Found problems: 1546
2005 China Team Selection Test, 3
Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
2006 MOP Homework, 6
Let $P$ be a convex polygon in the plane. A real number is assigned to each point in the plane so that the sum of the numbers assigned to the vertices of any polygon similar to $P$ is equal to $0$. Prove that all the assigned numbers are equal to $0$.
2003 Italy TST, 1
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. The line through $A$ parallel to $DF$ meets the line through $C$ parallel to $EF$ at $G$.
$(a)$ Prove that the quadrilateral $AICG$ is cyclic.
$(b)$ Prove that the points $B,I,G$ are collinear.
Indonesia MO Shortlist - geometry, g11.8
Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$.
(a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$.
(b) Show that $ A_1KML$ is a cyclic quadrilateral.
1996 Irish Math Olympiad, 4
In an acute-angled triangle $ ABC$, $ D,E,F$ are the feet of the altitudes from $ A,B,C$, respectively, and $ P,Q,R$ are the feet of the perpendiculars from $ A,B,C$ onto $ EF,FD,DE$, respectively. Prove that the lines $ AP,BQ,CR$ are concurrent.
1972 IMO Longlists, 43
A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.
2001 Austrian-Polish Competition, 8
The prism with the regular octagonal base and with all edges of the length equal to $1$ is given. The points $M_{1},M_{2},\cdots,M_{10}$ are the midpoints of all the faces of the prism. For the point $P$ from the inside of the prism denote by $P_{i}$ the intersection point (not equal to $M_{i}$) of the line $M_{i}P$ with the surface of the prism. Assume that the point $P$ is so chosen that all associated with $P$ points $P_{i}$ do not belong to any edge of the prism and on each face lies exactly one point $P_{i}$. Prove that \[\sum_{i=1}^{10}\frac{M_{i}P}{M_{i}P_{i}}=5\]
2006 South East Mathematical Olympiad, 1
[size=130]In $\triangle ABC$, $\angle A=60^\circ$. $\odot I$ is the incircle of $\triangle ABC$. $\odot I$ is tangent to sides $AB$, $AC$ at $D$, $E$, respectively. Line $DE$ intersects line $BI$ and $CI$ at $F$, $G$ respectively. Prove that [/size]$FG=\frac{BC}{2}$.
2010 Kazakhstan National Olympiad, 6
Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$
Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$
2001 Federal Competition For Advanced Students, Part 2, 3
A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.
2002 China Team Selection Test, 1
Given triangle $ ABC$ and $ AB\equal{}c$, $ AC\equal{}b$ and $ BC\equal{}a$ satisfying $ a \geq b \geq c$, $ BE$ and $ CF$ are two interior angle bisectors. $ P$ is a point inside triangle $ AEF$. $ R$ and $ Q$ are the projections of $ P$ on sides $ AB$ and $ AC$.
Prove that $ PR \plus{} PQ \plus{} RQ < b$.
2008 Regional Competition For Advanced Students, 3
Given is an acute angled triangle $ ABC$. Determine all points $ P$ inside the triangle with
\[1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2\]
2013 Iran MO (3rd Round), 3
Suppose line $\ell$ and four points $A,B,C,D$ lies on $\ell$. Suppose that circles $\omega_1 , \omega_2$ passes through $A,B$ and circles $\omega'_1 , \omega'_2$ passes through $C,D$. If $\omega_1 \perp \omega'_1$ and $\omega_2 \perp \omega'_2$ then prove that lines $O_1O'_2 , O_2O'_1 , \ell $ are concurrent where $O_1,O_2,O'_1,O'_2$ are center of $\omega_1 , \omega_2 , \omega'_1 , \omega'_2$.
2014 Singapore Senior Math Olympiad, 1
In the triangle $ABC$, the excircle opposite to the vertex $A$ with centre $I$ touches the side BC at D. (The circle also touches the sides of $AB$, $AC$ extended.) Let $M$ be the midpoint of $BC$ and $N$ the midpoint of $AD$. Prove that $I,M,N$ are collinear.
2004 China Team Selection Test, 3
In convex quadrilateral $ ABCD$, $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, $ DA\equal{}d$, $ AC\equal{}e$, $ BD\equal{}f$. If $ \max \{a,b,c,d,e,f \}\equal{}1$, then find the maximum value of $ abcd$.
2013 Baltic Way, 12
A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.
2012 Indonesia TST, 3
Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.
2003 All-Russian Olympiad, 3
In a triangle $ABC, O$ is the circumcenter and $I$ the incenter. The excircle $\omega_a$ touches rays $AB,AC$ and side $BC$ at $K,M,N$, respectively. Prove that if the midpoint $P$ of $KM$ lies on the circumcircle of $\triangle ABC$, then points $O,N, I$ lie on a line.
2013 ELMO Shortlist, 14
Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$.
Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$.
(a) Prove that $L$ is nonempty.
(b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$.
[i]Proposed by David Yang[/i]
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle and $I$ the center of its incircle. $P$ is a point inside $ABC$ such that $\angle PBA +\angle PCA = \angle PBC + \angle PCB$. Prove that $AP\geq AI$ with equality iff $P=I$.
2014 Contests, 1
Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$
1972 IMO Longlists, 45
Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at point $O$. Let a line through $O$ intersect segment $AB$ at $M$ and segment $CD$ at $N$. Prove that the segment $MN$ is not longer than at least one of the segments $AC$ and $BD$.
2004 Croatia Team Selection Test, 3
A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.
2007 Estonia National Olympiad, 3
A circle passing through the endpoints of the leg AB of an isosceles triangle ABC
intersects the base BC in point P. A line tangent to the circle in point B intersects the circumcircle of ABC in point Q. Prove that P lies on line AQ if and only if AQ and BC are perpendicular.
2014 Sharygin Geometry Olympiad, 7
A parallelogram $ABCD$ is given. The perpendicular from $C$ to $CD$ meets the perpendicular from $A$ to $BD$ at point $F$, and the perpendicular from $B$ to $AB$ meets the perpendicular bisector to $AC$ at point $E$. Find the ratio in which side $BC$ divides segment $EF$.