Found problems: 25757
1995 AMC 12/AHSME, 21
Two nonadjacent vertices of a rectangle are $(4,3)$ and $(-4,-3)$, and the coordinates of the other two vertices are integers. The number of such rectangles is
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
2013 Saudi Arabia BMO TST, 6
Let $ABC$ be a triangle with incenter $I,$ and let $D,E,F$ be the midpoints of sides $BC, CA, AB$, respectively. Lines $BI$ and $DE$ meet at $P $ and lines $CI$ and $DF$ meet at $Q$. Line $PQ$ meets sides $AB$ and $AC$ at $T$ and $S$, respectively. Prove that $AS = AT$
2017 Princeton University Math Competition, A2/B4
The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?
2022 Indonesia TST, G
In a nonisosceles triangle $ABC$, point $I$ is its incentre and $\Gamma$ is its circumcircle. Points $E$ and $D$ lie on $\Gamma$ and the circumcircle of triangle $BIC$ respectively such that $AE$ and $ID$ are both perpendicular to $BC$. Let $M$ be the midpoint of $BC$, $N$ be the midpoint of arc $BC$ on $\Gamma$ containing $A$, $F$ is the point of tangency of the $A-$excircle on $BC$, and $G$ is the intersection of line $DE$ with $\Gamma$. Prove that lines $GM$ and $NF$ intersect at a point located on $\Gamma$.
(Possibly proposed by Farras Faddila)
2006 Princeton University Math Competition, 3
Find the exact value of $\sin 36^o$.
2005 National High School Mathematics League, 1
In $\triangle ABC$, $AB>AC$, $l$ is tangent line of the circumscribed circle of $\triangle ABC$ that passes $A$. The circle with center $A$ and radius $AC$, intersects segment $AB$ at $D$, and line $l$ at $E, F$ ($F,B$ are on the same side). Prove that lines $DE, DF$ pass the incenter and an excenter of $\triangle ABC$ respectively.
2005 IMO, 1
Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.
Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent.
[i]Bogdan Enescu, Romania[/i]
1992 Vietnam Team Selection Test, 1
In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.
2020 China Girls Math Olympiad, 7
Let $O$ be the circumcenter of triangle $\triangle ABC$, where $\angle BAC=120^{\circ}$. The tangent at $A$ to $(ABC)$ meets the tangents at $B,C$ at $(ABC)$ at points $P,Q$ respectively. Let $H,I$ be the orthocenter and incenter of $\triangle OPQ$ respectively. Define $M,N$ as the midpoints of arc $\overarc{BAC}$ and $OI$ respectively, and let $MN$ meet $(ABC)$ again at $D$. Prove that $AD$ is perpendicular to $HI$.
2012 Indonesia MO, 4
Given a triangle $ABC$, let the bisector of $\angle BAC$ meets the side $BC$ and circumcircle of triangle $ABC$ at $D$ and $E$, respectively. Let $M$ and $N$ be the midpoints of $BD$ and $CE$, respectively. Circumcircle of triangle $ABD$ meets $AN$ at $Q$. Circle passing through $A$ that is tangent to $BC$ at $D$ meets line $AM$ and side $AC$ respectively at $P$ and $R$. Show that the four points $B,P,Q,R$ lie on the same line.
[i]Proposer: Fajar Yuliawan[/i]
2013 Austria Beginners' Competition, 4
Let $ABC$ be an acute-angled triangle and $D$ a point on the altitude through $C$. Let $E$, $F$, $G$ and $H$ be the midpoints of the segments $AD$, $BD$, $BC$ and $AC$. Show that $E$, $F$, $G$, and $H$ form a rectangle.
(G. Anegg, Innsbruck)
2006 Princeton University Math Competition, 5
$A, B$, and $C$ are vertices of a triangle, and $P$ is a point within the triangle. If angles $\angle BAP$, $\angle BCP$, and $\angle ABP$ are all $30^o$ and angle $\angle ACP$ is $45^o$, what is $\sin(\angle CBP)$?
2010 Greece National Olympiad, 3
A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively.
Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic.
Babis
2016 India Regional Mathematical Olympiad, 1
Let \(ABC\) be a triangle and \(D\) be the mid-point of \(BC\). Suppose the angle bisector of \(\angle ADC\) is tangent to the circumcircle of triangle \(ABD\) at \(D\). Prove that \(\angle A=90^{\circ}\).
2020 Regional Olympiad of Mexico Southeast, 5
Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.
2008 Regional Olympiad of Mexico Center Zone, 2
Let $ABC$ be a triangle with incenter $I $, the line $AI$ intersects $BC$ at $ L$ and the circumcircle of $ABC$ at $L'$. Show that the triangles $BLI$ and $L'IB$ are similar if and only if $AC = AB + BL$.
2019 Dürer Math Competition (First Round), P5
Let $ABC$ and $A'B'C'$ be similar triangles with different orientation such that their orthocenters coincide. Show that lines $AA′, BB′, CC′ are concurrent or parallel.
2002 Moldova National Olympiad, 3
The sides $ AB$,$ BC$ and $ CA$ of the triangle $ ABC$ are tangent to the incircle of the triangle $ ABC$ with center $ I$ at the points $ C_1$,$ A_1$ and $ B_1$, respectively.Let $ B_2$ be the midpoint of the side $ AC$.Prove that the lines $ B_1I$, $ A_1C_1$ and $ BB_2$ are concurrent.
2011 Morocco National Olympiad, 4
The diagonals of a trapezoid $ ABCD $ whose bases are $ [AB] $ and $ [CD] $ intersect at $P.$ Prove that
\[S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA},\]
Where $S_{XYZ} $ denotes the area of $\triangle XYZ $.
2022 Yasinsky Geometry Olympiad, 5
Point $X$ is chosen on side $AD$ of square $ABCD$. The inscribed circle of triangle $ABX$ touches $AX$, $BX$, and $AB$ at points $N$, $K$, and $F$, respectively. Prove that the ray $NK$ passes through the center $O$ of the square $ABCD$.
(Dmytro Shvetsov)
1989 Greece National Olympiad, 4
A trapezoid with bases $a,b$ and altitude $h$ is circumscribed around a circl.. Prove that $h^2\le ab$.
2014 Purple Comet Problems, 9
The diagram below shows a shaded region bounded by a semicircular arc of a large circle and two smaller semicircular arcs. The smallest semicircle has radius $8$, and the shaded region has area $180\pi$. Find the diameter of the large circle.
[asy]
import graph;
size(3cm);
fill((-22.5,0)..(0,22.5)..(22.5,0)--cycle,rgb(.76,.76,.76));
fill((6.5,0.1)..(14.5,-8)..(22.5,0.1)--cycle,rgb(.76,.76,.76));
fill((-22.5,-0.1)..(-8,14.5)..(6.5,-0.1)--cycle,white);
draw((-22.5,0)..(-8,14.5)..(6.5,0),linewidth(1.5));
draw((6.5,0)..(14.5,-8)..(22.5,0),linewidth(1.5));
draw(Circle((0,0),22.5),linewidth(1.5));
[/asy]
2023 Middle European Mathematical Olympiad, 5
We are given a convex quadrilateral $ABCD$ whose angles are not right. Assume there are points $P, Q, R, S$ on its sides $AB, BC, CD, DA$, respectively, such that $PS \parallel BD$, $SQ \perp BC$, $PR \perp CD$. Furthermore, assume that the lines $PR, SQ$, and $AC$ are concurrent. Prove thatthe points $P, Q, R, S$ are concyclic.
2023 Girls in Mathematics Tournament, 3
Let $ABC$ an acute triangle and $D$ and $E$ the feet of heights by $A$ and $B$, respectively, and let $M$ be the midpoint of $AC$. The circle that passes through $D$ and $B$ and is tangent to $BE$ in $B$ intersects the line $BM$ in $F, F\neq B$. Show that $FM$ is the angle bisector of $\angle AFD$.
2003 Purple Comet Problems, 22
In $\triangle ABC$, max $\{\angle A, \angle B \} = \angle C + 30^{\circ}$ and $\frac{R}{r} = \sqrt{3} + 1$, where $R$ is the radius of the circumcircle and $r$ is the radius of the incircle. Find $\angle C$ in degrees.