Found problems: 25757
2018 Regional Olympiad of Mexico Northwest, 3
Let $ABC$ be an acute triangle orthocenter angle $H$. Let $\omega_1$ be the circle tangent to $BC$ at $B$ and passing through $H$ and $\omega_2$ the circle tangent to $BC$ at $C$ and passing through through $H$. A line $\ell$ passing through $H$ intersects the circles $\omega_1$ and $\omega_2$ at points $D$ and $E$, respectively (with $D$ and $E$ other than $H$). Lines $BD$ and $CE$ intersect at $F$, the lines $\ell$ and $AF$ intersect at $X$ and the circles $\omega_1$ and $\omega_2$ intersect at the points $P$ and $H$. Prove that the points $A, H, P$ and $X$ are still on the same circle.
1992 IMO, 1
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
1982 IMO Longlists, 39
Let $S$ be the unit circle with center $O$ and let $P_1, P_2,\ldots, P_n$ be points of $S$ such that the sum of vectors $v_i=\stackrel{\longrightarrow}{OP_i}$ is the zero vector. Prove that the inequality $\sum_{i=1}^n XP_i \geq n$ holds for every point $X$.
2018 Moldova Team Selection Test, 7
Let the triangle $ABC $ with $m (\angle ABC)=60^{\circ} $ and $m (\angle BAC)=40^{\circ}$ . Points $D $ and $E $ are on the sides $(AB) $ and $(AC) $ such that $m (\angle DCB )=70^{\circ}$ and $m (\angle EBC)=40^{\circ}$ . $BE$ and $CD$ intersect in $F $ . Prove that $BC $ and $AF $ are perpendicular.
2014 Romania Team Selection Test, 1
Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.
2012 Israel National Olympiad, 1
In the picture below, the circles are tangent to each other and to the edges of the rectangle. The larger circle's radius equals 1. Determine the area of the rectangle.
[img]https://i.imgur.com/g3GUg4Z.png[/img]
2012 Paraguay Mathematical Olympiad, 3
Let $ABC$ be a triangle (right in $B$) inscribed in a semi-circumference of diameter $AC=10$. Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.
1994 Moldova Team Selection Test, 7
Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers:
$a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.
2025 USAMO, 4
Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.
2005 May Olympiad, 4
There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.
2014 JBMO TST - Turkey, 3
Let a line $\ell$ intersect the line $AB$ at $F$, the sides $AC$ and $BC$ of a triangle $ABC$ at $D$ and $E$, respectively and the internal bisector of the angle $BAC$ at $P$. Suppose that $F$ is at the opposite side of $A$ with respect to the line $BC$, $CD = CE$ and $P$ is in the interior the triangle $ABC$. Prove that
\[FB \cdot FA+CP^2 = CF^2 \iff AD \cdot BE = PD^2.\]
1957 AMC 12/AHSME, 25
The vertices of triangle $ PQR$ have coordinates as follows: $ P(0,a),\,Q(b,0),\,R(c,d),$ where $ a,\,b,\,c$ and $ d$ are positive. The origin and point $ R$ lie on opposite sides of $ PQ$. The area of triangle $ PQR$ may be found from the expression:
$ \textbf{(A)}\ \frac{ab \plus{} ac \plus{} bc \plus{} cd}{2} \qquad
\textbf{(B)}\ \frac{ac \plus{} bd \minus{} ab}{2}\qquad
\textbf{(C)}\ \frac{ab \minus{} ac \minus{} bd}{2}\qquad
\textbf{(D)}\ \frac{ac \plus{} bd \plus{} ab}{2}\qquad
\textbf{(E)}\ \frac{ac \plus{} bd \minus{} ab \minus{} cd}{2}$
Brazil L2 Finals (OBM) - geometry, 2011.5
Inside a square of side $16$ are placed $1000$ points. Show that it is possible to put a equilateral triangle of side $2\sqrt3$ in the plane so that it covers at least $16$ of these points.
May Olympiad L2 - geometry, 2003.4
Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$.
Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.
2008 Oral Moscow Geometry Olympiad, 4
A circle can be circumscribed around the quadrilateral $ABCD$. Point $P$ is the foot of the perpendicular drawn from point $A$ on line $BC$, and respectively $Q$ from $A$ on $DC$, $R$ from $D$ on $AB$ and $T$ from $D$ on $BC$ . Prove that points $P,Q,R$ and $T$ lie on the same circle.
(A. Myakishev)
2011 Iran Team Selection Test, 11
Let $ABC$ be a triangle and $A',B',C'$ be the midpoints of $BC,CA,AB$ respectively. Let $P$ and $P'$ be points in plane such that $PA=P'A',PB=P'B',PC=P'C'$. Prove that all $PP'$ pass through a fixed point.
2021 Kyiv City MO Round 1, 7.2
Andriy and Olesya take turns (Andriy starts) in a $2 \times 1$ rectangle, drawing horizontal segments of length $2$ or vertical segments of length $1$, as shown in the figure below.
[img]https://i.ibb.co/qWqWxgh/Kyiv-MO-2021-Round-1-7-2.png[/img]
After each move, the value $P$ is calculated - the total perimeter of all small rectangles that are formed (i.e., those inside which no other segment passes). The winner is the one after whose move $P$ is divisible by $2021$ for the first time. Who has a winning strategy?
[i]Proposed by Bogdan Rublov[/i]
1986 Swedish Mathematical Competition, 2
The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.
2024 Mexico National Olympiad, 3
Let $ABCDEF$ a convex hexagon, and let $A_1,B_1,C_1,D_1,E_1$ y $F_1$ be the midpoints of $AB,BC,CD,$ $DE,EF$ and $FA$, respectively. Construct points $A_2,B_2,C_2,D_2,E_2$ and $F_2$ in the interior of $A_1B_1C_1D_1E_1F_1$ such that both
1. The sides of the dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ are all equal and
2. $\angle A_1B_2B_1+\angle C_1D_2D_1+\angle E_1F_2F_1=\angle B_1C_2C_1+\angle D_1E_2E_1+\angle F_1A_2A_1=360^\circ$, where all these angles are less than $180 ^\circ$,
Prove that $A_2B_2C_2D_2E_2F_2$ is cyclic.
[b]Note:[/b] Dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ is shaped like a 6-pointed star, where the points are $A_1,B_1,C_1,D_1,E_1$ y $F_1$.
2003 Croatia National Olympiad, Problem 1
Let $a,b,c$ be the sides of triangle $ABC$ and let $\alpha,\beta,\gamma$ be the corresponding angles.
(a) If $\alpha=3\beta$, prove that $\left(a^2-b^2\right)(a-b)=bc^2$.
(b) Is the converse true?
2013 Saudi Arabia IMO TST, 1
Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.
2015 Iran Team Selection Test, 2
In triangle $ABC$(with incenter $I$) let the line parallel to $BC$ from $A$ intersect circumcircle of $\triangle ABC$ at $A_1$ let $AI\cap BC=D$ and $E$ is tangency point of incircle with $BC$ let $ EA_1\cap \odot (\triangle ADE)=T$ prove that $AI=TI$.
2008 SDMO (Middle School), 3
In the diagram, $AD:DB=1:1$, $BE:EC=1:2$, and $CF:FA=1:3$. If the area of triangle $ABC$ is $120$, then find the area of triangle $DEF$.
(will insert image here later)
2014 Sharygin Geometry Olympiad, 8
A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$?
(N. Beluhov, S. Gerdgikov)
1999 Korea Junior Math Olympiad, 1
There exists point $O$ inside a convex quadrilateral $ABCD$ satisfying $OA=OB$ and $OC=OD$, and $\angle AOB = \angle COD=90^{\circ}$. Consider two squares, (1)square having $AC$ as one side and located in the opposite side of $B$ and (2)square having $BD$ as one side and located in the opposite side of $E$. If the common part of these two squares is also a square, prove that $ABCD$ is an inscribed quadrilateral.