Found problems: 1049
2013 Moldova Team Selection Test, 3
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.
2006 Iran MO (3rd Round), 4
$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space.
1) Prove that Image of every line is a line.
2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)
2000 Iran MO (2nd round), 2
The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if
\[\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}\]
2002 CentroAmerican, 4
Let $ ABC$ be a triangle, $ D$ be the midpoint of $ BC$, $ E$ be a point on segment $ AC$ such that $ BE\equal{}2AD$ and $ F$ is the intersection point of $ AD$ with $ BE$. If $ \angle DAC\equal{}60^{\circ}$, find the measure of the angle $ FEA$.
2021 Taiwan APMO Preliminary First Round, 2
(a) Let the incenter of $\triangle ABC$ be $I$. We connect $I$ other $3$ vertices and divide $\triangle ABC$ into $3$ small triangles which has area $2,3$ and $4$. Find the area of the inscribed circle of $\triangle ABC$.
(b) Let $ABCD$ be a parallelogram. Point $E,F$ is on $AB,BC$ respectively. If $[AED]=7,[EBF]=3,[CDF]=6$, then find $[DEF].$ (Here $[XYZ]$ denotes the area of $XYZ$)
1991 Vietnam Team Selection Test, 1
1.) In the plane let us consider a set $S$ consisting of $n \geq 3$ distinct points satisfying the following three conditions:
[b]I.[/b] The distance between any two points $\in S$ is not greater than 1.
[b]II.[/b] For every point $A \in S$, there are exactly two “neighbor” points, i.e. two points $X, Y \in S$ for which $AX = AY = 1$.
[b]III. [/b] For arbitrary two points $A, B \in S$, let $A', A''$ be the two neighbors of $A, B', B''$ the two neighbors of $B$, then $A'AA'' = B'BB''$.
Is there such a set $S$ if $n = 1991$? If $n = 2000$ ? Explain your answer.
2007 Balkan MO Shortlist, G3
Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that
\[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\]
Prove that there exists a point $ M$ in the plane of the pentagon such that
\[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\]
Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.
2015 Bulgaria National Olympiad, 5
In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$, respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$. Prove that the lines $AQ, BP$ and $LT$ intersect in a point.
1997 Pre-Preparation Course Examination, 5
Let $H$ be the orthocenter of the triangle $ABC$ and $P$ an arbitrary point on circumcircle of triangle. $BH$ meets $AC$ at $E$. $PAQB$ and $PARC$ are two parallelograms and $AQ$ meets $HR$ at $X$. Show that $EX \parallel AP$.
2022 Oral Moscow Geometry Olympiad, 3
Extensions of opposite sides of a convex quadrilateral $ABCD$ intersect at points $P$ and $Q$. Points are marked on the sides of $ABCD$ (one per side), which are the vertices of a parallelogram with a side parallel to $PQ$. Prove that the intersection point of the diagonals of this parallelogram lies on one of the diagonals of quadrilateral $ABCD$.
(E. Bakaev)
2014 Czech-Polish-Slovak Junior Match, 1
On the plane circles $k$ and $\ell$ are intersected at points $C$ and $D$, where circle $k$ passes through the center $L$ of circle $\ell$. The straight line passing through point $D$ intersects circles $k$ and $\ell$ for the second time at points $A$ and $B$ respectively in such a way that $D$ is the interior point of segment $AB$. Show that $AB = AC$.
2022 All-Russian Olympiad, 7
Point $E$ is marked on side $BC$ of parallelogram $ABCD$, and on the side $AD$ - point $F$ so that the circumscribed circle of $ABE$ is tangent to line segment $CF$. Prove that the circumcircle of triangle $CDF$ is tangent to line $AE$.
2014 Contests, 2
Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.
2008 China Northern MO, 4
As shown in figure , it is known that $ABCD$ is parallelogram, $A,B,C$ lie on circle $\odot O_1$, $AD$ and $BD$ intersect $\odot O$ at points $E$ and $F$ respectively, $C,D,F$ lie on circle $\odot O_2$, $AD$ intersects $\odot O_2$ at point $G$. If the radii of circles $\odot O_1$, $\odot O_2$ are $R_1, R_2$ respectively, prove that $\frac{EG}{AD}=\frac{R_2^2}{R_1^2}$.
[img]https://cdn.artofproblemsolving.com/attachments/d/f/1d9925a77d4f3fe068bd24364fb396eaa9a27a.png[/img]
2007 BAMO, 2
The points of the plane are colored in black and white so that whenever three vertices of a parallelogram are the same color, the fourth vertex is that color, too. Prove that all the points of the plane are the same color.
1996 Bosnia and Herzegovina Team Selection Test, 3
Let $M$ be a point inside quadrilateral $ABCD$ such that $ABMD$ is parallelogram. If $\angle CBM = \angle CDM$ prove that $\angle ACD = \angle BCM$
2006 International Zhautykov Olympiad, 2
Let $ ABC$ be a triangle and $ K$ and $ L$ be two points on $ (AB)$, $ (AC)$ such that $ BK \equal{} CL$ and let $ P \equal{} CK\cap BL$. Let the parallel through $ P$ to the interior angle bisector of $ \angle BAC$ intersect $ AC$ in $ M$. Prove that $ CM \equal{} AB$.
2010 JBMO Shortlist, 2
Let $ABC$ be acute-angled triangle . A circle $\omega_1(O_1,R_1)$ passes through points $B$ and $C$ and meets the sides $AB$ and $AC$ at points $D$ and $E$ ,respectively .
Let $\omega_2(O_2,R_2)$ be the circumcircle of triangle $ADE$ . Prove that $O_1O_2$ is equal to the circumradius of triangle $ABC$ .
2022 Caucasus Mathematical Olympiad, 2
In parallelogram $ABCD$, points $E$ and $F$ on segments $AD$ and $CD$ are such that $\angle BCE=\angle BAF$. Points $K$ and $L$ on segments $AD$ and $CD$ are such that $AK=ED$ and $CL=FD$. Prove that $\angle BKD=\angle BLD$.
2010 Federal Competition For Advanced Students, Part 1, 4
The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$.
(a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side.
(b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side.
(c) For which inner point does the sum of the areas of the three small triangles attain a minimum?
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)[/i]
2011 Postal Coaching, 4
Consider $2011^2$ points arranged in the form of a $2011 \times 2011$ grid. What is the maximum number of points that can be chosen among them so that no four of them form the vertices of either an isosceles trapezium or a rectangle whose parallel sides are parallel to the grid lines?
1999 South africa National Olympiad, 3
The bisector of $\angle{BAD}$ in the parallellogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the centre of the circle passing through the points $C,\ K$ and $L$ lies on the circle passing through the points $B,\ C$ and $D$.
Oliforum Contest I 2008, 2
Let $ ABCD$ be a cyclic quadrilateral with $ AB>CD$ and $ BC>AD$. Take points $ X$ and $ Y$ on the sides $ AB$ and $ BC$, respectively, so that $ AX\equal{}CD$ and $ AD\equal{}CY$. Let $ M$ be the midpoint of $ XY$. Prove that $ AMC$ is a right angle.
2020 Junior Macedonian National Olympiad, 4
Let $ABC$ be an isosceles triangle with base $AC$. Points $D$ and $E$ are chosen on the sides $AC$ and $BC$, respectively, such that $CD = DE$. Let $H, J,$ and $K$ be the midpoints of $DE, AE,$ and $BD$, respectively. The circumcircle of triangle $DHK$ intersects $AD$ at point $F$, whereas the circumcircle of triangle $HEJ$ intersects $BE$ at $G$. The line through $K$ parallel to $AC$ intersects $AB$ at $I$. Let $IH \cap GF =$ {$M$}. Prove that $J, M,$ and $K$ are collinear points.
1983 IMO Longlists, 8
On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.