Found problems: 25757
2013 Sharygin Geometry Olympiad, 22
The common perpendiculars to the opposite sidelines of a nonplanar quadrilateral are mutually orthogonal. Prove that they intersect.
2004 South africa National Olympiad, 4
Let $A_1$ and $B_1$ be two points on the base $AB$ of isosceles triangle $ABC$ (with $\widehat{C}>60^\circ$) such that $\widehat{A_1CB_1}=\widehat{BAC}$. A circle externally tangent to the circumcircle of triangle $\triangle A_1B_1C$ is tangent also to rays $CA$ and $CB$ at points $A_2$ and $B_2$ respectively. Prove that $A_2B_2=2AB$.
2004 Iran MO (3rd Round), 25
Finitely many convex subsets of $\mathbb R^3$ are given, such that every three have non-empty intersection. Prove that there exists a line in $\mathbb R^3$ that intersects all of these subsets.
2013 India PRMO, 9
In a triangle $ABC$, let $H, I$ and $O$ be the orthocentre, incentre and circumcentre, respectively. If the points $B, H, I, C$ lie on a circle, what is the magnitude of $\angle BOC$ in degrees?
1992 Mexico National Olympiad, 3
Given $7$ points inside or on a regular hexagon, show that three of them form a triangle with area $\le 1/6$ the area of the hexagon.
2021 Czech-Polish-Slovak Junior Match, 5
A regular heptagon $ABCDEFG$ is given. The lines $AB$ and $CE$ intersect at $ P$. Find the measure of the angle $\angle PDG$.
2012 Iran MO (3rd Round), 4
The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$.
[i]Proposed By Pedram Safaei[/i]
2017 Korea - Final Round, 5
Let there be cyclic quadrilateral $ABCD$ with $L$ as the midpoint of $AB$ and $M$ as the midpoint of $CD$. Let $AC \cap BD = E$, and let rays $AB$ and $DC$ meet again at $F$. Let $LM \cap DE = P$. Let $Q$ be the foot of the perpendicular from $P$ to $EM$. If the orthocenter of $\triangle FLM$ is $E$, prove the following equality.
$$\frac{EP^2}{EQ} = \frac{1}{2} \left( \frac{BD^2}{DF} - \frac{BC^2}{CF} \right)$$
2007 District Olympiad, 1
Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13 BC$, find the measures of the angles of the triangle $\vartriangle ABC$.
2006 Baltic Way, 11
The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle?
II Soros Olympiad 1995 - 96 (Russia), 9.9
Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for second time at a point $K$, different from $A$. A circle passing through $A$, $K$ and the center of the first circle intersects the first one for second time at point $M$. Find the locus of points $M$.
Durer Math Competition CD 1st Round - geometry, 2010.D3
Prove that the diagonals of a quadrilateral are perpendicular to each other if and only if the midpoints of it's sides lie on a circle.
2022 VN Math Olympiad For High School Students, Problem 7
Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point.
Let $H_a, H_b, H_c$ be the orthocenter of triangles $TBC, TCA, TAB$, respectively.
a) Prove that: $T$ is the centroid of the $\triangle H_aH_bH_c$.
b) Denote $D, E, F$ respectively by the intersections of $H_cH_b$ and the segment $BC$, $H_cH_a$ and the segment $CA$, $H_aH_b$ and the segment $AB$. Prove that: the triangle $DEF$ is equilateral.
c) Prove that: the lines passing through $D, E, F$ and are respectively perpendicular to $BC, CA, AB$ are concurrent at a point. Let that point be $S$.
d) Prove that: $TS$ is parallel to the [i]Euler[/i] line of the triangle $ABC$.
2002 China Girls Math Olympiad, 4
Circles $O_1$ and $O_2$ interest at two points $ B$ and $ C,$ and $ BC$ is the diameter of circle $O_1.$ Construct a tangent line of circle $O_1$ at $ C$ and intersecting circle $O_2$ at another point $ A.$ We join $ AB$ to intersect circle $O_1$ at point $ E,$ then join $ CE$ and extend it to intersect circle $O_2$ at point $ F.$ Assume $ H$ is an arbitrary point on line segment $ AF.$ We join $ HE$ and extend it to intersect circle $O_1$ at point $ G,$ and then join $ BG$ and extend it to intersect the extend line of $ AC$ at point $ D.$ Prove that \[ \frac{AH}{HF} = \frac{AC}{CD}.\]
2023 VIASM Summer Challenge, Problem 4
Let $ABCD$ be a parallelogram and $P$ be an arbitrary point in the plane. Let $O$ be the intersection of two diagonals $AC$ and $BD.$ The circumcircles of triangles $POB$ and $POC$ intersect the circumcircles of triangle $OAD$ at $Q$ and $R,$ respectively $(Q,R \ne O).$ Construct the parallelograms $PQAM$ and $PRDN.$
Prove that: the circumcircle of triangle $MNP$ passes through $O.$
[i]Proposed by Tran Quang Hung ([url=https://artofproblemsolving.com/community/user/68918]buratinogigle[/url])[/i]
1987 Czech and Slovak Olympiad III A, 1
Given a trapezoid, divide it by a line into two quadrilaterals in such a way that both of them are cyclic with the same circumradius. Discuss conditions of solvability.
2004 Korea National Olympiad, 5
$A, B, C$, and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$, which is tangent to $BC$, be $O_1$. Let the centre of the scribed circle of triangle $ACD$, which is tangent to $CD$, be $O_2$.
(1) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$.
(2) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$, when $C$ is moving along arc $BD$.
2012 South africa National Olympiad, 2
Let $ABCD$ be a square and $X$ a point such that $A$ and $X$ are on opposite sides of $CD$. The lines $AX$ and $BX$ intersect $CD$ in $Y$ and $Z$ respectively. If the area of $ABCD$ is $1$ and the area of $XYZ$ is $\frac{2}{3}$, determine the length of $YZ$
2018 Olympic Revenge, 4
Let $\triangle ABC$ an acute triangle of incenter $I$ and incircle $\omega$. $\omega$ is tangent to $BC, CA$ and $AB$ at points $T_{A}, T_{B}$ and $T_{C}$, respectively. Let $l_{A}$ the line through $A$ and parallel to $BC$ and define $l_{B}$ and $l_{C}$ analogously. Let $L_{A}$ the second intersection point of $AI$ with the circumcircle of $\triangle ABC$ and define $L_{B}$ and $L_{C}$ analogously. Let $P_{A}=T_{B}T_{C}\cap l_{A}$ and define $P_{B}$ and $P_{C}$ analogously. Let $S_{A}=P_{B}T_{B}\cap P_{C}T_{C}$ and define $S_{B}$ and $S_{C}$ analogously. Prove that $S_{A}L_{A}, S_{B}L_{B}, S_{C}L_{C}$ are concurrent.
1981 USAMO, 3
If $A,B,C$ are the angles of a triangle, prove that
\[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\]
and determine when equality holds.
2000 Harvard-MIT Mathematics Tournament, 10
What is the total surface area of an ice cream cone, radius $R$, height $H$, with a spherical scoop of ice cream of radius $r$ on top? (Given $R<r$)
Russian TST 2014, P1
For what values of $k{}$ can a regular octagon with side-length $k{}$ be cut into $1 \times 2{}$ dominoes and rhombuses with side-length 1 and a $45^\circ{}$ angle?
2008 Romania Team Selection Test, 2
Let $ ABC$ be a triangle and let $ \mathcal{M}_{a}$, $ \mathcal{M}_{b}$, $ \mathcal{M}_{c}$ be the circles having as diameters the medians $ m_{a}$, $ m_{b}$, $ m_{c}$ of triangle $ ABC$, respectively. If two of these three circles are tangent to the incircle of $ ABC$, prove that the third is tangent as well.
2018 Harvard-MIT Mathematics Tournament, 7
Ben "One Hunna Dolla" Franklin is flying a kite $KITE$ such that $IE$ is the perpendicular bisector of $KT$. Let $IE$ meet $KT$ at $R$. The midpoints of $KI,IT,TE,EK$ are $A,N,M,D,$ respectively. Given that $[MAKE]=18,IT=10,[RAIN]=4,$ find $[DIME]$.
Note: $[X]$ denotes the area of the figure $X$.
2020 Yasinsky Geometry Olympiad, 1
The square $ABCD$ is divided into $8$ equal right triangles and the square $KLMN$, as shown in the figure. Find the area of the square $ABCD$ if $KL = 5, PS = 8$.
[img]https://1.bp.blogspot.com/-B2QIHvPcIx0/X4BhUTMDhSI/AAAAAAAAMj4/4h0_q1P6drskc5zSvtfTZUskarJjRp5LgCLcBGAsYHQ/s0/Yasinsky%2B2020%2Bp1.png[/img]