Found problems: 2023
1966 IMO Longlists, 50
For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality$S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$
2010 Kazakhstan National Olympiad, 5
Let $O$ be the circumcircle of acute triangle $ABC$, $AD$-altitude of $ABC$ ($ D \in BC$), $ AD \cap CO =E$, $M$-midpoint of $AE$, $F$-feet of perpendicular from $C$ to $AO$.
Proved that point of intersection $OM$ and $BC$ lies on circumcircle of triangle $BOF$
2000 JBMO ShortLists, 21
All the angles of the hexagon $ABCDEF$ are equal. Prove that
\[AB-DE=EF-BC=CD-FA \]
1993 All-Russian Olympiad, 2
From the symmetry center of two congruent intersecting circles, two rays are drawn that intersect the circles at four non-collinear points. Prove that these points lie on one circle.
2011 Switzerland - Final Round, 2
Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]
2005 Cono Sur Olympiad, 2
Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively.
(a) Show that $R$, $Q$, $W$, $S$ are collinear.
(b) Show that $MP=RS-QW$.
Swiss NMO - geometry, 2010.2
Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$.
Show that $ PMID$ ist cyclic.
2003 Hungary-Israel Binational, 2
Let $M$ be a point inside a triangle $ABC$ . The lines $AM , BM , CM$ intersect $BC, CA, AB$ at $A_{1}, B_{1}, C_{1}$, respectively. Assume that
$S_{MAC_{1}}+S_{MBA_{1}}+S_{MCB_{1}}= S_{MA_{1}C}+S_{MB_{1}A}+S_{MC_{1}B}$ . Prove that one of the lines $AA_{1}, BB_{1}, CC_{1}$ is a median of the triangle $ABC.$
2011 Mexico National Olympiad, 6
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circumferences intersecting at points $A$ and $B$. Let $C$ be a point on line $AB$ such that $B$ lies between $A$ and $C$. Let $P$ and $Q$ be points on $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively such that $CP$ and $CQ$ are tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, $P$ is not inside $\mathcal{C}_2$ and $Q$ is not inside $\mathcal{C}_1$. Line $PQ$ cuts $\mathcal{C}_1$ at $R$ and $\mathcal{C}_2$ at $S$, both points different from $P$, $Q$ and $B$. Suppose $CR$ cuts $\mathcal{C}_1$ again at $X$ and $CS$ cuts $\mathcal{C}_2$ again at $Y$. Let $Z$ be a point on line $XY$. Prove $SZ$ is parallel to $QX$ if and only if $PZ$ is parallel to $RX$.
1995 Hungary-Israel Binational, 2
Let $ P_1$, $ P_2$, $ P_3$, $ P_4$ be five distinct points on a circle. The distance of $ P$ from the line $ P_iP_k$ is denoted by $ d_{ik}$. Prove that $ d_{12}d_{34} \equal{} d_{13}d_{24}$.
2011 All-Russian Olympiad Regional Round, 10.7
Points $C_0$ and $B_0$ are the respective midpoints of sides $AB$ and $AC$ of a non-isosceles acute triangle $ABC$, $O$ is its circumscenter and $H$ is the orthocenter. Lines $BH$ and $OC_0$ intersect at $P$, while lines $CH$ and $OB_0$ intersect at $Q$. $OPHQ$ is rhombus. Prove that points $A$, $P$ and $Q$ are collinear. (Author: L. Emelyanov)
1982 Canada National Olympiad, 1
In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$, 2, 3, and 4 ($A_5 = A_1$). Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$.
[asy]
unitsize(1 cm);
pair O;
pair[] A, B;
O = (0,0);
A[1] = (0.5,-3);
A[2] = (2,0);
A[3] = (-0.2,0.5);
A[4] = (-1,0);
B[1] = A[2] - A[1];
B[2] = A[3] - A[2];
B[3] = A[4] - A[3];
B[4] = A[1] - A[4];
draw(A[1]--A[2]--A[3]--A[4]--cycle);
draw(B[1]--B[2]--B[3]--B[4]--cycle);
draw(O--B[1]);
draw(O--B[2]);
draw(O--B[3]);
draw(O--B[4]);
label("$A_1$", A[1], S);
label("$A_2$", A[2], E);
label("$A_3$", A[3], N);
label("$A_4$", A[4], W);
label("$B_1$", B[1], NE);
label("$B_2$", B[2], W);
label("$B_3$", B[3], SW);
label("$B_4$", B[4], S);
label("$O$", O, E);
[/asy]
2014 ELMO Shortlist, 6
Let $ABCD$ be a cyclic quadrilateral with center $O$.
Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$.
Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$.
Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$.
Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$.
[i]Proposed by Yang Liu[/i]
2007 Baltic Way, 11
In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements:
i) $P$ is the circumcentre of triangle $ABC$.
ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$.
iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$.
Prove that $S$ is the incentre of triangle $ABC$.
2006 India National Olympiad, 1
In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that
(1) $IO$ is perpendicular to $BI$;
(2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.
2009 ITAMO, 2
Let $ABC$ be an acute-angled scalene triangle and $\Gamma$ be its circumcircle. $K$ is the foot of the internal bisector of $\angle BAC$ on $BC$. Let $M$ be the midpoint of the arc $BC$ containing $A$. $MK$ intersect $\Gamma$ again at $A'$. $T$ is the intersection of the tangents at $A$ and $A'$. $R$ is the intersection of the perpendicular to $AK$ at $A$ and perpendicular to $A'K$ at $A'$. Show that $T, R$ and $K$ are collinear.
2007 China Team Selection Test, 1
Let convex quadrilateral $ ABCD$ be inscribed in a circle centers at $ O.$ The opposite sides $ BA,CD$ meet at $ H$, the diagonals $ AC,BD$ meet at $ G.$ Let $ O_{1},O_{2}$ be the circumcenters of triangles $ AGD,BGC.$ $ O_{1}O_{2}$ intersects $ OG$ at $ N.$ The line $ HG$ cuts the circumcircles of triangles $ AGD,BGC$ at $ P,Q$, respectively. Denote by $ M$ the midpoint of $ PQ.$ Prove that $ NO \equal{} NM.$
2006 Kyiv Mathematical Festival, 1
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Triangle $ABC$ and straight line $l$ are given at the plane. Construct using a compass and a ruler the straightline which is parallel to $l$ and bisects the area of triangle $ABC.$
2000 Macedonia National Olympiad, 1
Let $AB$ be a diameter of a circle with centre $O$, and $CD$ be a chord perpendicular to $AB$. A chord $AE$ intersects $CO$ at $M$, while $DE$ and $BC$ intersect at $N$. Prove that $CM:CO=CN:CB$.
1974 IMO Longlists, 44
We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.
1986 Iran MO (2nd round), 2
In a trapezoid $ABCD$, the legs $AB$ and $CD$ meet in $M$ and the diagonals $AC$ and $BD$ meet in $N.$ Let $AC=a$ and $BC=b.$ Find the area of triangles $AMD$ and $AND$ in terms of $a$ and $b.$
2012 Indonesia TST, 3
Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.
1984 IMO Longlists, 48
Let $ABC$ be a triangle with interior angle bisectors $AA_1, BB_1, CC_1$ and incenter $I$. If $\sigma[IA_1B] + \sigma[IB_1C] + \sigma[IC_1A] = \frac{1}{2}\sigma[ABC]$, where $\sigma[ABC]$ denotes the area of $ABC$, show that $ABC$ is isosceles.
1988 Romania Team Selection Test, 7
In the plane there are given the lines $\ell_1$, $\ell_2$, the circle $\mathcal{C}$ with its center on the line $\ell_1$ and a second circle $\mathcal{C}_1$ which is tangent to $\ell_1$, $\ell_2$ and $\mathcal{C}$. Find the locus of the tangent point between $\mathcal{C}$ and $\mathcal{C}_1$ while the center of $\mathcal{C}$ is variable on $\ell_1$.
[i]Mircea Becheanu[/i]
2013 Sharygin Geometry Olympiad, 2
Let $ABC$ be an isosceles triangle ($AC = BC$) with $\angle C = 20^\circ$. The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_1$ and $B_1$ respectively. Prove that the triangle $A_1OB_1$ (where $O$ is the circumcenter of $ABC$) is regular.