Found problems: 1049
2007 Junior Balkan Team Selection Tests - Romania, 1
Consider $ \rho$ a semicircle of diameter $ AB$. A parallel to $ AB$ cuts the semicircle at $ C, D$ such that $ AD$ separates $ B, C$. The parallel at $ AD$ through $ C$ intersects the semicircle the second time at $ E$. Let $ F$ be the intersection point of the lines $ BE$ and $ CD$. The parallel through $ F$ at $ AD$ cuts $ AB$ in $ P$. Prove that $ PC$ is tangent to $ \rho$.
[i]Author: Cosmin Pohoata[/i]
2014 Dutch Mathematical Olympiad, 2 juniors
Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles.
[asy]
unitsize(1.5 cm);
pair A, B, C, D, G, H;
A = (0,0);
B = (2,0);
D = (0.5,1.5);
C = B + D - A;
G = reflect(A,B)*(C) + C - B;
H = reflect(B,C)*(H) + A - B;
draw(H--A--D--C--G);
draw(interp(A,G,-0.1)--interp(A,G,1.1));
draw(interp(C,H,-0.1)--interp(C,H,1.1));
draw(D--G--H--cycle, dashed);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, E);
dot("$D$", D, NW);
dot("$G$", G, NE);
dot("$H$", H, SE);
[/asy]
Ukrainian From Tasks to Tasks - geometry, 2016.8
Let $ABCD$ be a convex quadrilateral. It is known that $S_{ABD} = 7$, $S_{BCD}= 5$ and $S_{ABC}= 3$. Inside the quadrilateral mark the point $X$ so that $ABCX$ is a parallelogram. Find $S_{ADX}$ and $S_{BDX}$.
2021 Kyiv City MO Round 1, 10.3
Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. Let point $C$ be such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ at points $X$ and $Y$, respectively. Prove that $CX = CY$.
[i]Proposed by Oleksii Masalitin[/i]
2020 Brazil Cono Sur TST, 2
Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.
2023 All-Russian Olympiad Regional Round, 10.8
The bisector of $\angle BAD$ of a parallelogram $ABCD$ meets $BC$ at $K$. The point $L$ lies on $AB$ such that $AL=CK$. The lines $AK$ and $CL$ meet at $M$. Let $(ALM)$ meet $AD$ after $D$ at $N$. Prove that $\angle CNL=90^{o}$
2013 Costa Rica - Final Round, G2
Consider the triangle $ABC$. Let $P, Q$ inside the angle $A$ such that $\angle BAP=\angle CAQ$ and $PBQC$ is a parallelogram. Show that $\angle ABP=\angle ACP.$
2012 China Western Mathematical Olympiad, 1
$O$ is the circumcenter of acute $\Delta ABC$, $H$ is the Orthocenter. $AD \bot BC$, $EF$ is the perpendicular bisector of $AO$,$D,E$ on the $BC$. Prove that the circumcircle of $\Delta ADE$ through the midpoint of $OH$.
2012 APMO, 4
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.
(Here we denote $XY$ the length of the line segment $XY$.)
2004 CHKMO, 3
Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.
1996 Argentina National Olympiad, 4
Let $ABCD$ be a parallelogram with center $O$ such that $\angle BAD <90^o$ and $\angle AOB> 90^o$. Consider points $A_1$ and $B_1$ on the rays $OA$ and $OB$ respectively, such that $A_1B_1$ is parallel to $AB$ and $\angle A_1B_1C = \frac12 \angle ABC$. Prove that $A_1D$ is perpendicular to $B_1C$.
2011 Rioplatense Mathematical Olympiad, Level 3, 4
We consider $\Gamma_1$ and $\Gamma_2$ two circles that intersect at points $P$ and $Q$ . Let $A , B$ and $C$ be points on the circle $\Gamma_1$ and $D , E$ and $F$ points on the circle $\Gamma_2$ so that the lines $A E$ and $B D$ intersect at $P$ and the lines $A F$ and $C D$ intersect at $Q$. Denote $M$ and $N$ the intersections of lines $A B$ and $D E$ and of lines $A C$ and $D F$ , respectively. Show that $A M D N$ is a parallelogram.
2010 Sharygin Geometry Olympiad, 18
A point $B$ lies on a chord $AC$ of circle $\omega.$ Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F,$ and the rays $AD$ and $CE$ do in a point $G.$ Prove that the line $FG$ passes through the midpoint of the segment $AC.$
2004 All-Russian Olympiad, 3
A triangle $ T$ is contained inside a point-symmetrical polygon $ M.$ The triangle $ T'$ is the mirror image of the triangle $ T$ with the reflection at one point $ P$, which inside the triangle $ T$ lies. Prove that at least one of the vertices of the triangle $ T'$ lies in inside or on the boundary of the polygon $ M.$
Kyiv City MO Seniors 2003+ geometry, 2021.10.3
Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$.
(Oleksii Masalitin)
2001 IMO Shortlist, 3
Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.
2012 Math Prize for Girls Olympiad, 1
Let $A_1A_2 \dots A_n$ be a polygon (not necessarily regular) with $n$ sides. Suppose there is a translation that maps each point $A_i$ to a point $B_i$ in the same plane. For convenience, define $A_0 = A_n$ and $B_0 = B_n$. Prove that
\[
\sum_{i=1}^{n} (A_{i-1} B_{i})^2 = \sum_{i=1}^{n} (B_{i-1} A_{i})^2 \, .
\]
1984 Poland - Second Round, 2
We construct similar isosceles triangles on the sides of the triangle $ ABC $: triangle $ APB $ outside the triangle $ ABC $ ($ AP = PB $), triangle $ CQA $ outside the triangle $ ABC $ ($ CQ = QA $), triangle $ CRB $ inside the triangle $ ABC $ ($ CR = RB $). Prove that $ APRQ $ is a parallelogram or that the points $ A, P, R, Q $ lie on a straight line.
2013 Online Math Open Problems, 17
Let $ABXC$ be a parallelogram. Points $K,P,Q$ lie on $\overline{BC}$ in this order such that $BK = \frac{1}{3} KC$ and $BP = PQ = QC = \frac{1}{3} BC$. Rays $XP$ and $XQ$ meet $\overline{AB}$ and $\overline{AC}$ at $D$ and $E$, respectively. Suppose that $\overline{AK} \perp \overline{BC}$, $EK-DK=9$ and $BC=60$. Find $AB+AC$.
[i]Proposed by Evan Chen[/i]
2005 China Team Selection Test, 2
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.
(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
2001 JBMO ShortLists, 12
Consider the triangle $ABC$ with $\angle A= 90^{\circ}$ and $\angle B \not= \angle C$. A circle $\mathcal{C}(O,R)$ passes through $B$ and $C$ and intersects the sides $AB$ and $AC$ at $D$ and $E$, respectively. Let $S$ be the foot of the perpendicular from $A$ to $BC$ and let $K$ be the intersection point of $AS$ with the segment $DE$. If $M$ is the midpoint of $BC$, prove that $AKOM$ is a parallelogram.
2015 District Olympiad, 1
Consider the parallelogram $ ABCD, $ whose diagonals intersect at $ O. $ The bisector of the angle $ \angle DAC $ and that of $ \angle DBC $ intersect each other at $ T. $ Moreover, $ \overrightarrow{TD} +\overrightarrow{TC} =\overrightarrow{TO} . $
Find the angles of the triangle $ ABT. $
2013 Polish MO Finals, 3
Given is a quadrilateral $ABCD$ in which we can inscribe circle. The segments $AB, BC, CD$ and $DA$ are the diameters of the circles $o1, o2, o3$ and $o4$, respectively. Prove that there exists a circle tangent to all of the circles $o1, o2, o3$ and $o4$.
2002 AMC 12/AHSME, 23
In $ \triangle{ABC}$, we have $ AB\equal{}1$ and $ AC\equal{}2$. Side $ BC$ and the median from $ A$ to $ BC$ have the same length. What is $ BC$?
$ \textbf{(A)}\ \frac{1\plus{}\sqrt2}{2} \qquad
\textbf{(B)}\ \frac{1\plus{}\sqrt3}{2} \qquad
\textbf{(C)}\ \sqrt2 \qquad
\textbf{(D)}\ \frac{3}{2} \qquad
\textbf{(E)}\ \sqrt3$