Found problems: 1546
1986 China Team Selection Test, 4
Given a triangle $ABC$ for which $C=90$ degrees, prove that given $n$ points inside it, we can name them $P_1, P_2 , \ldots , P_n$ in some way such that:
$\sum^{n-1}_{k=1} \left( P_K P_{k+1} \right)^2 \leq AB^2$ (the sum is over the consecutive square of the segments from $1$ up to $n-1$).
[i]Edited by orl.[/i]
2013 South africa National Olympiad, 6
Let $ABC$ be an acute-angled triangle with $AC \neq BC$, and let $O$ be the circumcentre and $F$ the foot of the altitude through $C$. Furthermore, let $X$ and $Y$ be the feet of the perpendiculars dropped from $A$ and $B$ respectively to (the extension of) $CO$. The line $FO$ intersects the circumcircle of $FXY$ a second time at $P$. Prove that $OP<OF$.
2005 Tournament of Towns, 4
$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$.
[i](5 points)[/i]
2000 Baltic Way, 5
Let $ ABC$ be a triangle such that \[ \frac{BC}{AB \minus{} BC}\equal{}\frac{AB \plus{} BC}{AC}\] Determine the ratio $ \angle A : \angle C$.
1970 IMO Longlists, 32
Let there be given an acute angle $\angle AOB = 3\alpha$, where $\overline{OA}= \overline{OB}$. The point $A$ is the center of a circle with radius $\overline{OA}$. A line $s$ parallel to $OA$ passes through $B$. Inside the given angle a variable line $t$ is drawn through $O$. It meets the circle in $O$ and $C$ and the given line $s$ in $D$, where $\angle AOC = x$. Starting from an arbitrarily chosen position $t_0$ of $t$, the series $t_0, t_1, t_2, \ldots$ is determined by defining $\overline{BD_{i+1}}=\overline{OC_i}$ for each $i$ (in which $C_i$ and $D_i$ denote the positions of $C$ and $D$, corresponding to $t_i$). Making use of the graphical representations of $BD$ and $OC$ as functions of $x$, determine the behavior of $t_i$ for $i\to \infty$.
2007 Tournament Of Towns, 6
Let $P$ and $Q$ be two convex polygons. Let $h$ be the length of the projection of $Q$ onto a line perpendicular to a side of $P$ which is of length $p$. Define $f(P,Q)$ to be the sum of the products $hp$ over all sides of $P$. Prove that $f(P,Q) = f(Q, P)$.
2008 Germany Team Selection Test, 1
Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \left\{A,B,C\right\}$ can be mapped to the set $ \left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?
1976 IMO Longlists, 17
Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are $\sqrt{3} :\sqrt{3} :2$.
2009 Hong Kong TST, 3
Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE \equal{} A'$, $ CE\cap DA \equal{} B'$, $ DA\cap EB \equal{} C'$, $ EB\cap AC \equal{} D'$ and $ AC\cap BD \equal{} E'$. Suppose also that $ eABD'\cap eAC'E \equal{} A''$, $ eBCE'\cap eBD'A \equal{} B''$, $ eCDA'\cap eCE'B \equal{} C''$, $ eDEB'\cap eDA'C \equal{} D''$, $ eEAC'\cap eEB'D \equal{} E''$. Prove that $ AA'', BB'', CC'', DD'', EE''$ are concurrent. (Here $ l_1\cap l_2 \equal{} P$ means that $ P$ is the intersection of lines $ l_1$ and $ l_2$. Also $ eA_1A_2A_3\cap eB_1B_2B_3 \equal{} Q$ means that $ Q$ is the intersection of the circumcircles of $ \Delta A_1A_2A_3$ and $ \Delta B_1B_2B_3$.)
2011 Albania National Olympiad, 5
The triangle $ABC$ acute with gravity center $M$ is such that $\angle AMB = 2 \angle ACB$. Prove that:
[b](a)[/b] $AB^4=AC^4+BC^4-AC^2 \cdot BC^2,$
[b](b)[/b] $\angle ACB \geq 60^o$.
2001 Czech-Polish-Slovak Match, 2
A triangle $ABC$ has acute angles at $A$ and $B$. Isosceles triangles $ACD$ and $BCE$ with bases $AC$ and $BC$ are constructed externally to triangle $ABC$ such that $\angle ADC = \angle ABC$ and $\angle BEC = \angle BAC$. Let $S$ be the circumcenter of $\triangle ABC$. Prove that the length of the polygonal line $DSE$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.
2003 Indonesia MO, 2
Let $ABCD$ be a quadrilateral, and $P,Q,R,S$ are the midpoints of $AB, BC, CD, DA$ respectively. Let $O$ be the intersection between $PR$ and $QS$. Prove that $PO = OR$ and $QO = OS$.
2007 India National Olympiad, 1
In a triangle $ ABC$ right-angled at $ C$ , the median through $ B$ bisects the angle between $ BA$ and the bisector of $ \angle B$. Prove that
\[ \frac{5}{2} < \frac{AB}{BC} < 3\]
2010 Sharygin Geometry Olympiad, 25
For two different regular icosahedrons it is known that some six of their vertices are vertices of a regular octahedron. Find the ratio of the edges of these icosahedrons.
2008 Sharygin Geometry Olympiad, 2
(F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$. Determine the ratio $ CK/LB$.
1981 Vietnam National Olympiad, 1
Prove that a triangle $ABC$ is right-angled if and only if
\[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]
1989 IMO Longlists, 1
If in a convex quadrilateral $ ABCD, E$ and $ F$ are the midpoints of the sides $ BC$ and $ DA$ respectively. Show that the sum of the areas of the triangles $ EDA$ and $ FBC$ is equal to the area of the quadrangle.
1972 IMO Longlists, 10
Given five points in the plane, no three of which are collinear, prove that there can be found at least two obtuse-angled triangles with vertices at the given points. Construct an example in which there are exactly two such triangles.
2013 India Regional Mathematical Olympiad, 1
Let $ABC$ be an isosceles triangle with $AB=AC$ and let $\Gamma$ denote its circumcircle. A point $D$ is on arc $AB$ of $\Gamma$ not containing $C$. A point $E$ is on arc $AC$ of $\Gamma$ not containing $B$. If $AD=CE$ prove that $BE$ is parallel to $AD$.
1974 IMO Longlists, 19
Prove that there exists, for $n \geq 4$, a set $S$ of $3n$ equal circles in space that can be partitioned into three subsets $s_5, s_4$, and $s_3$, each containing $n$ circles, such that each circle in $s_r$ touches exactly $r$ circles in $S.$
1987 IMO Longlists, 23
A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance $d$ from it. By distance is meant the shortest distance measured over the curved survace of the lampshade. Prove that the area of the lampshade is $d^2(2\theta + \sqrt 3)$ where $\cot \frac {\theta}{2} = \frac{3}{\theta}.$
2014 Sharygin Geometry Olympiad, 1
A right-angled triangle $ABC$ is given. Its catheus $AB$ is the base of a regular triangle $ADB$ lying in the exterior of $ABC$, and its hypotenuse $AC$ is the base of a regular triangle $AEC$ lying in the interior of $ABC$. Lines $DE$ and $AB$ meet at point $M$. The whole configuration except points $A$ and $B$ was erased. Restore the point $M$.
1978 IMO Longlists, 20
Let $O$ be the center of a circle. Let $OU,OV$ be perpendicular radii of the circle. The chord $PQ$ passes through the midpoint $M$ of $UV$. Let $W$ be a point such that $PM = PW$, where $U, V,M,W$ are collinear. Let $R$ be a point such that $PR = MQ$, where $R$ lies on the line $PW$. Prove that $MR = UV$.
[u]Alternative version:[/u] A circle $S$ is given with center $O$ and radius $r$. Let $M$ be a point whose distance from $O$ is $\frac{r}{\sqrt{2}}$. Let $PMQ$ be a chord of $S$. The point $N$ is defined by $\overrightarrow{PN} =\overrightarrow{MQ}$. Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$. Prove that $MR =\sqrt{2}r$.
2005 Postal Coaching, 2
Let $< \Gamma _j >$ be a sequnce of concentric circles such that the sequence $< R_j >$ , where $R_j$ denotes the radius of $\Gamma_j$, is increasing and $R_j \longrightarrow \infty$ as $j \longrightarrow \infty$. Let $A_1 B_1 C_1$ be a triangle inscribed in $\Gamma _1$. extend the rays $\vec{A_i B_1} , \vec{B_1 C_1 }, \vec{C_1 A_1}$ to meet $\Gamma_2$ in $B_2, C_2$and $A_2$ respectively and form the triangle $A_2 B_2 C_2$. Continue this process. Show that the sequence of triangles $< A_n B_n C_n >$ tends to an equilateral triangle as $n \longrightarrow \infty$
2011 Tuymaada Olympiad, 3
An excircle of triangle $ABC$ touches the side $AB$ at $P$ and the extensions of sides $AC$ and $BC$ at $Q$ and $R$, respectively. Prove that if the midpoint of $PQ$ lies on the circumcircle of $ABC$, then the midpoint of $PR$ also lies on that circumcircle.