Found problems: 1546
2012 All-Russian Olympiad, 2
The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.
2007 Estonia National Olympiad, 1
Consider a cylinder and a cone with a common base such that the volume of the
part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.
1998 Vietnam Team Selection Test, 2
In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$. The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation: \[ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) \] is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.
1999 IberoAmerican, 3
Let $A$ and $B$ points in the plane and $C$ a point in the perpendiclar bisector of $AB$. It is constructed a sequence of points $C_1,C_2,\dots, C_n,\dots$ in the following way: $C_1=C$ and for $n\geq1$, if $C_n$ does not belongs to $AB$, then $C_{n+1}$ is the circumcentre of the triangle $\triangle{ABC_n}$.
Find all the points $C$ such that the sequence $C_1,C_2,\dots$ is defined for all $n$ and turns eventually periodic.
Note: A sequence $C_1,C_2, \dots$ is called eventually periodic if there exist positive integers $k$ and $p$ such that $C_{n+p}=c_n$ for all $n\geq{k}$.
1994 Baltic Way, 14
Let $\alpha,\beta,\gamma$ be the angles of a triangle opposite to its sides with lengths $a,b,c$ respectively. Prove the inequality
\[a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)\]
1982 IMO Longlists, 39
Let $S$ be the unit circle with center $O$ and let $P_1, P_2,\ldots, P_n$ be points of $S$ such that the sum of vectors $v_i=\stackrel{\longrightarrow}{OP_i}$ is the zero vector. Prove that the inequality $\sum_{i=1}^n XP_i \geq n$ holds for every point $X$.
2005 Tuymaada Olympiad, 4
In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle.
[i]Proposed by L. Emelyanov[/i]
2001 Tournament Of Towns, 6
Let $AH_A$, $BH_B$ and $CH_C$ be the altitudes of triangle $\triangle ABC$. Prove that the triangle whose vertices are the intersection points of the altitudes of triangles $\triangle AH_BH_C$, $\triangle BH_AH_C$ and $\triangle CH_AH_B$ is equal to triangle $\triangle H_AH_BH_C$.
2001 Tournament Of Towns, 5
Nine points are drawn on the surface of a regular tetrahedron with an edge of $1$ cm. Prove that among these points there are two located at a distance (in space) no greater than $0.5$ cm.
2002 Iran MO (2nd round), 4
Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$
1994 IberoAmerican, 1
Let $A,\ B$ and $C$ be given points on a circumference $K$ such that the triangle $\triangle{ABC}$ is acute. Let $P$ be a point in the interior of $K$. $X,\ Y$ and $Z$ be the other intersection of $AP, BP$ and $CP$ with the circumference. Determine the position of $P$ such that $\triangle{XYZ}$ is equilateral.
2015 IFYM, Sozopol, 2
Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.
2012 Sharygin Geometry Olympiad, 13
Points $A, B$ are given. Find the locus of points $C$ such that $C$, the midpoints of $AC, BC$ and the centroid of triangle $ABC$ are concyclic.
1997 IberoAmerican, 3
Let $P = \{P_1, P_2, ..., P_{1997}\}$ be a set of $1997$ points in the interior of a circle of radius 1, where $P_1$ is the center of the circle. For each $k=1.\ldots,1997$, let $x_k$ be the distance of $P_k$ to the point of $P$ closer to $P_k$, but different from it. Show that $(x_1)^2 + (x_2)^2 + ... + (x_{1997})^2 \le 9.$
2010 China Team Selection Test, 1
Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.
2002 Kurschak Competition, 1
We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$, $O$, $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$, then there must be another vertex on this circle as well.
2001 Tournament Of Towns, 4
Let $n\ge2$ be an integer. A regular $(2n+1)-gon$ is divided in to $2n-1$ triangles by diagonals which do not meet except at the vertices. Prove that at least three of these triangles are isosceles.
2006 Estonia National Olympiad, 4
Triangle $ ABC$ is isosceles with $ AC \equal{} BC$ and $ \angle{C} \equal{} 120^o$. Points $ D$ and $ E$ are chosen on segment $ AB$ so that $ |AD| \equal{} |DE| \equal{} |EB|$. Find the sizes of the angles of triangle $ CDE$.
2003 Baltic Way, 14
Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.
2012 CentroAmerican, 2
Let $\gamma$ be the circumcircle of the acute triangle $ABC$. Let $P$ be the midpoint of the minor arc $BC$. The parallel to $AB$ through $P$ cuts $BC, AC$ and $\gamma$ at points $R,S$ and $T$, respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$. Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$.
2007 Estonia Math Open Senior Contests, 8
Tangents $ l_1$ and $ l_2$ common to circles $ c_1$ and $ c_2$ intersect at point $ P$, whereby tangent points remain to different sides from $ P$ on both tangent lines. Through some point $ T$, tangents $ p_1$ and $ p_2$ to circle $ c_1$ and tangents $ p_3$ and $ p_4$ to circle $ c_2$ are drawn. The intersection points of $ l_1$ with lines $ p_1, p_2, p_3, p_4$ are $ A_1, B_1, C_1, D_1$, respectively, whereby the order of points on $ l_1$ is: $ A_1, B_1, P, C_1, D_1$. Analogously, the intersection points of $ l_2$ with lines $ p_1, p_2, p_3, p_4$ are $ A_2, B_2, C_2, D_2$, respectively. Prove that if both quadrangles $ A_1A_2D_1D_2$ and $ B_1B_2C_1C_2$ are cyclic then radii of $ c_1$ and $ c_2$ are equal.
2004 South East Mathematical Olympiad, 6
ABC is an isosceles triangle with AB=AC. Point D lies on side BC. Point F is inside $\triangle$ABC and lies on the circumcircle of triangle ADC. The circumcircle of triangle BDF intersects side AB at point E. Prove that $CD\cdot EF+DF\cdot AE=BD\cdot AF$.
2015 IFYM, Sozopol, 1
Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.
2011 Serbia National Math Olympiad, 3
Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$. $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$, respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$, and $M$ is midpoint of $AB$. Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$.
2001 India IMO Training Camp, 3
In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that:
\[AL+BM+CN \leq 3(AD+BE+CF)\]
When does equality occur?