Found problems: 1546
1994 Kurschak Competition, 1
The ratio of the sides of a parallelogram is $\lambda>1$. Given $\lambda$, determine the maximum of the acute angle subtended by the diagonals of the parallelogram.
2012 Sharygin Geometry Olympiad, 10
In a convex quadrilateral all sidelengths and all angles are pairwise different.
a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side?
b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?
2013 Iran MO (3rd Round), 2
We define the distance between two circles $\omega ,\omega '$by the length of the common external tangent of the circles and show it by $d(\omega , \omega ')$. If two circles doesn't have a common external tangent then the distance between them is undefined. A point is also a circle with radius $0$ and the distance between two cirlces can be zero.
(a) [b]Centroid.[/b] $n$ circles $\omega_1,\dots, \omega_n$ are fixed on the plane. Prove that there exists a unique circle $\overline \omega$ such that for each circle $\omega$ on the plane the square of distance between $\omega$ and $\overline \omega$ minus the sum of squares of distances of $\omega$ from each of the $\omega_i$s $1\leq i \leq n$ is constant, in other words:\[d(\omega,\overline \omega)^2-\frac{1}{n}{\sum_{i=1}}^n d(\omega_i,\omega)^2= constant\]
(b) [b]Perpendicular Bisector.[/b] Suppose that the circle $\omega$ has the same distance from $\omega_1,\omega_2$. Consider $\omega_3$ a circle tangent to both of the common external tangents of $\omega_1,\omega_2$. Prove that the distance of $\omega$ from centroid of $\omega_1 , \omega_2$ is not more than the distance of $\omega$ and $\omega_3$. (If the distances are all defined)
(c) [b]Circumcentre.[/b] Let $C$ be the set of all circles that each of them has the same distance from fixed circles $\omega_1,\omega_2,\omega_3$. Prove that there exists a point on the plane which is the external homothety center of each two elements of $C$.
(d) [b]Regular Tetrahedron.[/b] Does there exist 4 circles on the plane which the distance between each two of them equals to $1$?
Time allowed for this problem was 150 minutes.
1989 China Team Selection Test, 1
A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.
2008 Cono Sur Olympiad, 2
Let $P$ be a point in the interior of triangle $ABC$. Let $X$, $Y$, and $Z$ be points on sides $BC$, $AC$, and $AB$ respectively, such that
$<PXC=<PYA=<PZB$.
Let $U$, $V$, and $W$ be points on sides $BC$, $AC$, and $AB$, respectively, or on their extensions if necessary, with $X$ in between $B$ and $U$, $Y$ in between $C$ and $V$, and $Z$ in between $A$ and $W$, such that $PU=2PX$, $PV=2PY$, and $PW=2PZ$. If the area of triangle $XYZ$ is $1$, find the area of triangle $UVW$.
1983 IMO Longlists, 75
Find the sum of the fiftieth powers of all sides and diagonals of a regular $100$-gon inscribed in a circle of radius $R.$
2010 Tournament Of Towns, 3
Is it possible to cover the surface of a regular octahedron by several regular hexagons without gaps and overlaps? (A regular octahedron has $6$ vertices, each face is an equilateral triangle, each vertex belongs to $4$ faces.)
2002 Mediterranean Mathematics Olympiad, 3
In an acute-angled triangle $ABC$, $M$ and $N$ are points on the sides $AC$ and $BC$ respectively, and $K$ the midpoint of $MN$. The circumcircles of triangles $ACN$ and $BCM$ meet again at a point $D$. Prove that the line $CD$ contains the circumcenter $O$ of $\triangle ABC$ if and only if $K$ is on the perpendicular bisector of $AB.$
2005 China Team Selection Test, 2
In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.
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.
2012 Mexico National Olympiad, 6
Consider an acute triangle $ABC$ with circumcircle $\mathcal{C}$. Let $H$ be the orthocenter of $ABC$ and $M$ the midpoint of $BC$. Lines $AH$, $BH$ and $CH$ cut $\mathcal{C}$ again at points $D$, $E$, and $F$ respectively; line $MH$ cuts $\mathcal{C}$ at $J$ such that $H$ lies between $J$ and $M$. Let $K$ and $L$ be the incenters of triangles $DEJ$ and $DFJ$ respectively. Prove $KL$ is parallel to $BC$.
2013 Iran MO (3rd Round), 8
Let $A_1A_2A_3A_4A_5$ be a convex 5-gon in which the coordinates of all of it's vertices are rational. For each $1\leq i \leq 5$ define $B_i$ the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i+3}A_{i+4}$.
($A_i=A_{i+5}$) Prove that at most 3 lines from the lines $A_iB_i$ ($1\leq i \leq 5$) are concurrent.
Time allowed for this problem was 75 minutes.
2008 Sharygin Geometry Olympiad, 3
(R.Pirkuliev) Prove the inequality
\[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}},
\]
where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.
2014 Contests, 2
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.
[i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]
2011 Croatia Team Selection Test, 3
Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.
1997 Federal Competition For Advanced Students, P2, 3
Let be given a triangle $ ABC$. Points $ P$ on side $ AC$ and $ Y$ on the production of $ CB$ beyond $ B$ are chosen so that $ Y$ subtends equal angles with $ AP$ and $ PC$. Similarly, $ Q$ on side $ BC$ and $ X$ on the production of $ AC$ beyond $ C$ are such that $ X$ subtends equal angles with $ BQ$ and $ QC$. Lines $ YP$ and $ XB$ meet at $ R$, $ XQ$ and $ YA$ meet at $ S$, and $ XB$ and $ YA$ meet at $ D$. Prove that $ PQRS$ is a parallelogram if and only if $ ACBD$ is a cyclic quadrilateral.
1972 IMO Longlists, 39
How many tangents to the curve $y = x^3-3x\:\: (y = x^3 + px)$ can be drawn from different points in the plane?
1996 Austrian-Polish Competition, 2
A convex hexagon $ ABCDEF$ satisfies the following conditions:
1) $ AB\parallel DE$, $ BC\parallel EF$, and $ CD\parallel FA$.
2) The distances between these pairs of parallel lines are the same.
3) $ \angle FAB \equal{} \angle CDE \equal{} 90^\circ$
Prove that the diagonals $ BE$ and $ CF$ of the hexagon intersect with angle $ 45$ degrees.
$ \bullet$ Thank you dear [b]Babis Stergiou[/b] for your translation. :P
2007 Ukraine Team Selection Test, 10
Find all positive integers $ n$ such that acute-angled $ \triangle ABC$ with $ \angle BAC<\frac{\pi}{4}$ could be divided into $ n$ quadrilateral. Every quadrilateral is inscribed in circle and radiuses of circles are in geometric progression.
[hide] be carefull ! :lol: [/hide]
2012 Portugal MO, 2
In triangle $[ABC]$, the bissector of the angle $\angle{BAC}$ intersects the side $[BC]$ at $D$. Suppose that $\overline{AD}=\overline{CD}$. Find the lengths $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ that minimize the perimeter of $[ABC]$, given that all the sides of the triangles $[ABC]$ and $[ADC]$ have integer lengths.
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 $AC.$
2006 China Team Selection Test, 1
$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$).
Prove that $l_1 || l_2$.
2012 Indonesia TST, 2
Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.
1996 Kurschak Competition, 1
Prove that in a trapezoid with perpendicular diagonals, the product of the legs is at least as much as the product of the bases.
2011 Albania National Olympiad, 3
In a convex quadrilateral $ABCD$ ,$\angle ABC$ and $\angle BCD$ are $\geq 120^o$. Prove that $|AC|$ + $|BD| \geq |AB|+|BC|+|CD|$. (With $|XY|$ we understand the length of the segment $XY$).