Found problems: 1581
1981 Tournament Of Towns, (008) 2
$M$ is a finite set of points in a plane. Point $O$ in the plane is called an “almost centre of symmetry” of set $M$ if it is possible to remove from $M$ one point in such a way that among the remaining members $O$ is the centre of symmetry in the usual sense. How many such “almost centres of symmetry” may a finite point set in a plane have? Indicate all such points.
(V Prasolov, Moscow)
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.
2011 Harvard-MIT Mathematics Tournament, 7
Let $ABCD$ be a quadrilateral inscribed in the unit circle such that $\angle BAD$ is $30$ degrees. Let $m$ denote the minimum value of $CP + PQ + CQ$, where $P$ and $Q$ may be any points lying along rays $AB$ and $AD$, respectively. Determine the maximum value of $m$.
2004 Bundeswettbewerb Mathematik, 2
Let $k$ be a positive integer. In a circle with radius $1$, finitely many chords are drawn. You know that every diameter of the circle intersects at most $k$ of these chords.
Prove that the sum of the lengths of all these chords is less than $k \cdot \pi$.
2013 India IMO Training Camp, 3
For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.
2010 Romanian Masters In Mathematics, 3
Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent.
[i]Pavel Kozhevnikov, Russia[/i]
2010 All-Russian Olympiad, 2
Into triangle $ABC$ gives point $K$ lies on bisector of $ \angle BAC$. Line $CK$ intersect circumcircle $ \omega$ of triangle $ABC$ at $M \neq C$. Circle $ \Omega$ passes through $A$, touch $CM$ at $K$ and intersect segment $AB$ at $P \neq A$ and $\omega $ at $Q \neq A$.
Prove, that $P$, $Q$, $M$ lies at one line.
2012 China Team Selection Test, 3
Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial
\[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\]
with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality
\[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]
2005 Bundeswettbewerb Mathematik, 3
Two circles $k_1$ and $k_2$ intersect at two points $A$ and $B$. Some line through the point $B$ meets the circle $k_1$ at a point $C$ (apart from $B$), and the circle $k_2$ at a point $E$ (apart from $B$). Another line through the point $B$ meets the circle $k_1$ at a point $D$ (apart from $B$), and the circle $k_2$ at a point $F$ (apart from $B$). Assume that the point $B$ lies between the points $C$ and $E$ and between the points $D$ and $F$.
Finally, let $M$ and $N$ be the midpoints of the segments $CE$ and $DF$.
Prove that the triangles $ACD$, $AEF$ and $AMN$ are similar to each other.
1998 National Olympiad First Round, 36
$ ABCD$ is a $ 4\times 4$ square. $ E$ is the midpoint of $ \left[AB\right]$. $ M$ is an arbitrary point on $ \left[AC\right]$. How many different points $ M$ are there such that $ \left|EM\right|\plus{}\left|MB\right|$ is an integer?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$
2007 Romania Team Selection Test, 4
The points $M, N, P$ are chosen on the sides $BC, CA, AB$ of a triangle $\Delta ABC$, such that the triangle $\Delta MNP$ is acute-angled. We denote with $x$ the length of the shortest altitude of the triangle $\Delta ABC$, and with $X$ the length of the longest altitudes of the triangle $\Delta MNP$. Prove that $x \leq 2X$.
2005 Rioplatense Mathematical Olympiad, Level 3, 2
In trapezoid $ABCD$, the sum of the lengths of the bases $AB$ and $CD$ is equal to the length of the diagonal $BD$. Let $M$ denote the midpoint of $BC$, and let $E$ denote the reflection of $C$ about the line $DM$. Prove that $\angle AEB=\angle ACD$.
2024 Canadian Junior Mathematical Olympiad, 3
Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.
2011 USA Team Selection Test, 7
Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.
2013 Today's Calculation Of Integral, 868
In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation.
(1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$.
(2) Find the volume of the common part of $V_1$ and $V_2$.
2011 China Team Selection Test, 1
Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.
1967 IMO Longlists, 54
Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?
2011 Iran MO (2nd Round), 3
The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.
2011 Turkey MO (2nd round), 2
Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.
2019 Tuymaada Olympiad, 8
In $\triangle ABC$ $\angle B$ is obtuse and $AB \ne BC$. Let $O$ is the circumcenter and $\omega$ is the circumcircle of this triangle. $N$ is the midpoint of arc $ABC$. The circumcircle of $\triangle BON$ intersects $AC$ on points $X$ and $Y$. Let $BX \cap \omega = P \ne B$ and $BY \cap \omega = Q \ne B$. Prove that $P, Q$ and reflection of $N$ with respect to line $AC$ are collinear.
2011 Tokyo Instutute Of Technology Entrance Examination, 4
On a plane, given a square $D$ with side length 1 and a line which intersects with $D$. For the solid obtained by a rotation of $D$ about the line as the axis, answer the following questions:
(1) Suppose that the line $l$ on a plane the same with $D$ isn't parallel to any edges. Prove that the line by which the volume of the solid is maximized has only intersection point with $D$. Note that the line as axis of rotation is parallel to $l$.
(2) Find the possible maximum volume for which all solid formed by the rotation axis as line intersecting with $D$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 4[/i]
2018 Korea Junior Math Olympiad, 5
Let there be an acute scalene triangle $ABC$ with circumcenter $O$. Denote $D,E$ be the reflection of $O$ with respect to $AB,AC$, respectively. The circumcircle of $ADE$ meets $AB$, $AC$, the circumcircle of $ABC$ at points $K,L,M$, respectively, and they are all distinct from $A$. Prove that the lines $BC,KL,AM$ are concurrent.
2007 International Zhautykov Olympiad, 3
Let $ABCDEF$ be a convex hexagon and it`s diagonals have one common point $M$. It is known that the circumcenters of triangles $MAB,MBC,MCD,MDE,MEF,MFA$ lie on a circle.
Show that the quadrilaterals $ABDE,BCEF,CDFA$ have equal areas.
2008 Junior Balkan Team Selection Tests - Romania, 2
Let $ m,n$ be two natural nonzero numbers and sets $ A \equal{} \{ 1,2,...,n\}, B \equal{} \{1,2,...,m\}$. We say that subset $ S$ of Cartesian product $ A \times B$ has property $ (j)$ if $ (a \minus{} x)(b \minus{} y)\le 0$ for each pairs $ (a,b),(x,y) \in S$. Prove that every set $ S$ with propery $ (j)$ has at most $ m \plus{} n \minus{} 1$ elements.
[color=#FF0000]The statement was edited, in order to reflect the actual problem asked. The sign of the inequality was inadvertently reversed into $ (a \minus{} x)(b \minus{} y)\ge 0$, and that accounts for the following two posts.[/color]
2008 Sharygin Geometry Olympiad, 2
(V.Protasov, 8) For a given pair of circles, construct two concentric circles such that both are tangent to the given two. What is the number of solutions, depending on location of the circles?