Found problems: 25757
2018 Purple Comet Problems, 20
Let $ABCD$ be a square with side length $6$. Circles $X, Y$ , and $Z$ are congruent circles with centers inside the square such that $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$, $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$, and $Z$ is tangent to side $\overline{CD}$ and both circles $X$ and $Y$ . The radius of the circle $X$ can be written $m -\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
2012 Oral Moscow Geometry Olympiad, 6
Tangents drawn to the circumscribed circle of an acute-angled triangle $ABC$ at points $A$ and $C$, intersect at point $Z$. Let $AA_1, CC_1$ be altitudes. Line $A_1C_1$ intersects $ZA, ZC$ at points $X$ and $Y$, respectively. Prove that the circumscribed circles of the triangles $ABC$ and $XYZ$ are tangent.
2007 Moldova Team Selection Test, 1
Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.
1979 Brazil National Olympiad, 5
[list=i]
[*] ABCD is a square with side 1. M is the midpoint of AB, and N is the midpoint of BC. The lines CM and DN meet at I. Find the area of the triangle CIN.
[*] The midpoints of the sides AB, BC, CD, DA of the parallelogram ABCD are M, N, P, Q respectively. Each midpoint is joined to the two vertices not on its side. Show that the area outside the resulting 8-pointed star is $\frac{2}{5}$ the area of the parallelogram.
[*] ABC is a triangle with CA = CB and centroid G. Show that the area of AGB is $\frac{1}{3}$ of the area of ABC.
[*] Is (ii) true for all convex quadrilaterals ABCD?
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2009 Vietnam Team Selection Test, 1
Let an acute triangle $ ABC$ with curcumcircle $ (O)$. Call $ A_1,B_1,C_1$ are foots of perpendicular line from $ A,B,C$ to opposite side. $ A_2,B_2,C_2$ are reflect points of $ A_1,B_1,C_1$ over midpoints of $ BC,CA,AB$ respectively. Circle $ (AB_2C_2),(BC_2A_2),(CA_2B_2)$ cut $ (O)$ at $ A_3,B_3,C_3$ respectively.
Prove that: $ A_1A_3,B_1B_3,C_1C_3$ are concurent.
2014 NIMO Problems, 3
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
2010 Dutch BxMO TST, 1
Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$.
(a) Prove that $ABMD$ is a rhombus.
(b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.
1997 Chile National Olympiad, 3
Let $ ABCD $ be a quadrilateral, whose diagonals intersect at $ O $. The triangles $ \triangle AOB $, $ \triangle BOC $, $ \triangle COD $ have areas $1, 2, 4$, respectively. Find the area of $ \triangle AOD $ and prove that $ ABCD $ is a trapezoid.
2021 China Team Selection Test, 5
Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.
1990 India Regional Mathematical Olympiad, 8
If the circumcenter and centroid of a triangle coincide, prove that it must be equilateral.
2002 Bundeswettbewerb Mathematik, 4
In an acute-angled triangle $ABC$, we consider the feet $H_a$ and $H_b$ of the altitudes from $A$ and $B$, and the intersections $W_a$ and $W_b$ of the angle bisectors from $A$ and $B$ with the opposite sides $BC$ and $CA$ respectively. Show that the centre of the incircle $I$ of triangle $ABC$ lies on the segment $H_aH_b$ if and only if the centre of the circumcircle $O$ of triangle $ABC$ lies on the segment $W_aW_b$.
1979 IMO Longlists, 47
Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that
\[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\]
Determine the angles of triangle $PQR.$
2009 Stanford Mathematics Tournament, 6
Equilateral triangle $ABC$ has side lengths of $24$. Points $D$, $E$, and $F$ lies on sides $BC$, $CA$, $AB$ such that ${AD}\perp{BC}$, ${DE}\perp{AC}$, and ${EF}\perp{AB}$.
$G$ is the intersection of $AD$ and $EF$. Find the area of quadrilateral $BFGD$
2019 Serbia National Math Olympiad, 4
For a $\triangle ABC$ , let $A_1$ be the symmetric point of the intersection of angle bisector of $\angle BAC$ and $BC$ , where center of the symmetry is the midpoint of side $BC$, In the same way we define $B_1 $ ( on $AC$ ) and $C_1$ (on $AB$). Intersection of circumcircle of $\triangle A_1B_1C_1$ and line $AB$ is the set $\{Z,C_1 \}$, with $BC$ is the set $\{X,A_1\}$ and with $CA$ is the set $\{Y,B_1\}$. If the perpendicular lines from $X,Y,Z$ on $BC,CA$ and $ AB$ , respectively are concurrent , prove that $\triangle ABC$ is isosceles.
2008 Harvard-MIT Mathematics Tournament, 16
Point $ A$ lies at $ (0, 4)$ and point $ B$ lies at $ (3, 8)$. Find the $ x$-coordinate of the point $ X$ on the $ x$-axis maximizing $ \angle AXB$.
2009 Sharygin Geometry Olympiad, 2
Given nonisosceles triangle $ ABC$. Consider three segments passing through different vertices of this triangle and bisecting its perimeter. Are the lengths of these segments certainly different?
2016 Iranian Geometry Olympiad, 2
Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$.
by Iman Maghsoudi
2005 Postal Coaching, 23
Let $\Gamma$ be the incircle of an equilateral triangle $ABC$ of side length $2$ units.
(a) Show that for all points $P$ on $\Gamma$, $PA^2 +PB^2 +PC^2 = 5$.
(b) Show that for all points $P$ on $\Gamma$, it is possible to construct a triangle of sides equal to $PA,PB,PC$ and whose area is equal to $\frac{\sqrt{3}}{4}$ units.
2006 Junior Balkan Team Selection Tests - Romania, 1
Let $ABCD$ be a cyclic quadrilateral of area 8. If there exists a point $O$ in the plane of the quadrilateral such that $OA+OB+OC+OD = 8$, prove that $ABCD$ is an isosceles trapezoid.
2018 SIMO, Q2
Given $\triangle ABC$, let $I,O,\Gamma$ denote its incenter, circumcenter and circumcircle respecitvely. Let $AI$ intersect $\Gamma$ at $M(\neq A)$. Circle $\omega$ is tangent to $AB$, $AC$ and $\Gamma$ internally at $T$ (i.e. the mixtilinear incircle opposite $A$). Let the tangents at $A$ and $T$ to $\Gamma$ meet at $P$, and let $PI$ and $TM$ intersect at $Q$. Prove that $QA$ and $MO$ intersect at a point on $\Gamma$.
1973 AMC 12/AHSME, 32
The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $ \sqrt{15}$ is
$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 9/2 \qquad
\textbf{(C)}\ 27/2 \qquad
\textbf{(D)}\ \frac{9\sqrt3}{2} \qquad
\textbf{(E)}\ \text{none of these}$
2006 Bosnia and Herzegovina Junior BMO TST, 2
In an acute triangle $ABC$, $\angle C = 60^o$. If $AA'$ and $BB'$ are two of the altitudes and $C_1$ is the midpoint of $AB$, prove that triangle $C_1A'B'$ is equilateral.
2014 BMT Spring, P1
Let a simple polygon be defined as a polygon in which no consecutive sides are parallel and no two non-consecutive sides share a common point. Given that all vertices of a simple polygon $P$ are lattice points (in a Cartesian coordinate system, each vertex has integer coordinates), and each side of $P$ has integer length, prove that the perimeter must be even.
2016 NIMO Problems, 6
Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$ and $BC>AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt n,$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.
[i]Proposed by David Altizio and Evan Chen[/i]
1990 IMO Longlists, 6
Let $S, T$ be the circumcenter and centroid of triangle $ABC$, respectively. $M$ is a point in the plane of triangle $ABC$ such that $90^\circ \leq \angle SMT < 180^\circ$. $A_1, B_1, C_1$ are the intersections of $AM, BM, CM$ with the circumcircle of triangle $ABC$ respectively. Prove that $MA_1 + MB_1 + MC_1 \geq MA + MB + MC.$