Found problems: 1546
2000 Spain Mathematical Olympiad, 3
Two circles $C_1$ and $C_2$ with the respective radii $r_1$ and $r_2$ intersect in $A$ and $B.$ A variable line $r$ through $B$ meets $C_1$ and $C_2$ again at $P_r$ and $Q_r$ respectively. Prove that there exists a point $M,$ depending only on $C_1$ and $C_2,$ such that the perpendicular bisector of each segment $P_rQ_r$ passes through $M.$
2001 Bundeswettbewerb Mathematik, 4
A square $ R$ of sidelength $ 250$ lies inside a square $ Q$ of sidelength $ 500$. Prove that: One can always find two points $ A$ and $ B$ on the perimeter of $ Q$ such that the segment $ AB$ has no common point with the square $ R$, and the length of this segment $ AB$ is greater than $ 521$.
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 $ABC$.
2010 Contests, 2
Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of
\[AB^2 + 2AC^2 - 3AD^2.\]
2014 Dutch IMO TST, 3
Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.
2014 Vietnam Team Selection Test, 4
a. Let $ABC$ be a triangle with altitude $AD$ and $P$ a variable point on $AD$. Lines $PB$ and $AC$ intersect each other at $E$, lines $PC$ and $AB$ intersect each other at $F.$ Suppose $AEDF$ is a quadrilateral inscribed . Prove that \[\frac{PA}{PD}=(\tan B+\tan C)\cot \frac{A}{2}.\]
b. Let $ABC$ be a triangle with orthocentre $H$ and $P$ a variable point on $AH$. The line through $C$ perpendicular to $AC$ meets $BP$ at $M$, The line through $B$ perpendicular to $AB$ meets $CP$ at $N.$ $K$ is the projection of $A$on $MN$. Prove that $\angle BKC+\angle MAN$ is invariant .
1978 IMO Longlists, 54
Let $p, q$ and $r$ be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes $\alpha, \beta$, and $\gamma$, containing $p, q$, and $r$ respectively, that are perpendicular to each other $(\alpha\perp\beta, \beta\perp\gamma, \gamma\perp \alpha).$
2008 Sharygin Geometry Olympiad, 6
(B.Frenkin) Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.
2003 Czech-Polish-Slovak Match, 2
In an acute-angled triangle $ABC$ the angle at $B$ is greater than $45^\circ$. Points $D,E, F$ are the feet of the altitudes from $A,B,C$ respectively, and $K$ is the point on segment $AF$ such that $\angle DKF = \angle KEF$.
(a) Show that such a point $K$ always exists.
(b) Prove that $KD^2 = FD^2 + AF \cdot BF$.
1990 China National Olympiad, 1
Given a convex quadrilateral $ABCD$, side $AB$ is not parallel to side $CD$. The circle $O_1$ passing through $A$ and $B$ is tangent to side $CD$ at $P$. The circle $O_2$ passing through $C$ and $D$ is tangent to side $AB$ at $Q$. Circle $O_1$ and circle $O_2$ meet at $E$ and $F$. Prove that $EF$ bisects segment $PQ$ if and only if $BC\parallel AD$.
2013 CentroAmerican, 2
Let $ABC$ be an acute triangle and let $\Gamma$ be its circumcircle. The bisector of $\angle{A}$ intersects $BC$ at $D$, $\Gamma$ at $K$ (different from $A$), and the line through $B$ tangent to $\Gamma$ at $X$. Show that $K$ is the midpoint of $AX$ if and only if $\frac{AD}{DC}=\sqrt{2}$.
2017 Bundeswettbewerb Mathematik, 3
Let $M$ be the incenter of the tangential quadrilateral $A_1A_2A_3A_4$. Let line $g_1$ through $A_1$ be perpendicular to $A_1M$; define $g_2,g_3$ and $g_4$ similarly. The lines $g_1,g_2,g_3$ and $g_4$ define another quadrilateral $B_1B_2B_3B_4$ having $B_1$ be the intersection of $g_1$ and $g_2$; similarly $B_2,B_3$ and $B_4$ are intersections of $g_2$ and $g_3$, $g_3$ and $g_4$, resp. $g_4$ and $g_1$.
Prove that the diagonals of quadrilateral $B_1B_2B_3B_4$ intersect in point $M$.
[asy]
import graph; size(15cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-9.773972777861085,xmax=12.231603726660566,ymin=-3.9255487671791487,ymax=7.37238601960895;
pair M=(2.,2.), A_4=(-1.6391623316400197,1.2875505916864178), A_1=(3.068893183992864,-0.5728665455336459), A_2=(4.30385937824148,2.2922812065339455), A_3=(2.221541124684679,4.978916319940133), B_4=(-0.9482172571022687,-2.24176848577888), B_1=(4.5873184669543345,0.057960746374459436), B_2=(3.9796042717514277,4.848169684238838), B_3=(-2.4295496490492385,5.324816563638236);
draw(circle(M,2.),linewidth(0.8)); draw(A_4--A_1,linewidth(0.8)); draw(A_1--A_2,linewidth(0.8)); draw(A_2--A_3,linewidth(0.8)); draw(A_3--A_4,linewidth(0.8)); draw(M--A_3,linewidth(0.8)+dotted); draw(M--A_2,linewidth(0.8)+dotted); draw(M--A_1,linewidth(0.8)+dotted); draw(M--A_4,linewidth(0.8)+dotted); draw((xmin,-0.07436970390935019*xmin+5.144131675605378)--(xmax,-0.07436970390935019*xmax+5.144131675605378),linewidth(0.8)); draw((xmin,-7.882338401302275*xmin+36.2167572574517)--(xmax,-7.882338401302275*xmax+36.2167572574517),linewidth(0.8)); draw((xmin,0.4154483588930812*xmin-1.847833182441644)--(xmax,0.4154483588930812*xmax-1.847833182441644),linewidth(0.8)); draw((xmin,-5.107958950031516*xmin-7.085223310768749)--(xmax,-5.107958950031516*xmax-7.085223310768749),linewidth(0.8));
dot(M,linewidth(3.pt)+ds); label("$M$",(2.0593440948136896,2.0872038897020024),NE*lsf); dot(A_4,linewidth(3.pt)+ds); label("$A_4$",(-2.6355449660387147,1.085078446888477),NE*lsf); dot(A_1,linewidth(3.pt)+ds); label("$A_1$",(3.1575637581709772,-1.2486383377457595),NE*lsf); dot(A_2,linewidth(3.pt)+ds); label("$A_2$",(4.502882845783654,2.30684782237346),NE*lsf); dot(A_3,linewidth(3.pt)+ds); label("$A_3$",(2.169166061149418,5.203402184478307),NE*lsf); label("$g_3$",(-9.691606303109287,5.354407388189934),NE*lsf); label("$g_2$",(3.0889250292111465,6.727181967386543),NE*lsf); label("$g_1$",(-4.763345563793459,-3.4725331560442676),NE*lsf); label("$g_4$",(-2.663000457622647,6.878187171098171),NE*lsf); dot(B_4,linewidth(3.pt)+ds); label("$B_4$",(-1.5647807942653595,-3.0332452907013523),NE*lsf); dot(B_1,linewidth(3.pt)+ds); label("$B_1$",(4.955898456918535,-0.6583452686912173),NE*lsf); dot(B_2,linewidth(3.pt)+ds); label("$B_2$",(4.104778217816637,5.0661247265586455),NE*lsf); dot(B_3,linewidth(3.pt)+ds); label("$B_3$",(-3.4454819677647146,5.656417795613188),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]
1986 IMO Longlists, 17
We call a tetrahedron right-faced if each of its faces is a right-angled triangle.
[i](a)[/i] Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra.
[i](b)[/i] Prove that a tetrahedron with vertices $A_1,A_2,A_3,A_4$ is right-faced if and only if there exist four distinct real numbers $c_1, c_2, c_3$, and $c_4$ such that the edges $A_jA_k$ have lengths $A_jA_k=\sqrt{|c_j-c_k|}$ for $1\leq j < k \leq 4.$
2014 Contests, 2
A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$.
(Day 1, 2nd problem
authors: Michal Rolínek, Jaroslav Švrček)
2008 Sharygin Geometry Olympiad, 4
(A.Zaslavsky) Given three points $ C_0$, $ C_1$, $ C_2$ on the line $ l$. Find the locus of incenters of triangles $ ABC$ such that points $ A$, $ B$ lie on $ l$ and the feet of the median, the bisector and the altitude from $ C$ coincide with $ C_0$, $ C_1$, $ C_2$.
2011 India IMO Training Camp, 1
Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that:
$a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area.
$b) a\cdot AP=b\cdot BP=c\cdot PC.$
2004 South East Mathematical Olympiad, 2
In $\triangle$ABC, points D, M lie on side BC and AB respectively, point P lies on segment AD. Line DM intersects segments BP, AC (extended part), PC (extended part) at E, F and N respectively. Show that if DE=DF, then DM=DN.
1982 Vietnam National Olympiad, 3
Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.
2006 Italy TST, 2
Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that
\[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\]
if and only if $ABC$ is acute-angled.
2002 India IMO Training Camp, 7
Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.
2004 China Team Selection Test, 1
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$
2001 Canada National Olympiad, 5
Let $P_0$, $P_1$, $P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle P_{i-1} P_{i-2} P_{i-3}$.
(1) Prove that the points $P_1, P_5, P_9, P_{13},\cdots$ are collinear.
(2) Let $x$ be the distance from $P_1$ to $P_{1001}$, and let $y$ be the distance from $P_{1001}$ to $P_{2001}$. Determine all values of $t$ for which $\sqrt[500]{ \frac xy}$ is an integer.
2000 Cono Sur Olympiad, 1
In square $ABCD$ (labeled clockwise), let $P$ be any point on $BC$ and construct square $APRS$ (labeled clockwise). Prove that line $CR$ is tangent to the circumcircle of triangle $ABC$.
1979 IMO Longlists, 79
Let $S$ be a unit circle and $K$ a subset of $S$ consisting of several closed arcs. Let $K$ satisfy the following properties:
$(\text{i})$ $K$ contains three points $A,B,C$, that are the vertices of an acute-angled triangle
$(\text{ii})$ For every point $A$ that belongs to $K$ its diametrically opposite point $A'$ and all points $B$ on an arc of length $\frac{1}{9}$ with center $A'$ do not belong to $K$.
Prove that there are three points $E,F,G$ on $S$ that are vertices of an equilateral triangle and that do not belong to $K$.
1989 IMO Longlists, 6
The circles $ c_1$ and $ c_2$ are tangent at the point $ A.$ A straight line $ l$ through $ A$ intersects $ c_1$ and $ c_2$ at points $ C_1$ and $ C_2$ respectively. A circle $ c,$ which contains $ C_1$ and $ C_2,$ meets $ c_1$ and $ c_2$ at points $ B_1$ and $ B_2$ respectively. Let $ \omega$ be the circle circumscribed around triangle $ AB_1B_2.$ The circle $ k$ tangent to $ \omega$ at the point $ A$ meets $ c_1$ and $ c_2$ at the points $ D_1$ and $ D_2$ respectively. Prove that
[b](a)[/b] the points $ C_1,C_2,D_1,D_2$ are concyclic or collinear,
[b](b)[/b] the points $ B_1,B_2,D_1,D_2$ are concyclic if and only if $ AC_1$ and $ AC_2$ are diameters of $ c_1$ and $ c_2.$