Found problems: 25757
2018 CMIMC Geometry, 2
Let $ABCD$ be a square of side length $1$, and let $P$ be a variable point on $\overline{CD}$. Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$. The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the length of this segment?
2010 Costa Rica - Final Round, 1
Consider points $D,E$ and $F$ on sides $BC,AC$ and $AB$, respectively, of a triangle $ABC$, such that $AD, BE$ and $CF$ concurr at a point $G$. The parallel through $G$ to $BC$ cuts $DF$ and $DE$ at $H$ and $I$, respectively. Show that triangles $AHG$ and $AIG$ have the same areas.
2010 ELMO Problems, 1
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three.
[i]Carl Lian.[/i]
1991 Putnam, A1
The rectangle with vertices $(0,0)$, $(0,3)$, $(2,0)$ and $(2,3)$ is rotated clockwise through a right angle about the point $(2,0)$, then about $(5,0)$, then about $(7,0$), and finally about $(10,0)$. The net effect is to translate it a distance $10$ along the $x$-axis. The point initially at $(1,1)$ traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the $x$-axis and the lines parallel to the $y$-axis through $(1,0)$ and $(11,0)$).
2000 Croatia National Olympiad, Problem 2
The incircle of a triangle $ABC$ touches $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Find the angles of $\triangle A_1B_1C_1$ in terms of the angles of $\triangle ABC$.
1972 IMO Longlists, 39
How many tangents to the curve $y = x^3-3x\:\: (y = x^3 + px)$ can be drawn from different points in the plane?
2004 Iran MO (3rd Round), 9
Let $ABC$ be a triangle, and $O$ the center of its circumcircle.
Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively.
Prove that $\measuredangle ROS=\measuredangle BAC$.
2006 Italy TST, 1
The circles $\gamma_1$ and $\gamma_2$ intersect at the points $Q$ and $R$ and internally touch a circle $\gamma$ at $A_1$ and $A_2$ respectively. Let $P$ be an arbitrary point on $\gamma$. Segments $PA_1$ and $PA_2$ meet $\gamma_1$ and $\gamma_2$ again at $B_1$ and $B_2$ respectively.
a) Prove that the tangent to $\gamma_{1}$ at $B_{1}$ and the tangent to $\gamma_{2}$ at $B_{2}$ are parallel.
b) Prove that $B_{1}B_{2}$ is the common tangent to $\gamma_{1}$ and $\gamma_{2}$ iff $P$ lies on $QR$.
2005 Putnam, A6
Let $n$ be given, $n\ge 4,$ and suppose that $P_1,P_2,\dots,P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i.$ What is the probability that at least one of the vertex angles of this polygon is acute.?
2014 India PRMO, 3
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals.
If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?
1986 Tournament Of Towns, (126) 1
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
2020 Sharygin Geometry Olympiad, 10
Given are a closed broken line $A_1A_2\ldots A_n$ and a circle $\omega$ which touches each of lines $A_1A_2,A_2A_3,\ldots,A_nA_1$. Call the link [i]good[/i], if it touches $\omega$, and [i]bad[/i] otherwise (i.e. if the extension of this link touches $\omega$). Prove that the number of bad links is even.
2022 Moldova Team Selection Test, 4
In the acute triangle $ABC$ the point $M$ is on the side $BC$. The inscribed circle of triangle $ABM$ touches the sides $BM$, $MA$ and $AB$ in points $D$, $E$ and $F$, and the inscribed circle of triangle $ACM$ touches the sides $CM$, $MA$ and $AC$ in points $X$, $Y$ and $Z$. The lines $FD$ and $ZX$ intersect in point $H$. Prove that lines $AH$, $XY$ and $DE$ are concurrent.
KoMaL A Problems 2023/2024, A. 871
Let $ABC$ be an obtuse triangle, and let $H$ denote its orthocenter. Let $\omega_A$ denote the circle with center $A$ and radius $AH$. Let $\omega_B$ and $\omega_C$ be defined in a similar way. For all points $X$ in the plane of triangle $ABC$ let circle $\Omega(X)$ be defined in the following way (if possible): take the polars of point $X$ with respect to circles $\omega_A$, $\omega_B$ and $\omega_C$, and let $\Omega(X)$ be the circumcircle of the triangle defined by these three lines.
With a possible exception of finitely many points find the locus of points $X$ for which point $X$ lies on circle $\Omega(X)$.
[i]Proposed by Vilmos Molnár-Szabó, Budapest[/i]
2017-IMOC, G2
Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$
2020 Romanian Master of Mathematics Shortlist, G1
The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in terms of the angles of the triangle $ABC$.
[i]Iran, Shayan Talaei[/i]
2014 Taiwan TST Round 3, 4
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.
2021 Pan-African, 6
Let $ABCD$ be a trapezoid which is not a parallelogram, such that $AD$ is parallel to $BC$.
Let $O=BD\cap AC$ and $S$ be the second intersection of the circumcircles of triangles $AOB$ and $DOC$.
Prove that the circumcircles of triangles $ASD$ and $BSC$ are tangent.
2016 Sharygin Geometry Olympiad, P10
Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ ABC$.
2010 Iran MO (2nd Round), 5
In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.
1955 Poland - Second Round, 5
Given a triangle $ ABC $. Find the rectangle of smallest area containing the triangle.
2014 NIMO Summer Contest, 14
Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$.
Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Evan Chen[/i]
2010 Putnam, B2
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$
2009 Italy TST, 2
$ABC$ is a triangle in the plane. Find the locus of point $P$ for which $PA,PB,PC$ form a triangle whose area is equal to one third of the area of triangle $ABC$.
2011 Bundeswettbewerb Mathematik, 4
Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.