Found problems: 1546
1991 Vietnam Team Selection Test, 1
Let $T$ be an arbitrary tetrahedron satisfying the following conditions:
[b]I.[/b] Each its side has length not greater than 1,
[b]II.[/b] Each of its faces is a right triangle.
Let $s(T) = S^2_{ABC} + S^2_{BCD} + S^2_{CDA} + S^2_{DAB}$. Find the maximal possible value of $s(T)$.
1994 All-Russian Olympiad Regional Round, 9.6
Point $ P$ is taken inside a right angle $ KLM$. A circle $ S_1$ with center $ O_1$ is tangent to the rays $ LK,LP$ of angle $ KLP$ at $ A,D$ respectively. A circle $ S_2$ with center $ O_2$ is tangent to the rays of angle $ MLP$, touching $ LP$ at $ B$. Suppose $ A,B,O_1$ are collinear. Let $ O_2D,KL$ meet at $ C$. Prove that $ BC$ bisects angle $ ABD$.
2004 Postal Coaching, 9
Let $ABCDEF$ be a regular hexagon of side lengths $1$ and $O$ its centre, Join $O$ cto each of the six vertices , thus getting $12$ unit line segments in all. Find the number of closed paths from (i) $O$ to $O$ (ii) $A$ to $A$ each of length $2004$
2003 India IMO Training Camp, 4
There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.
1989 IMO Longlists, 23
Let $ ABC$ be a triangle. Prove that there is a unique point $ U$ in the plane of $ ABC$ such that there exist real numbers $ \alpha, \beta, \gamma, \delta$ not all zero, such that
\[ \alpha PL^2 \plus{} \beta PM^2 \plus{} \gamma PN^2 \plus{} \delta UP^2\]
is constant for all points $ P$ of the plane, where $ L,M,N$ are the feet of the perpendiculars from $ P$ to $ BC,CA,AB$ respectively. Identify $ U.$
1977 Canada National Olympiad, 5
A right circular cone has base radius 1 cm and slant height 3 cm is given. $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone and back to $P$ is drawn (see diagram). What is the minimum distance from the vertex $V$ to this path?
[asy]
import graph;
unitsize(1 cm);
filldraw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(Circle((0,0),1)),gray(0.9),nullpen);
draw(yscale(0.3)*(arc((0,0),1.5,0,180)),dashed);
draw(yscale(0.3)*(arc((0,0),1.5,180,360)));
draw((1.5,0)--(0,4)--(-1.5,0));
draw((0,0)--(1.5,0),Arrows);
draw(((1.5,0) + (0.3,0.1))--((0,4) + (0.3,0.1)),Arrows);
draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,0,180)),dashed);
draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,180,360)));
label("$V$", (0,4), N);
label("1 cm", (0.75,-0.5), N);
label("$P$", (-1.5,0), SW);
label("3 cm", (1.7,2));
[/asy]
Estonia Open Senior - geometry, 2007.2.3
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.
1972 IMO Longlists, 36
A finite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.
1979 IMO Longlists, 81
Let $\Pi$ be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped $P_0$ can be decomposed into parallelepipeds $P_1,P_2, . . . ,P_N\in \Pi$, prove that $P_0\in \Pi$.
1988 China National Olympiad, 2
Given two circles $C_1,C_2$ with common center, the radius of $C_2$ is twice the radius of $C_1$. Quadrilateral $A_1A_2A_3A_4$ is inscribed in $C_1$. The extension of $A_4A_1$ meets $C_2$ at $B_1$; the extension of $A_1A_2$ meets $C_2$ at $B_2$; the extension of $A_2A_3$ meets $C_2$ at $B_3$; the extension of $A_3A_4$ meets $C_2$ at $B_4$. Prove that $P(B_1B_2B_3B_4)\ge 2P(A_1A_2A_3A_4)$, and in what case the equality holds? ($P(X)$ denotes the perimeter of quadrilateral $X$)
1990 IMO Longlists, 28
Let $ABC$ be an arbitrary acute triangle. Circle $\Gamma$ satisfies the following conditions:
(i) Circle $\Gamma$ intersects all three sides of triangle $ABC.$
(ii) In the convex hexagon formed by above six intersections, the three pairs of opposite sides are parallel respectively. (The hexagon maybe degenerate, that is, two or more vertices are coincide. In this case, "opposite sides are parallel" is defined through limit opinion.)
Find the locus of the center of circle $\Gamma$, and explain how to construct the locus.
2010 South africa National Olympiad, 5
(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle $\alpha$. Determine the smallest number of lines for which this is possible.
(b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections.
1997 Pre-Preparation Course Examination, 5
Let $H$ be the orthocenter of the triangle $ABC$ and $P$ an arbitrary point on circumcircle of triangle. $BH$ meets $AC$ at $E$. $PAQB$ and $PARC$ are two parallelograms and $AQ$ meets $HR$ at $X$. Show that $EX \parallel AP$.
2010 Postal Coaching, 1
Let $\gamma,\Gamma$ be two concentric circles with radii $r,R$ with $r<R$. Let $ABCD$ be a cyclic quadrilateral inscribed in $\gamma$. If $\overrightarrow{AB}$ denotes the Ray starting from $A$ and extending indefinitely in $B's$ direction then Let $\overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CD} , \overrightarrow{DA}$ meet $\Gamma$ at the points $C_1,D_1,A_1,B_1$ respectively. Prove that
\[\frac{[A_1B_1C_1D_1]}{[ABCD]} \ge \frac{R^2}{r^2}\]
where $[.]$ denotes area.
1997 China National Olympiad, 2
Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}$ respectively for $k\ge 2$.
Consider the sequence of quadrilaterals $A_iB_iC_iD_i$.
i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not?
ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?
1967 IMO Longlists, 39
Show that the triangle whose angles satisfy the equality \[ \frac{sin^2(A) + sin^2(B) + sin^2(C)}{cos^2(A) + cos^2(B) + cos^2(C)} = 2 \] is a rectangular triangle.
2001 Tournament Of Towns, 3
Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $\triangle ABC$. The segments $AY$ and $CX$ intersect at the point $Z$. Given that $AY = YC$ and $AB = ZC$, prove that the points $B$, $X$, $Z$, and $Y$ lie on the same circle.
2013 Baltic Way, 15
Four circles in a plane have a common center. Their radii form a strictly increasing arithmetic progression. Prove that there is no square with each vertex lying on a different circle.
2010 Korea - Final Round, 2
Let $ I$ be the incentre and $ O$ the circumcentre of a given acute triangle $ ABC$. The incircle is tangent to $ BC$ at $ D$. Assume that $ \angle B < \angle C$ and the segments $ AO$ and $ HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$. Let the intersection of the line $ OD$ and $ AH$ be $ E$. If the midpoint of $ CI$ is $ F$, prove that $ E,F,I,O$ are concyclic.
2006 Tuymaada Olympiad, 2
Let $ABC$ be a triangle, $G$ it`s centroid, $H$ it`s orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$). It is known that $MG=R$, where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$.
[i]Proposed by F. Bakharev[/i]
2010 Serbia National Math Olympiad, 1
Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$.
[i]Proposed by Dusan Djukic[/i]
2006 MOP Homework, 3
There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps.
what does barycenter of n distinct points mean?
1992 Cono Sur Olympiad, 2
Let $P$ be a point outside the circle $C$. Find two points $Q$ and $R$ on the circle, such that $P,Q$ and $R$ are collinear and $Q$ is the midpopint of the segmenet $PR$. (Discuss the number of solutions).
2004 Romania National Olympiad, 1
Let $n \geq 3$ be an integer and $F$ be the focus of the parabola $y^2=2px$. A regular polygon $A_1 A_2 \ldots A_n$ has the center in $F$ and none of its vertices lie on $Ox$. $\left( FA_1 \right., \left( FA_2 \right., \ldots, \left( FA_n \right.$ intersect the parabola at $B_1,B_2,\ldots,B_n$.
Prove that \[ FB_1 + FB_2 + \ldots + FB_n > np . \]
[i]Calin Popescu[/i]
1996 Taiwan National Olympiad, 3
Let be given points $A,B$ on a circle and let $P$ be a variable point on that circle. Let point $M$ be determined by $P$ as the point that is either on segment $PA$ with $AM=MP+PB$ or on segment $PB$ with $AP+MP=PB$. Find the locus of points $M$.