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\begin{document}
\begin{center}
\textsc{\LARGE Geometry Camp 2009} \\ \bigskip
\textit{\Large Problem set}
\end{center}
\Lcen{Easy \fn{This does not mean that they are not "problems". You will not be able to solve them unless you pay enough attention, and try hard enough!}}
\pr{0} In $\tr ABC$ $AB=AC$ iff\fn{\emph{iff} means if and only if. When some problem includes this, it asks to prove "both ways". \emph{This is important!}} $\an B=\an C$ (Kidding!)
\pr{1} Prove that a quadrilateral in which both pairs of opposite angles are equal has to be a parallelogram.
% 35
\pr{2}\ns{1}\footnote{Asterisk ($\star$) indicates the difficulty level of the problems. Note that: "difficulty" is relative, and the difficult level of double asterisk problem of "Easy" set is not the same as that of "Not so easy!!!" set.}(a) $P,Q,R$ are points on the sides $BC,CA,AB$ of $\tr ABC$. Prove that the perpendiculars to the sides at these points meet in a common point iff \[ BP^2+CQ^2+AR^2=PC^2+QA^2+RB^2 \]
% 40
(b) Given a triangle $ABC$, Let $L,M,N$ be the feet of the perpendiculars from the point $K$ to the sides $BC,CA,AB$ respectively. Prove that the perpendiculars from $A,B,C$ to $MN,NL,LM$ respectively are concurrent.
\pr{3} \fn{Yes...the opposite statement of "the power of point" theorem is true, and undoubtedly very important.} (a) Let two segments $AB$ and $CD$ intersect at $P$. Prove that $AP\cd PB=CP\cd PD$ iff A,B,C,D are concyclic.
\\(b) Let $A,B,T$ be three distinct points on $\omega$, and $P$ be a point on the extension of $BA$. Prove that $PT$ is tangent to $\omega$ iff $PT^2=PA\cd PB$.
\pr{4} The convex hexagon $ABCDEF$ is inscribed in a circle. Prove that the diagonals $AD,BE,CF$ are concurrent iff \[ AB\cd CD\cd EF=BC \cd DE\cd FA\]
%96
\pr{5} The circles $S_1$ and $S_2$ intersects A. Through $A$ any pair of straight lines $BAC$ and $B_0AC_0$ are drawn with $B,B_0$ on $S_1$ and $C,C_0$ on $S_2$. Prove that the chord $BB_0$ of $S_1$ and $C_0C$ of $S_2$ are inclined at a constant angle.
%107
\pr{6} \ns{1} The centers of two circles of unequal radii $r_1,r_2$ are a distance $r_{12}$ apart. Calculate the distance between the two centers of similitude.\footnote{Plane Euclidean page 115.}
%116
\pr{7} In $\tr ABC$, $P$ lies on $BC$ (possibly on the extensions). Prove that, \[ \frac{BP}{PC}=\frac{AB \times \sin \an BAP}{AC \times \sin \an PAC}\]
%124
\pr{8}\ns{2} $A,B$ are two fixed points. $P$ moves so that $\frac{PA}{PB}$ is constant. Prove that the locus of $P$ is a circle. \fn{The circle of Appolonius.}
%125
\pr{9} Let $H$ is the orthocenter of $\tr ABC$ and $D$ be the foot of perpendicular from $A$ on $BC$. Prove that $AH=a \cot A=2R \cos A$ ($R$ circumradius), and $HD=2R\cos B \cos C$.
\pr{10}\ns{1}Let the extension of $BO$ meet the circumcircle of $\tr ABC$ at $Q$. Prove that $AQCH$ is the parallelogram. ($O$ circumcenter, $H$ orthocenter)
%133
\pr{11} $area(ABC)=\sqrt{rr_1r_2r_3}$.
%142
\pr{12} \ns{2} $r_1+r_2+r_3=4R+r$. ($R,r, r_1 ,r_2 ,r_3 $ are the radius of circumcircle, incircle, and the excircles respectively.)
\pr{13}\fn{Trigonometric form of Ceva's Theorem, both of the forms are handy proving concurrency.} Let $\triangle ABC$ be a triangle, and let $P$, $Q$, $R$ be any points in the
plane distinct from $A, B, C,$ respectively. Then $AP, BQ,
CR$ are
concurrent if and only if
\[
\frac{\sin\an CAP}{\sin\an PAB} \cd \frac{\sin\an ABQ}{\sin\an QBC} \cd
\frac{\sin\an BCR}{\sin\an RCA} = 1.
\]
\pr{14}\fn{This is a very important lemma. Don't forget!}(a) Suppose the angle bisector of $\an BAC$ intersect the circumcircle of $\tr ABC$ at $X \neq A$. Let $I$ be a point on the line segment $AX$. Then $I$ is the incenter of $\tr ABC$ if and only if $XI = XB = XC=XI_a$.
\\ (b) Find the length $AX$ in terms of the side lengths and angles, using trigonometry.
\pr{15}\ns{2}Two circles intersect at the
points $A$ and $B$. Tangent lines drawn to
both of the circles at the point $A$
intersect the circles at the points $M$ and
$N$. The lines $BM$ and $BN$ intersect the
circles once more at the points $P$ and $Q$
respectively.
Prove that the segments $MP$ and $NQ$ are equal.
%v12_1 ex 5 tot
\pr{16} (Perpendicular Lemma)\fn{More applications: \url{http://www.math.ust.hk/excalibur/v12_n3.pdf}} On a plane, for distinct
points $R, S, X, Y$ we have $RX^2-SX^2 =
RY^2 - SY^2$ if and only if $RS \perp XY$.
%
\pr{17}\ns{1}Two circles meet at $P$ and $Q$. A line intersects segment PQ and meets the circles at the points $A, B, C, D$ in that order. Prove that $\an APB =
\an CQD$.
%1998,p-125
\pr{18}\ns{2}A convex quadrilateral $ABCD$ is given for which $\an ABC+\an BCD <\pi$. The comon point of the lines $AB$ and $CD$ is $E$. Prove that $\an ABC=\an ADC$ iff \[AC^2=CD \cd CE-AB\cd AE\]
\pr{19}\ns{1} The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet at $E$, while the diagonals $AC$ and $BD$ meet at $F$. The circumcircle of the triangles $AFD$ and $BFC$ meet again at $H$. Prove that $\an EHF=\frac \pi2$.
%95-96
\pr{20}Prove that the radical axis of two intersecting circles passes through the intersection points.
%porselov 64
\pr{21} Given three circles in plane whose centers do not lie on one line. Let us draw
radical axes for each pair of these circles. Prove that all the three radical axes meet at one point. \fn{This point is called the radical center of the three circles.}
%porselov 64
\pr{22}\ns{2}Let $M$ be the midpoint of the altitude $BE$ in $\tr ABC$ and suppose that the excircle opposite $B$ touches $AC$ at $Y$. Then the line $MY$ goes through the incenter $I$.
\pr{23}\ns{1}\fn{If the incircle touches $AB$ at $Z$, then we can also deduce that $B,Z,I,P,X$ are concyclic! This is indeed a very useful and surprizing result.} Let $I$ be the incenter of $\tr ABC$. Then $AI$ is the bisector of
angle $A$. If $X$ and $Y$ are the points of contact of the incircle on $BC$ and $AC$ then prove that the lines $AI, XY$ and the perpendicular from $B$ to $AI$ are concurrent at a point $P$.
%episodes.46
\pr{24}\ns{1} Let $r,R$ be the inradius and circumradius of $\tr ABC$. Prove that \[ \cos A+\cos B + \cos C=1+\frac{r}{R}\]
\pr{25}\ns{1}If $h_a,h_b,h_c$ are the lengths of the altitudes of $\tr ABC$, whose incircle has center $I$ and radius $r$. Prove that
\\ (a) \[ \frac{r}{h_a}+\frac{r}{h_b}+\frac{r}{h_c}=1\]
\\(b) \[ h_a+h_b+h_c \geq 9r. \]
\pr{26} (Leibniz's\fn{Have you heard of Leibniz's identity? If you haven't, ask someone!} Inequality) In a triangle $ABC$ with circumradius $R$ prove that $9R^2\geq a^2+b^2+c^2$.
\pr{26}\ns{2}Let $ABC$ be an acute triangle. Let $D$ be on side $BC$ such that $AD\perp BC$. Let $H$ be a point on the segment $AD$
different from $A$ and $D$. Let line $BH$ intersect side $AC$ at $E$ and line $CH$
intersect side $AB$ at $F$. Prove that $\an EDA =\an FDA$.
%hk training canada 03
\pr{27}\ns{2}The circles $S_1$ and $S_2$ intersect at $M$ and $N$. Show that if vertices $A$ and $C$ of a rectangle $ABCD$ lie on $S_1$ while vertices $B$ and $D$ lie on $S_2$, then the intersection of the diagonals of the rectangle lies on the line $MN$.
%po len russia
\pr{28}
The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$. Let $O$ be the intersection of $AC$ and $BD$. Circle $S_2$ passes through $D$ and $O$, intersecting $AD$ and $CD$ at $M$ and $N$, respectively.
Lines $OM$ and $AB$ intersect at $R$, lines $ON$ and $BC$ intersect at $T$;
and $R$ and $T$ lie on the same side of line $BD$ as $A$.
Prove that $O, R,T$, and $B$ are concyclic.
%2001, p 12
\bigskip
\Lcen{Medium}
\pr{1} Two circles with centers $A$ and $B$ intersect at right angles. Their common chord meets $AB$ at $F$. $DE$ is a chord of the first circle passing through $B$. Prove that $A,D,E,F$ are concyclic.
%108
\pr{2} \ns{1} $L$ is the midpoint of the side $BC$ of $\tr ABC$. The circle through $L$ which touches $AB$ at $B$ and the circle through $L$ which touches $AC$ at $C$ meet at $D$. Prove thar $LA \cd LD=LB^2$
%108
\pr{3} The centers of two circles $S_1,S_2$ and their common tangents intersect at $T$. $AP$ and $AQ$ are the tangents at $A$ to the two circles. Prove that $AT$ bisects $\an PAQ$.
%116
\pr{4} Two circles $AP_1Q_1$ and $AP_2Q_2$ cut at $A$. The lines $P_1P_2,Q_1Q_2$ are their common tangents. Prove that the circles $AP_1P_2$ and $AQ_1Q_2$ touch each other.
%117
\pr{5}\fn{Brocard angle}\ns{2}Prove that there exists a point $P$ inside $\tr ABC$ such that $\an BA\Omega=\an CA\Omega=\an AC \Omega= \omega$. Prove that \[\cot \omega=\cot A+\cot B+\cot B\]
%125
\pr{6}\ns{1}Let the extension of $DG$ meet the circumcircle at $D_2$. Prove that $GD=\frac 12 D_2G$. ($G$ centroid,$D$ feet of perpendicular from $A$.)
%135
\pr{7} \ns{1} If $O_A,O_B,O_C$ are the reflections of $O$ in the sides $BC,CA,AB$ respectively, prove that $AO_A,BO_B,CO_C$ are concurrent. ($O$ cirumcenter).
%135
\pr{8} On sides $AB$ and $AC$ of triangle $ABC$ there are given points $D,E$ such that $DE$ is tangent of circle inscribed in triangle $ABC$ and $DE \parallel BC$. Prove
\[AB + BC + CA\geq 8DE\]
\pr{9} For every triangle $ABC$, let $D,E,F$ be a point located on segment $BC,CA,AB$, respectively. Let $P$ be the intersection of $AD$ and $EF$. Prove that:
\[
\frac {AB}{AF}\times DC + \frac {AC}{AE}\times DB = \frac {AD}{AP}\times BC
\]
%http://www.mathlinks.ro/viewtopic.php?t=294062
\pr{10}\ns{1} Prove that in any triangle $ABC$ ,
\[ \sqrt {r_a} + \sqrt {r_b} + \sqrt {r_c}\ge3\sqrt {3r}\]
Where $r,r_a,r_b,r_c$ are the radius of the incircle, $A-,B-,C-$ excircles of $\tr ABC$.
%http://www.mathlinks.ro/viewtopic.php?t=303038
\pr{11}\ns{2} Let $ABC$ be a triangle and let $P$ be a point on the angle bisector $AD$, with $D$ on $BC$. Let $E$, $F$ and $G$ be the intersections of $AP$, $BP$ and $CP$ with the circumcircle of the triangle, respectively. Let $H$ be the intersection of $EF$ and $AC$, and let $I$ be the intersection of $EG$ and $AB$. Determine the geometric place of the intersection of $BH$ and $CI$ when $P$ varies.
%http://www.mathlinks.ro/viewtopic.php?t=283704
\pr{12}\ns{1} A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD + BC = AB$.
%IMO 1985/1
\pr{13}\ns{1}A circumference $\alpha$ intersects with circumference $\beta$ in points $A$ and $B$. There is a tangent line to both circumferences $\alpha$ and $\beta$ which intersects them in points $C$ and $D$ respectively. Points $C$, $D$, $B$ ($B$ is closer to the tangent line) lie on the circumference $\gamma$. Prove, that the radius of circumference $\gamma$ is the geometric mean of the radiuses of the circumferences $\alpha$ and $\beta$.
%http://www.mathlinks.ro/viewtopic.php?p=1511237#1511237
\pr{14}\fn{This problem is a real gem. It has multiple solutions with unique and very,very beautiful ideas. This is USAMO 97.} Let $ABC$ be a triangle, and draw isosceles triangles $\triangle DBC$, $\triangle AEC$, $\triangle ABF$ external to $\triangle ABC$ (with $BC, CA, AB$ as their respective bases).
Prove that the lines through $A,B,C$ perpendicular to $EF,FD,DE$, respectively, are concurrent.
%USAMO 97
\pr{15} \ns{2} $ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that:
\[
\sin \left( \frac {A}{2} \right) \cdot \sin \left( \frac {B}{2} \right) + \sin \left( \frac {B}{2} \right) \cdot \sin \left( \frac {C}{2} \right) + \sin \left( \frac {C}{2} \right) \cdot \sin \left( \frac {A}{2} \right) \leq \frac {5}{8} + \frac {r}{4 \cdot R}.
\]
%http://www.mathlinks.ro/viewtopic.php?t=60393&sid=482ebc9e1d36aea6bde2fabe1c7be480
\pr{16}\ns{1}Let $ABCD$ be a cyclic quadrilateral. Prove that the incenters of
triangles $\triangle ABC, \triangle BCD, \triangle CDA,
\triangle DAB$ form a rectangle.
%unbound 68
\pr{17} \ns{1} Let $ABCD$ be a cyclic quadrilateral.
Prove that the sum of the inradii of $\triangle ABC$ and
$\triangle CDA$ equals the sum of the inradii of $\triangle BCD$
and $\triangle DAB$.
%unbound 68
\pr{18}\ns{2}\fn{This is IMO 1978/4. A great problem if you want to learn some homothety. Recommended reading for Homothety: \url{http://www.math.ust.hk/excalibur/v9_n4.pdf}} In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$
%
\pr{19}\ns{1} Let $O$ be the point of intersection of the diagonals $AC$ and $BD$ of the quadrilateral $ABCD$ with $AB = BC$ and $CD = DA$. Again, let $N$ and $K$ be the feet of perpendiculars from $D$ and $B$ to $AB$ and $CD$, respectively.
Prove that the points $N$, $O$, and $K$ are collinear.
%http://www.mathlinks.ro/viewtopic.php?p=1415913#1415913
\pr{20}\ns{1} In triangle $ABC$, let $AK, BL, CM$ be the altitudes and $H$ the orthocenter. Let $P$ be the midpoint of $AH$. If $BH$ and $M K$ meet at $S$, and $LP$ and $AM$ meet at $T$, show that $TS$ is perpendicular to $BC$.
%1998-1999,p64
\pr{21}\ns{2} Let $D,E,F$ be the points on the sides $BC,CA,AB$ respectively, of $\tr ABC$. Let $P,Q,R$ be the second intersection of $AD,BE,CF$ respectively, with the cricumcircle of $\tr ABC$. Show that \[\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\geq 9\]
%1998,p97
\pr{22}\ns{1} Let $ABC$ be an acute triangle, $AD,BE,CZ$ be its altitudes and $H$ its orthocenter. Let $AI,AI'$ be the internal and external bisectors of angles $A$. Let $M,N$ be the midpoints of $BC,AH$, respectively. Prove that
\begin{itemize}
\item $MN$ is perpendicular to $EZ$.
\item If $MN$ cuts the segments $AI,AI'$ at the points $K,L$, then $KL=AH$.
\end{itemize}
%95j p28
\pr{23}\ns{2}Points $D$ and $E$ lie on sides $AB$ and $AC$ of triangle $ABC$ such
that $DE \parallel BC$. Let $P$ be an arbitrary point inside $ABC$. The lines
$PB$ and $PC$ intersect $DE$ at $F$ and $G$, respectively. If $O_1$ is the
circumcenter of $PDG$ and $O_2$ is the circumcenter of $PFE$, show
that $AP \parallel O_1O_2$.
%95-96
\pr{24}\ns{2} Let $ABCDEF$ be a convex hexagon such that $AB=BC,CD=DE,EF=FA$. Prove that \[\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\geq \frac 32 . \]
%ISL 97-7
\pr{25}\ns{2} \textit{Nagel Point $N$: If the Excircles of $ABC$ touch $BC,CA,AB$ at $D,E,F$, then the intersection point of $AD,BE,CF$ is called the Nagel Point $N$.} Prove that
\begin{itemize}
\item $I,G,N$ are collinear. ($G$ centroid, $I$ incenter.)
\item $GN=2\cd IG$.
\item \textit{Speiker center $S$: The incircle of the medial triangle is called the Speiker circle, and it's center is Speiker center $S$.} Prove that $S$ is the midpoint of $IN$.
\end{itemize}
%episodes
\pr{26}\ns{1}(a) (\textit{Archemides' Theorem})Let $M$ be the midpoint of the arc $ACB$ on the circumcircle of $\tr ABC$ and let $MD$ be the perpendicular to the longer of $AC$ and $BC$ (say $AC$). Then $D$ bisects the polygonal path $ACB$ that is $AD = DC + CB$.
\\ (b) Let $C'$ be the midpoint of side $AB$. Prove that $CD$ is parallel to the angle bisector of $\an C$.
\\(c) In the same way define $B'E$, $A'F$, and prove that $C'E,B'E,A'F$ are concurrent at the incenter of $\tr ABC$.
\pr{27}\ns{1} If three cevians $AD,BE,CF$ of $\tr ABC$ are concurrent at $P$. Prove that \[ \frac{AD}{AP}+\frac{BE}{BP}+\frac{CF}{CP} \geq \frac 92 .\]
%episodes
\pr{28}\ns{1} Let $ABCD$ be a convex quadrilateral such that $\an DAB = \an ABC = \an BCD$. Let $H$ and $O$ denote the orthocenter and circumcenter of $ABC$. Prove that $D,O,H$ are collinear.
%polen bul 97
\bigskip
\Lcen{\ff Not so easy!!! \fn{Hey...Don't panic!!! These are simply some of the coolest Olympiad problems. They are worth trying even if you can not solve them.}}\e
\pr{1}\ns{1} Let $I$ and $G$ be the incenter and the centroid of the given triangle $ABC.$ Let $M,\,N,\,P$ be the midpoint of $BC,\,CA,\,AB,$ respectively and let $J$ be the incenter of triangle $MNP.$
Then we have: $I,\,G,\,J$ are collinear and $GI = 2\cdot GJ$
%http://www.mathlinks.ro/viewtopic.php?t=156502
\pr{2} \ns{2} Let $D. E, F$ be the feet of the angle bisectors of angles $A, B, C$, respectively, of triangle $ABC$, and let $K_a, K_b, K_c$ be the points of contact of the tangents to the incircle of $ABC$ through $D, E, F$ (that is, the tangent lines not containing sides of the triangle).
Prove that the lines joining $K_a, K_b, K_c$ to the midpoints of $BC, CA. AB$ respectively, pass through a single point on the incircle of $ABC$.
%http://www.mathlinks.ro/viewtopic.php?t=283779
\pr{3} In triangle $ABC$, with $AB > BC$, $BM$ is a median and $BL$ an angle bisector. The line through $M$ parallel to $AB$ meets $BL$ at $D$ and the line through $L$ parallel to $BC$ meets $BP$ at $E$.
Prove that $ED \perp BL$.
%http://www.mathlinks.ro/viewtopic.php?p=1532155#1532155
\pr{4}\ns{2} Let $O$ be the circumcircle of a $\Delta ABC$ and let $I$ be its incenter, for a point $P$ of the plane let $f(P)$ be the point obtained by reflecting $P'$ by the midpoint of $OI$, with $P'$ the homothety of $P$ with center $O$ and ratio $\frac {R}{r}$ with $r$ the inradii and $R$ the circumradii,(understand it by $\frac {OP}{OP'} = \frac {R}{r}$). Let $A_1$, $B_1$ and $C_1$ the midpoints of $BC$, $AC$ and $AB$, respectively. Show that the rays $A_1f(A)$, $B_1f(B)$ and $C_1f(C)$ concur on the incircle.
%http://www.mathlinks.ro/viewtopic.php?t=283715
\pr{5}\ns{2}Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is cyclic, $QC_{1}\parallel CA$, and the points $C_{1}$ and $Q$ lie on opposite sides of the line $AB$. Construct a point $B_{1}$ in such a way that the convex quadrilateral $APCB_{1}$ is cyclic, $QB_{1}\parallel BA$, and the points $B_{1}$ and $Q$ lie on opposite sides of the line $AC$.
Prove that the points $B_{1}$, $C_{1}$, $P$, and $Q$ lie on a circle.
%http://www.mathlinks.ro/viewtopic.php?p=213011#213011
\pr{6}\ns{3}Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC,$ respectively, such that $\frac {AE}{ED} = \frac {BF}{FC}.$ Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T,$ respectively. Prove that the circumcircles of triangles $SAE,$ $SBF,$ $TCF,$ and $TDE$ pass through a common point.
%http://www.mathlinks.ro/viewtopic.php?p=490691#490691
\pr{7}\ns{3} Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that
\[
8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; ,
\]
with equality if and only if triangle $ABC$ is equilateral.
%http://www.mathlinks.ro/viewtopic.php?p=825515#p825515
\pr{8}\ns{1} Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.
%http://www.mathlinks.ro/viewtopic.php?p=336205#336205
\pr{9}\ns{1}Let $ABC$ be an acute, scalene triangle, and let $M$, $N$, and $P$ be the midpoints of $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. Let the perpendicular bisectors of $\overline{AB}$ and $\overline{AC}$ intersect ray $AM$ in points $D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in point $F$, inside of triangle $ABC$. Prove that points $A$, $N$, $F$, and $P$ all lie on one circle.
%http://www.mathlinks.ro/viewtopic.php?p=1116181#1116181
\pr{10}\ns{2}$ABCDEF$ is a convex hexagon with $AB = BC$, $CD = DE$,$EF = FA$.
Prove that
\[
\frac {{BC}} {{BE}} + \frac {{DE}} {{DA}} + \frac {{FA}} {{FC}} \geqslant \frac {3} {2}
\]
%http://www.mathlinks.ro/viewtopic.php?t=283183
\pr{11} \ns{2} An acute triangle $ABC$ is given. Points $A_1$ and $A_2$ are taken on the side $BC$ (with $A_2$ between $A_1$ and $C$), $B_1$ and $B_2$ on the side $AC$ (with $B_2$ between $B_1$ and $A$), and $C_1$ and $C_2$ on the side $AB$ (with $C_2$ between $C_1$ and $B$) so that
\[
\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1.
\]
The lines $AA_1,BB_1,$ and $CC_1$ bound a triangle, and the lines $AA_2,BB_2,$ and $CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
%http://www.mathlinks.ro/viewtopic.php?t=219895
\pr{12}\ns{1}The incircle $k$ of a non-isosceles triangle $ABC$ touches the sides $AB,BC,CA$
at $C_{1},A_{1},B_{1}$ , respectively. Lines $AA_{1},BB_{1},CC_{1}$ intersect $k$ again at
$A_{2},B_{2},C_{2}$, respectively. Let $A_{1}A_{3}$ and $B_{1},B_{3}$ be the bisectors of the
angles in △$A_{1}B_{1}C_{1}$ ($A_{3}$ ∈ $B_{1}C_{1}, B_{3}$ ∈ $A_{1}C_{1}$). Prove that:
$(a)$ $A_{2}A_{3}$ is a bisector of $\angle B_{1}A_{2}C_{1}$;
$(b)$ If the circumcircles of △$A_{1}A_{2}A_{3}$ and △$B_{1}B_{2}B_{3}$ intersect at $P$ and
$Q$, then the incenter $I$ of △$ABC$ lies on the line $PQ$.
%http://www.mathlinks.ro/viewtopic.php?p=1516814#1516814
\pr{13}\ns{1} Let $ABC$ be a triangle, $O$ its circumcenter, $S$ its centroid, and $H$ its orthocenter. Denote by $A_1, B_1$ and $C_1$ the centers of the circles circumscribed about the triangles $CHB, CHA$ and $AHB$ respectively.
Prove that the triangle $ABC$ is congruent to the triangle $A_1B_1C_1$ and that the nine-point circle of $ABC$ is also the nine-point circle of $A_1B_1C_1$.
%http://www.mathlinks.ro/viewtopic.php?t=280779
\pr{14}\ns{1}Let $ABC$ be a triangle and $K$ and $L$ be two points on $(AB)$, $(AC)$ such that $BK = CL$ and let $P = CK\cap BL$. Let the parallel through $P$ to the interior angle bisector of $\angle BAC$ intersect $AC$ in $M$. Prove that $CM = AB$.
%http://www.mathlinks.ro/viewtopic.php?p=1510314#1510314
\pr{15} Let $ABC$ be an acute-angled triangle, and let $H$ be its orthocenter. Let $D$ be the foot of the altitude from $B$ to $AC$, and let $E$ be the reflection of $A$ on $D$. The circumcircle of triangle $BCE$ intersects the median from $A$ in an interior point $F$. Prove that $A, D, H$ and $F$ are concyclic.
%http://www.mathlinks.ro/viewtopic.php?t=255138
\pr{16} \ns{1} Let $ABC$ be triangle with $AB \neq AC$. Point $E$ is such that $AE=BE$ and $BE\perp BC$. Point $F$ is such that $AF=CF$ and $CD \perp BC$. Let $D$ be the point on the line $BC$ such that $AD$ is tangent to the circumcircle of triangle ABC. Prove that $D,E,F$ are collinear.
%Russia 2003
\pr{17} \ns{1} Let $ABC$ be an isosceles triangle with $AB = AC$. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$. $Q$ is an arbitrary point on $BC$ different from $B$ and $C$. $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E, Q, F$ are distinct and collinear. Prove that $OQ$ is perpendicular to $EF$ if and only if $QE = QF$.
%IMO 94/2
\pr{18}\ns{1} An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$, and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$. Prove that the points $\, M, N, P, Q \,$ lie on a common circle.
%USAMO 90/5
\pr{19}\ns{2} Let $O$ and $I$ be the circumcenter and incenter of triangle $\triangle ABC$,
respectively. Let $\omega_A$ be the excircle of triangle $\triangle ABC$
opposite
to $A$; let it be tangent to $AB$, $AC$, $BC$
at $K,M,N$, respectively.
Assume that the midpoint of segment $KM$ lies on the circumcircle
of triangle $\triangle ABC$. Prove that $O,N,I$ are collinear.
%
\pr{20}\ns{1} Let $O$ be the center of circle $\omega$. Two equal chords of $AB$ and $CD$ of $\omega$ intersects at $L$ such that $AL>LB$, and $DL>LC$. Let $M$ and $N$ be points on $AL$ and $DL$ respectively such that $\an ALC=2\an MON$. Prove that the chord of $\omega$ passing through $M$ and $N$ is congruent to $AB$ and $CD$.
% Around the world 1999,p24
\pr{21}\ns{2}Let $O$ be the center of the excircle of $\tr ABC$ opposite to $A$. Let $M$ be the midpoint of $AC$, and let $P$ be the intersection of lines $MO$ and $BC$. Prove that if $\an BAC=2 \an ACB$, then $AB=BP$.
% Around the world 1999,p29
\pr{22}\ns{1}\fn{There is a very ingenious solution using Pole-Polar. For further study:\url{http://www.math.ust.hk/excalibur/v11_n3.pdf}} Let $ABCD$ be a cyclic
quadrilateral. Let $AB\cap CD = P$\fn{$AB\cap CD=X$ means that the intersection point of $AB$ and $CD$ is $X$. This sign is very common in problem literature.} and
$AD\cap BC = Q$. Let the tangents from $Q$
meet the circumcircle of $ABCD$ at $E$
and $F$. Prove that $P, E, F$ are collinear.
%pole polar
\bigskip
\lcen{IMO 2009: \textit{Why don't we try them?}}
\pr{2} (\textit{Day 1}) Let $ABC$ be a triangle with circumcentre $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ respectively. Let $K,L$ and $M$ be the midpoints of the segments $BP,CQ$ and $PQ$. respectively, and let $\Gamma$ be the circle passing through $K,L$ and $M$. Suppose that the line $PQ$ is tangent to the circle $\Gamma$. Prove that $OP = OQ.$
\pr{4} (\textit{Day 2}) Let $ABC$ be a triangle with $AB = AC$ . The angle bisectors of $\angle C AB$ and $\angle AB C$ meet the sides $B C$ and $C A$ at $D$ and $E$ , respectively. Let $K$ be the incentre of triangle $ADC$. Suppose that $\angle B E K = 45^\circ$ . Find all possible values of $\angle C AB$ . (Don't forget to check the cases, anyway.)
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