hypotenuse $AB$ of a right triangle $ABC$. Let $O_1$ and $O_2$ be the circumcenters of the $ADC$ and $DBC$ triangles, respectively. a) Prove that $\angle O_1DO_2$ is right. b) Prove that $AB$ is tangent to the circle of diameter $O_1O_2$ .

1996 Singapore Team Selection Test, 1
Tags: geometry , incenter , equal segments , right angle
Let $C, B, E$ be three points on a straight line $\ell$ in that order. Suppose that $A$ and $D$ are two points on the same side of $\ell$ such that (i) $\angle ACE = \angle CDE = 90^o$ and (ii) $CA = CB = CD$. Let $F$ be the point of intersection of the segment $AB$ and the circumcircle of $\vartriangle ADC$. Prove that $F$ is the incentre of $\vartriangle CDE$.

2014 Costa Rica - Final Round, 5
Tags: geometry , excircle , incircle , right angle
Let $ABC$ be a triangle, with $A'$, $B'$, and $C'$ the points of tangency of the incircle with $BC$, $CA$, and $AB$ respectively. Let $X$ be the intersection of the excircle with respect to $A$ with $AB$, and $M$ the midpoint of $BC$. Let $D$ be the intersection of $XM$ with $B'C'$. Show that $\angle C'A'D' = 90^o$.

2018 Oral Moscow Geometry Olympiad, 2
Tags: diagonal , geometry , trapezoid , perpendicular , right angle
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)

2015 BMT Spring, 3
Tags: geometry , right angle
A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$. .

2009 Regional Olympiad of Mexico Center Zone, 1
Tags: geometry , right angle
Let $\Gamma$ be a circle with the center $O$ and let $A$, $A ^ \prime $ be two diametrically opposite points in $\Gamma$. Let $P$ be the midpoint of $OA ^ \prime$ and $\ell$ a line that passes through $P$, different from the line $AA ^ \prime$ and different from the line perpendicular on $AA ^ \prime$. Let $B$ and $C$ be the intersection points of $\ell$ with $\Gamma$, let $H$ be the foot of the altitude from $A$ on $BC$, let $M$ be the midpoint of $BC$, and let $D$ be the intersection of the line $A ^ \prime M$ with $AH$. Show that the angle $\angle ADO = 90 ^ \circ $.

2021 Saudi Arabia IMO TST, 5
Tags: geometry , right angle , circumradius , incenter
Let $ABC$ be a non isosceles triangle with incenter $I$ . The circumcircle of the triangle $ABC$ has radius $R$. Let $AL$ be the external angle bisector of $\angle BAC $with $L \in BC$. Let $K$ be the point on perpendicular bisector of $BC$ such that $IL \perp IK$.Prove that $OK=3R$.

2019 Grand Duchy of Lithuania, 3
Tags: geometry , right angle , perpendicular , orthocenter , circumcircle
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. The perpendicular bisector of segment $CH$ intersects the sides $AC$ and $BC$ in points $X$ and $Y$ , respectively. The lines $XO$ and $YO$ intersect the side $AB$ in points $P$ and $Q$, respectively. Prove that if $XP + Y Q = AB + XY$ then $\angle OHC = 90^o$.

Croatia MO (HMO) - geometry, 2020.3
Tags: geometry , right angle , perpendicular
Given a triangle $ABC$ such that $AB<AC$ . On sides $AB$ and $BC$, points $P$ and $Q$ are marked respectively such that the lines $AQ$ and $CP$ are perpendicular, and the circle inscribed in the triangle $ABC$ touches the length $PQ$. The line $CP$ intersects the circle circumscribed around the triangle $ABC$ at the points $C$ and $T$. If the lines $CA,PQ$ and $BT$ intersect at one point, prove that the angle $\angle CAB$ is right.

1999 Swedish Mathematical Competition, 2
Tags: right angle , circles , geometry
Circle $C$ center $O$ touches externally circle $C'$ center $O'$. A line touches $C$ at $A$ and $C'$ at $B$. $P$ is the midpoint of $AB$. Show that $\angle OPO' = 90^o$.

2011 Oral Moscow Geometry Olympiad, 2
Tags: geometry , isosceles , equal segments , right angle , angle
In an isosceles triangle $ABC$ ($AB=AC$) on the side $BC$, point $M$ is marked so that the segment $CM$ is equal to the altitude of the triangle drawn on this side, and on the side $AB$, point $K$ is marked so that the angle $\angle KMC$ is right. Find the angle $\angle ACK$.

1998 Tuymaada Olympiad, 8
Tags: geometry , 3d geometry , pyramid , right angle , solid geometry
Given the pyramid $ABCD$. Let $O$ be the midpoint of edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.

2004 Germany Team Selection Test, 2
Tags: geometry , combinatorial geometry , polygon , extremal combinatorics , right angle , angle
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles. [i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]

2012 Peru MO (ONEM), 4
Tags: common tangent , geometry , mixtilinear , circles , right angle
In a circle $S$, a chord $AB$ is drawn and let $M$ be the midpoint of the arc $AB$. Let $P$ be a point in segment $AB$ other than its midpoint. The extension of the segment $MP$ cuts $S$ in $Q$. Let $S_1$ be the circle that is tangent to the AP segments and $MP$, and also is tangent to $S$, and let $S_2$ be the circle that is tangent to the segments $BP$ and $MP$, and also tangent to $S$. The common outer tangent lines to the circles $S_1$ and $S_2$ are cut at $C$. Prove that $\angle MQC = 90^o$.

2015 Saudi Arabia JBMO TST, 3
Tags: geometry , circumcircle , right triangle , right angle
A right triangle $ABC$ with $\angle C=90^o$ is inscribed in a circle. Suppose that $K$ is the midpoint of the arc $BC$ that does not contain $A$. Let $N$ be the midpoint of the segment $AC$, and $M$ be the intersection point of the ray $KN$ and the circle.The tangents to the circle drawn at $A$ and $C$ meet at $E$. prove that $\angle EMK = 90^o$

2016 Denmark MO - Mohr Contest, 3
Tags: geometry , right angle , equal segments , area
Prove that all quadrilaterals $ABCD$ where $\angle B = \angle D = 90^o$, $|AB| = |BC|$ and $|AD| + |DC| = 1$, have the same area. [img]https://1.bp.blogspot.com/-55lHuAKYEtI/XzRzDdRGDPI/AAAAAAAAMUk/n8lYt3fzFaAB410PQI4nMEz7cSSrfHEgQCLcBGAsYHQ/s0/2016%2Bmohr%2Bp3.png[/img]

2020 Tournament Of Towns, 5
Tags: right angle , trapezoid , equal segments , trapezium , geometry
Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle. Alexandr Yuran

2017 Auckland Mathematical Olympiad, 1
Tags: right angle , geometry
A $6$ meter ladder rests against a vertical wall. The midpoint of the ladder is twice as far from the ground as it is from the wall. At what height on the wall does the ladder reach?

2019 Saudi Arabia Pre-TST + Training Tests, 1.3
Tags: geometry , trapezoid , right angle , incircle , equal segments
Let $ABCD$ be a trapezoid with $\angle A = \angle B = 90^o$ and a point $E$ lies on the segment $CD$. Denote $(\omega)$ as incircle of triangle $ABE$ and it is tangent to $AB,AE,BE$ respectively at $P, F,K$. Suppose that $KF$ cuts $BC,AD$ at $M,N$ and $PM,PN$ cut $(\omega)$ at $H, T$. Prove that $PH = PT$.

2006 Oral Moscow Geometry Olympiad, 3
Tags: geometry , geometric inequality , inradius , right angle
On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $C', A'$ and $B'$ are selected, respectively, so that the angle $A'C'B'$ is right. Prove that the segment $A'B'$ is longer than the diameter of the inscribed circle of the triangle $ABC$. (M. Volchkevich)

1978 Vietnam National Olympiad, 5
Tags: maximum , right angle , geometry
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance $a$ apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length $c$ and negligible width which can pass through the bend?