Found problems: 3349
1987 IMO, 2
In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.
2002 Estonia Team Selection Test, 5
Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .
1967 IMO Longlists, 55
Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]
2012 Turkey MO (2nd round), 6
Let $B$ and $D$ be points on segments $[AE]$ and $[AF]$ respectively. Excircles of triangles $ABF$ and $ADE$ touching sides $BF$ and $DE$ is the same, and its center is $I$. $BF$ and $DE$ intersects at $C$. Let $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$ be the circumcenters of triangles $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$ respectively.
[b]a) [/b] Show that points $P_1, P_2, P_3, P_4$ concylic and points $Q_1, Q_2, Q_3, Q_4$ concylic.
[b]b) [/b] Denote centers of theese circles as $O_1$ and $O_2$. Prove that $O_1, O_2$ and $I$ are collinear.
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 $ABC$.
1993 IMO Shortlist, 8
The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC.$ If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC,$ prove that
\[ DE \geq \frac{2 \cdot \sqrt{2} \cdot S}{\sqrt{a^2 + b^2 + c^2 + 4 \cdot \sqrt{3} \cdot S}}. \]
2003 China Second Round Olympiad, 1
From point $P$ outside a circle draw two tangents to the circle touching at $A$ and $B$. Draw a secant line intersecting the circle at points $C$ and $D$, with $C$ between $P$ and $D$. Choose point $Q$ on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.
JBMO Geometry Collection, 1999
Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$).
[i]Greece[/i]
2014 Contests, 1
Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.
2008 Hong kong National Olympiad, 3
$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$.
Remark: the original question has missed the condition $ AB \neq AC$
2012 Regional Olympiad of Mexico Center Zone, 4
On an acute triangle $ABC$ we draw the internal bisector of $<ABC$, $BE$, and the altitude $AD$, ($D$ on $BC$), show that $<CDE$ it's bigger than 45 degrees.
1991 IMO Shortlist, 19
Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0$. Prove that $ \alpha \equal{} \frac {2}{3}$.
2008 Junior Balkan Team Selection Tests - Romania, 3
Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B',C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.
1969 Canada National Olympiad, 9
Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to $\sqrt{2}$.
2011 Finnish National High School Mathematics Competition, 3
Points $D$ and $E$ divides the base $BC$ of an isosceles triangle $ABC$ into three equal parts and $D$ is between $B$ and $E.$ Show that $\angle BAD<\angle DAE.$
2005 Georgia Team Selection Test, 2
In triangle $ ABC$ we have $ \angle{ACB} \equal{} 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD \equal{} AC$ and $ DB \equal{} DC$. Prove that $ \angle{BAC} \equal{} 3\angle{BAD}$.
2009 Belarus Team Selection Test, 3
Points $T,P,H$ lie on the side $BC,AC,AB$ respectively of triangle $ABC$, so that $BP$ and $AT$ are angle bisectors and $CH$ is an altitude of $ABC$. Given that the midpoint of $CH$ belongs to the segment $PT,$ find the value of $\cos A + \cos B$
I. Voronovich
2005 South East Mathematical Olympiad, 5
Line $l$ tangents unit circle $S$ in point $P$. Point $A$ and circle $S$ are on the same side of $l$, and the distance from $A$ to $l$ is $h$ ($h > 2$). Two tangents of circle $S$ are drawn from $A$, and intersect line $l$ at points $B$ and $C$ respectively.
Find the value of $PB \cdot PC$.
2013 Tuymaada Olympiad, 2
Points $X$ and $Y$ inside the rhombus $ABCD$ are such that $Y$ is inside the convex quadrilateral $BXDC$ and $2\angle XBY = 2\angle XDY = \angle ABC$. Prove that the lines $AX$ and $CY$ are parallel.
[i]S. Berlov[/i]
2009 AMC 12/AHSME, 19
Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $ A$ and $ B$, respectively. Each polygon had a side length of $ 2$. Which of the following is true?
$ \textbf{(A)}\ A\equal{}\frac{25}{49}B\qquad \textbf{(B)}\ A\equal{}\frac{5}{7}B\qquad \textbf{(C)}\ A\equal{}B\qquad \textbf{(D)}\ A\equal{}\frac{7}{5}B\qquad \textbf{(E)}\ A\equal{}\frac{49}{25}B$
2008 Peru Iberoamerican Team Selection Test, P2
Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic.
[i]Proposed by John Cuya, Peru[/i]
2009 China Team Selection Test, 1
Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$
1984 Canada National Olympiad, 4
An acute triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$.
2011 AIME Problems, 13
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$, then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.
2007 Germany Team Selection Test, 3
Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove:
\[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F
\]
When does equality occur?