Found problems: 1049
2008 Balkan MO, 1
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.
1997 Austrian-Polish Competition, 1
Let $P$ be the intersection of lines $l_1$ and $l_2$. Let $S_1$ and $S_2$ be two
circles externally tangent at $P$ and both tangent to $l_1$, and let $T_1$
and $T_2$ be two circles externally tangent at $P$ and both tangent to $l_2$.
Let $A$ be the second intersection of $S_1$ and $T_1, B$ that of $S_1$ and $T_2,
C$ that of $S_2$ and $T_1$, and $D$ that of $S_2$ and $T_2$. Show that the points $A,B,C,D$ are concyclic if and only if $l_1$ and $l_2$ are perpendicular.
2013 Saint Petersburg Mathematical Olympiad, 3
Let $M$ and $N$ are midpoint of edges $AB$ and $CD$ of the tetrahedron $ABCD$, $AN=DM$ and $CM=BN$. Prove that $AC=BD$.
S. Berlov
2008 National Olympiad First Round, 29
$[AB]$ and $[CD]$ are not parallel in the convex quadrilateral $ABCD$. Let $E$ and $F$ be the midpoints of $[AD]$ and $[BC]$, respectively. If $|CD|=12$, $|AB|=22$, and $|EF|=x$, what is the sum of integer values of $x$?
$
\textbf{(A)}\ 110
\qquad\textbf{(B)}\ 114
\qquad\textbf{(C)}\ 118
\qquad\textbf{(D)}\ 121
\qquad\textbf{(E)}\ \text{None of the above}
$
2003 China Western Mathematical Olympiad, 2
A circle can be inscribed in the convex quadrilateral $ ABCD$. The incircle touches the sides $ AB, BC, CD, DA$ at $ A_1, B_1, C_1, D_1$ respectively. The points $ E, F, G, H$ are the midpoints of $ A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $ EFGH$ is a rectangle if and only if $ A, B, C, D$ are concyclic.
2019 Saudi Arabia JBMO TST, 1
Let $E$ be a point lies inside the parallelogram $ABCD$ such that $\angle BCE = \angle BAE$.
Prove that the circumcenters of triangles $ABE,BCE,CDE,DAE$ are concyclic.
2016 CMIMC, 6
In parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at a point $P$ inside the parallelogram. If $PB=700$ and $PD=821$, what is $AC$?
2012 All-Russian Olympiad, 3
Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.
2001 Switzerland Team Selection Test, 3
In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.
2001 All-Russian Olympiad Regional Round, 9.3
In parallelogram $ABCD$, points $M$ and $N$ are selected on sides $AB$ and $BC$ respectively so that $AM = NC$, $Q$ is the intersection point of segments $AN$ and $CM$. Prove that $DQ$ is the bisector of angle $D$.
2022 International Zhautykov Olympiad, 3
In parallelogram $ABCD$ with acute angle $A$ a point $N$ is chosen on the segment $AD$, and a point $M$ on the segment $CN$ so that $AB = BM = CM$. Point $K$ is the reflection of $N$ in line $MD$. The line $MK$ meets the segment $AD$ at point $L$. Let $P$ be the common point of the circumcircles of $AMD$ and $CNK$ such that $A$ and $P$ share the same side of the line $MK$. Prove that $\angle CPM = \angle DPL$.
2014 Serbia JBMO TST, 3
Consider parallelogram $ABCD$, with acute angle at $A$, $AC$ and $BD$ intersect at $E$. Circumscribed circle of triangle $ACD$ intersects $AB$, $BC$ and $BD$ at $K$, $L$ and $P$ (in that order). Then, circumscribed circle of triangle $CEL$ intersects $BD$ at $M$. Prove that: $$KD*KM=KL*PC$$
2019 Costa Rica - Final Round, 2
Consider the parallelogram $ABCD$, with $\angle ABC = 60$ and sides $AB =\sqrt3$, $BC = 1$. Let $\omega$ be the circle of center $B$ and radius $BA$, and let $\tau$ be the circle of center $D$ and radius $DA$. Determine the area of the region between the circumferences $\omega$ and $\tau$, within the parallelogram $ABCD$ (the area of the shaded region).
[img]https://cdn.artofproblemsolving.com/attachments/5/a/02b17ec644289d95b6fce78cb5f1ecb3d3ba5b.png[/img]
2004 Germany Team Selection Test, 1
The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$.
Show that the line $CX$ bisects the angle $ACN$.
2010 Vietnam National Olympiad, 3
In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$
such that $BC$ not is the diameter.Consider a point $A$ varies in
$(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$
respective is intersect of $BC$ and internal and external bisector
of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through
orthocenter of $\triangle ABC$
and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.
1/Prove that $MN$ pass through a fixed point
2/Determint the place of $A$ such that $S_{AMN}$ has maxium value
2007 Danube Mathematical Competition, 2
Let $ ABCD$ be an inscribed quadrilateral and let $ E$ be the midpoint of the diagonal $ BD$. Let $ \Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be the circumcircles of triangles $ AEB$, $ BEC$, $ CED$ and $ DEA$ respectively. Prove that if $ \Gamma_4$ is tangent to the line $ CD$, then $ \Gamma_1,\Gamma_2,\Gamma_3$ are tangent to the lines $ BC,AB,AD$ respectively.
2015 Sharygin Geometry Olympiad, P4
In a parallelogram $ABCD$ the trisectors of angles $A$ and $B$ are drawn. Let $O$ be the common points of the trisectors nearest to $AB$. Let $AO$ meet the second trisector of angle $B$ at point $A_1$, and let $BO$ meet the second trisector of angle $A$ at point $B_1$. Let $M$ be the midpoint of $A_1B_1$. Line $MO$ meets $AB$ at point $N$ Prove that triangle $A_1B_1N$ is equilateral.
1998 Vietnam National Olympiad, 2
Let be given a tetrahedron whose circumcenter is $O$. Draw diameters $AA_{1},BB_{1},CC_{1},DD_{1}$ of the circumsphere of $ABCD$. Let $A_{0},B_{0},C_{0},D_{0}$ be the centroids of triangle $BCD,CDA,DAB,ABC$. Prove that $A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}$ are concurrent at a point, say, $F$. Prove that the line through $F$ and a midpoint of a side of $ABCD$ is perpendicular to the opposite side.
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
\[
\sum_{i=1}^{n} (A_{i-1} B_{i})^2 = \sum_{i=1}^{n} (B_{i-1} A_{i})^2 \, .
\]
2015 Junior Balkan Team Selection Tests - Romania, 1
Let $ABC$ be an acute triangle with $AB \neq AC$ . Also let $M$ be the midpoint of the side $BC$ , $H$ the orthocenter of the triangle $ABC$ , $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$ . Prove that $O_1AMO_2$ is a parallelogram .
1978 Austrian-Polish Competition, 2
A parallelogram is inscribed into a regular hexagon so that the centers of symmetry of both figures coincide. Prove that the area of the parallelogram does not exceed $2/3$ the area of the hexagon.
2003 Hong kong National Olympiad, 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.
2007 Austria Beginners' Competition, 4
Consider a parallelogram $ABCD$ such that the midpoint $M$ of the side $CD$ lies on the angle bisector of $\angle BAD$. Show that $\angle AMB$ is a right angle.
2003 IMAR Test, 3
The exinscribed circle of a triangle $ABC$ corresponding to its vertex $A$ touches the sidelines $AB$ and $AC$ in the points $M$ and $P$, respectively, and touches its side $BC$ in the point $N$. Show that if the midpoint of the segment $MP$ lies on the circumcircle of triangle $ABC$, then the points $O$, $N$, $I$ are collinear, where $I$ is the incenter and $O$ is the circumcenter of triangle $ABC$.
1972 USAMO, 5
A given convex pentagon $ ABCDE$ has the property that the area of each of five triangles $ ABC, BCD, CDE, DEA$, and $ EAB$ is unity [i](equal to 1)[/i]. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.