Found problems: 1546
2001 Tournament Of Towns, 3
Point $A$ lies inside an angle with vertex $M$. A ray issuing from point $A$ is reflected in one side of the angle at point $B$, then in the other side at point $C$ and then returns back to point $A$ (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle $\triangle BCM$ lies on line $AM$.
2012 Sharygin Geometry Olympiad, 11
Given triangle $ABC$ and point $P$. Points $A', B', C'$ are the projections of $P$ to $BC, CA, AB$. A line passing through $P$ and parallel to $AB$ meets the circumcircle of triangle $PA'B'$ for the second time in point $C_{1}$. Points $A_{1}, B_{1}$ are defined similarly. Prove that
a) lines $AA_{1}, BB_{1}, CC_{1}$ concur;
b) triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.
2014 Sharygin Geometry Olympiad, 9
Two circles $\omega_1$ and $\omega_2$ touching externally at point $L$ are inscribed into angle $BAC$. Circle $\omega_1$ touches ray $AB$ at point $E$, and circle $\omega_2$ touches ray $AC$ at point $M$. Line $EL$ meets $\omega_2$ for the second time at point $Q$. Prove that $MQ\parallel AL$.
2005 Tournament of Towns, 2
The altitudes $AD$ and $BE$ of triangle $ABC$ meet at its orthocentre $H$. The midpoints of $AB$ and $CH$ are $X$ and $Y$, respectively. Prove that $XY$ is perpendicular to $DE$.
[i](5 points)[/i]
2010 Tournament Of Towns, 2
Karlson and Smidge divide a cake in a shape of a square in the following way. First, Karlson places a candle on the cake (chooses some interior point). Then Smidge makes a straight cut from the candle to the boundary in the direction of his choice. Then Karlson makes a straight cut from the candle to the boundary in the direction perpendicular to Smidge's cut. As a result, the cake is split into two pieces; Smidge gets the smaller one. Smidge wants to get a piece which is no less than a quarter of the cake. Can Karlson prevent Smidge from getting the piece of that size?
2009 Polish MO Finals, 1
Each vertex of a convex hexagon is the center of a circle whose radius is equal to the shorter side of the hexagon that contains the vertex. Show that if the intersection of all six circles (including their boundaries) is not empty, then the hexagon is regular.
1972 IMO Longlists, 12
A circle $k = (S, r)$ is given and a hexagon $AA'BB'CC'$ inscribed in it. The lengths of sides of the hexagon satisfy $AA' = A'B, BB' = B'C, CC' = C'A$. Prove that the area $P$ of triangle $ABC$ is not greater than the area $P'$ of triangle $A'B'C'$. When does $P = P'$ hold?
2009 South africa National Olympiad, 2
Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?
2003 China Team Selection Test, 3
(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that:
\[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \]
(2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?
2009 Italy TST, 2
$ABC$ is a triangle in the plane. Find the locus of point $P$ for which $PA,PB,PC$ form a triangle whose area is equal to one third of the area of triangle $ABC$.
2009 Greece National Olympiad, 4
Consider pairwise distinct complex numbers $z_1,z_2,z_3,z_4,z_5,z_6$ whose images $A_1,A_2,A_3,A_4,A_5,A_6$ respectively are succesive points on the circle centered at $O(0,0)$ and having radius $r>0.$
If $w$ is a root of the equation $z^2+z+1=0$ and the next equalities hold \[z_1w^2+z_3w+z_5=0 \\ z_2w^2+z_4w+z_6=0\] prove that
[b]a)[/b] Triangle $A_1A_3A_5$ is equilateral
[b]b)[/b] \[|z_1-z_2|+|z_2-z_3|+|z_3-z_4|+|z_4-z_5|+z_5-z_6|+|z_6-z_1|=3|z_1-z_4|=3|z_2-z_5|=3|z_3-z_6|.\]
1990 IMO Longlists, 91
Quadrilateral $ABCD$ has an inscribed circle with center $O$. Knowing that $AB = CD$, and $M, K$ are the midpoints of $BC, AD$ respectively. Prove that $OM = OK.$
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.
1995 Vietnam Team Selection Test, 1
Let be given a triangle $ ABC$ with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$. Six distinct points $ A_1$, $ A_2$, $ B_1$, $ B_2$, $ C_1$, $ C_2$ not coinciding with $ A$, $ B$, $ C$ are chosen so that $ A_1$, $ A_2$ lie on line $ BC$; $ B_1$, $ B_2$ lie on $ CA$ and $ C_1$, $ C_2$ lie on $ AB$. Let $ \alpha$, $ \beta$, $ \gamma$ three real numbers satisfy $ \overrightarrow{A_1A_2} \equal{} \frac {\alpha}{a}\overrightarrow{BC}$, $ \overrightarrow{B_1B_2} \equal{} \frac {\beta}{b}\overrightarrow{CA}$, $ \overrightarrow{C_1C_2} \equal{} \frac {\gamma}{c}\overrightarrow{AB}$. Let $ d_A$, $ d_B$, $ d_C$ be respectively the radical axes of the circumcircles of the pairs of triangles $ AB_1C_1$ and $ AB_2C_2$; $ BC_1A_1$ and $ BC_2A_2$; $ CA_1B_1$ and $ CA_2B_2$. Prove that $ d_A$, $ d_B$ and $ d_C$ are concurrent if and only if $ \alpha a \plus{} \beta b \plus{} \gamma c \neq 0$.
1990 APMO, 5
Show that for every integer $n \geq 6$, there exists a convex hexagon which can be dissected into exactly $n$ congruent triangles.
1982 IMO Longlists, 5
Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.
2010 Sharygin Geometry Olympiad, 15
Let $AA_1, BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC.$ $AA_1$ meets $B_1C_1$ in a point $K.$ The circumcircles of triangles $A_1KC_1$ and $A_1KB_1$ intersect the lines $AB$ and $AC$ for the second time at points $N$ and $L$ respectively. Prove that
[b]a)[/b] The sum of diameters of these two circles is equal to $BC,$
[b] b)[/b] $\frac{A_1N}{BB_1} + \frac{A_1L}{CC_1}=1.$
2007 Brazil National Olympiad, 3
Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.
2010 Contests, 1
In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.
2012 Baltic Way, 15
The circumcentre $O$ of a given cyclic quadrilateral $ABCD$ lies inside the quadrilateral but not on the diagonal $AC$. The diagonals of the quadrilateral intersect at $I$. The circumcircle of the triangle $AOI$ meets the sides $AD$ and $AB$ at points $P$ and $Q$, respectively; the circumcircle of the triangle $COI$ meets the sides $CB$ and $CD$ at points $R$ and $S$, respectively. Prove that $PQRS$ is a parallelogram.
1982 IMO Longlists, 10
Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that
\[\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.\]
1988 IMO Longlists, 58
For a convex polygon $P$ in the plane let $P'$ denote the convex polygon with vertices at the midpoints of the sides of $P.$ Given an integer $n \geq 3,$ determine sharp bounds for the ratio
\[ \frac{\text{area } P'}{\text{area } P}, \] over all convex $n$-gons $P.$
1998 Iran MO (3rd Round), 2
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE$ and $EF = FA$. Prove that
\[\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.\]
2008 China Girls Math Olympiad, 3
Determine the least real number $ a$ greater than $ 1$ such that for any point $ P$ in the interior of the square $ ABCD$, the area ratio between two of the triangles $ PAB$, $ PBC$, $ PCD$, $ PDA$ lies in the interval $ \left[\frac {1}{a},a\right]$.
2002 Italy TST, 1
Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.