geometry problems from Brazilian Cono Sur (OMCS) + IMO + IberoAmerican (OIM) Team Selection Tests (TST) with aops links in the names
(IMO TST only those not in IMO Shortlist)
[3p per day]
collected inside aops: here
IMO / IberoAmerican TST
1.1 Let $ABC$ be a triangle and $L$ its circumscribed circle. The internal bisector of angle $A$ meets $BC$ at point $P$. Let $L_1$ be the circle tangent to $AP,BP$ and $L$. Similarly, let $L_2$ be the circle tangent to $AP,CP$ and $L$. Prove that the tangency points of $L_1$ and $L_2$ with $AP$ coincide.
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly.
(a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$.
(b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.
In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.
Let $L$ be a circle with center $O$ and tangent to sides $AB$ and $AC$ of a triangle $ABC$ in points $E$ and $F$, respectively. Let the perpendicular from $O$ to $BC$ meet $EF$ at $D$. Prove that $A,D$ and $M$ are collinear, where $M$ is the midpoint of $BC$.
Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.
Let $BD$ and $CE$ be the bisectors of the interior angles $\angle B$ and $\angle C$, respectively ($D\in AC$, $E\in AB$). Consider the circumcircle of $ABC$ with center $O$ and the excircle corresponding to the side $BC$ with center $I_a$. These two circles intersect at points $P$ and $Q$.
(a) Prove that $PQ$ is parallel to $DE$.
(b) Prove that $I_aO$ is perpendicular to $DE$.
In a triangle $ABC$, the bisector of the angle at $A$ of a triangle $ABC$ intersects the segment $BC$ and the circumcircle of $ABC$ at points $A_1$ and $A_2$, respectively. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that
$$\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.$$
Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠ $AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.
Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$.
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ and $Q$ be points on the segments $AB$ and $AC$, respectively, such that $BP : PQ : QC = AC : CB : BA$.
Prove that the points $A$, $P$, $Q$ and $O$ lie on one circle.
Alternative formulation
Let $O$ be the center of the circumcircle of a triangle $ABC$. If $P$ and $Q$ are points on the sides $AB$ and $AC$, respectively, satisfying $\frac{BP}{PQ}=\frac{CA}{BC}$ and $\frac{CQ}{PQ}=\frac{AB}{BC}$, then show that the points $A$, $P$, $Q$ and $O$ lie on one circle.
In a triangle $ABC,$ the internal and external bisectors of the angle $A$ intersect the line $BC$ at $D$ and $E$ respectively. The line $AC$ meets the circle with diameter $DE$ again at $F.$ The tangent line to the circle $ABF$ at $A$ meets the circle with diameter $DE$ again at $G.$ Show that $AF = AG.$
Let $I$ be the incenter of a triangle $ABC$ with $\angle BAC=60^\circ$. A line through $I$ parallel to $AC$ intersects $AB$ at $F$. Let $P$ be the point on the side $BC$ such that $3BP=BC$. Prove that $\angle BFP=\frac12\angle ABC$.
Determine the locus of points $M$ in the plane of a given rhombus $ABCD$ such that $MA\cdot MC+MB\cdot MD=AB^2$.
Let $AB$ be a chord, not a diameter, of a circle with center $O$. The smallest arc $AB$ is divided into three congruent arcs $AC$, $CD$, $DB$. The chord AB is also divided into three equal segments $AC'$, $C'D'$, $D'B$. Let $P$ be the intersection point of between the lines $CC'$ and $DD'$. Prove that $\angle APB = \frac13 \angle AOB$.
Let $A, B, C, D, E$ points in circle of radius r, in that order, such that $AC = BD = CE = r$. The points $H_1, H_2, H_3$ are the orthocenters of the triangles $ACD$, $BCD$ and $BCE$, respectively. Prove that $H_1H_2H_3$ is a right triangle .
Let $ABC$ be an acute triangle and $D$ a point on the side $AB$. The circumcircle of triangle $BCD$ cuts the side $AC$ again at $E$ .The circumcircle of triangle $ACD$ cuts the side $BC$ again at $F$. If $O$ is the circumcenter of the triangle $CEF$. Prove that $OD$ is perpendicular to $AB$.
Given two circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively intesrecting at two points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. The lines $XA$ and $YA$ intersect $\omega_2$ again in $Z$ and $W$, respectively, such that $A$ is between $X,Z$ and $A$ is between $Y,W$. Let $M$ be the midpoint of $O_1O_2$, S be the midpoint of $XA$ and $T$ be the midpoint of $WA$. Prove that $MS = MT$ if, and only if, the points $X, Y, Z$ and $W$ are concyclic.
Find a triangle $ABC$ with a point $D$ on side $AB$ such that the measures of $AB, BC, CA$ and $CD$ are all integers and $\frac{AD}{DB}=\frac{9}{7}$, or prove that such a triangle does not exist.
Let $ABCD$ be a convex cyclic quadrilateral with $AD > BC$, A$B$ not being diameter and $C D$ belonging to the smallest arc $AB$ of the circumcircle. The rays $AD$ and $BC$ are cut at $K$, the diagonals $AC$ and $BD$ are cut at $P$ and the line $KP$ cuts the side $AB$ at point $L$. Prove that angle $\angle ALK$ is acute.
In a triangle $ABC$, points $H, L, K$ are chosen on the sides $AB, BC, AC$, respectively, so that $CH \perp AB$, $HL \parallel AC$ and $HK \parallel BC$. In the triangle $BHL$, let $P, Q$ be the feet of the heights from the vertices $B$ and $H$. In the triangle $AKH$, let $R, S$ be the feet of the heights from the vertices $A$ and $H$. Show that the four points $P, Q, R, S$ are collinear.
Cono Sur TST
since 2013 serves also as OMCPLP TST / Lusophon / Portuguese Language
The bisector of angle $B$ in a triangle $ABC$ intersects side $AC$ at point $D$. Let $E$ be a point on side $BC$ such that $3\angle CAE =2\angle BAE$. The segments $BD$ and $AE$ intersect at point $P$. If $ED=AD=AP$, determine the angles of the triangle .
Let $ABCD$ be a parallelogram, $H$ the orthocenter of the triangle $ABD$ and $O$ the circumcenter of the triangle $BCD$. Prove that the points $H, O$ and $C$ are collinear.
Let $L$ and $M$, respectively be the intersections of the internal and external bisectors of angle $C$ of the triangle $ABC$ and the line AB. If $CL = CM$, prove that $AC^2+ BC^2= 4R^2$ , where $R$ is the radius of the circumscribed circle of triangle $ABC$.
Let $AD$ be the bisector of the angle $A$ of the triangle $ABC$. Consider the points $M, N$ on the ray $\overrightarrow{AB}$ and $\overrightarrow{BC}$ respectively and such as $\angle MDA = \angle ABC$ and $\angle NDA =\angle BCA$. Let $P = AD \cap MN$. prove that $AD^3 = AB \cdot AC \cdot AP$.
Let $A_1, B_1, C_1$ be the intersection points of the extensions of the medians of the triangle $ABC$ with the circumcircle of triangle $ABC$. Prove that if $A_1B_1C_1$ is equilateral, then $ABC$ is equilateral.
The convex quadrilateral $ABCD$ is inscribed in a circle with radius $5 $ cm. If $AB=8$ cm, $AC = 3\sqrt{10}$ cm, $CD=6$ cm and $\angle ADC<90^o$, calculate the area od the quadrilateral.
Given one circlewich center $O$, the two tangents from a point $S$, outside the circle , intersect it at points $P$ and $Q$. The points at which the line $SO$ intersects the circle are denoted $A$ and $B$, where $B$ is between $S$ and $A$. Finally, where $X$ is an interior point in the smallest arc $PB$, we denote $C$ and $D$ the intersection points of the line $OS$ with $QX$ and $PX$, respectively. Prove that $\frac{1}{AC}+\frac{1}{AD}=\frac{2}{AB}$.
In the convex quadrilateral $ABCD$ with sides $AD$ and $BC$ not parallel, lie $M$ and $P$ on sides $AB$ and $CD$, respectively, such that$$\frac{MA}{MB}=\frac{PD}{PC}$$and let $Q$ be any point on the side $AD$ . A line parallel to $MP$ passing through $Q$ cuts the parallels to $BC$ passing through $A$ and $D$ at points $X$ and $Y$ , respectively. Prove that $MX$, $P Y$ and $BC$ are concurrent.
Points $D, E$ and $F$ are on sides $BC, CA$ and $AB$ of triangle $ABC$, respectively, so that segments $AD, BE$ and $CF$ pass through the same point $G$. Prove that if quadrilaterals $AFGE$, $BDGE$ and $CEGD$ are tangential and the circles inscribed with these three quadrilaterals are tangent in pairs, then the triangle $ABC$ is equilateral.
Let $P$ be a point on the small arc $AB$ of the circumscribed circle $\Gamma$ of square $ABCD$. Segments $AC$ and $PD$ intersect at $Q$ and $AB$ and $PC$ at $R$. Show that $QR$ is the bisector of the ́ angle $\angle PQB$.
(a) Let $ABCD$ be a quadrilateral and $X$ a point on the plane such that the perimeters of the triangles $ABX$, $BCX$, $CDX$ and $DAX$ are the same. Show that $ABCD$ is tangential.
(b) Show that there is a tangential quadrilateral $ABCD$ such that none point $X$ of the plane satisfies the conditions of item (a).
Let $I$ be the incenter of the triangle $ABC$ and $M$ the midpoint of the side $AB$. Knowing that $CI = MI$, calculate the minimum measure of the angle $\angle CIM$.
The circles $S_1$ and $S_2$ intersect at two points $A$ and $B$. The straight line that passes through $A$ and is parallel to the straight line connecting the centers of the two circles cuts $S_1$ at $C\ne A$ and cuts $S_2$ at $D \ne A$. The circle $S_3$ of diameter $CD$ cuts $S_1$ at $P\ne C$ and $S_2$ at $Q\ne D$. Prove that the lines $CP$, $DQ$ and $AB$ are concurrent.
Let $ABCDEF$ be a convex hexagon such that each of the diagonals $AD, BE$, and $CF$ divide the hexagon into two regions of equal areas. Prove that $AD,BE$ and $CF$ are concurrent.
It there is a convex pentagon such that any of the internal bisectors of its angles and one of its diagonals?
original wording
PS. Google Translation doesn't make sense.
Let $H$ be the point of intersection of the altitudes $BB_1$ and $CC_1$ of triangle $ABC$. Let $\ell$ be the line that passes through $A$ and is perpendicular to $AC$. Prove that the lines $BC$, $B_1C_1$ and $\ell$ are concurrent if and only if $H$ is the midpoint of $BB_1$.
Let $ABC$ be a triangle and $P$ a point on the median of side $BC$. Let $D$ be the intersection of $AC$ and $BP$ and $ E$ be the intersection of $AB$ and $CP$. if the inradii of the triangles $BEP$ and $CDP$ are equal, prove that $AB = AC$.
The incircle of triangle $ABC$ is tangent to sides $BC, CA, AB$ at points $K, L,M$, respectively. Let $P$ be the intersection of the internal bisector of the angle $\angle ACB$ with the line $MK$. Show that the lines $AP$ and $LK$ are parallel.
The quadrilateral $ABCD$ is cyclic. Let $L$ be the incenter of the triangle $ABC$ and $M$ be the incenter of the triangle $BCD$. The point $R$ is the intersection ̧between the line perpendicular to $AC$ that passes through $L$ and the line perpendicular to $BD$ that passes through $M$. Prove that the $LMR$ triangle is isosceles.
Let $ABCD$ a rectangle and $E,F$ points in the segments $BC$ and $DC$, respectively, such that $\angle DAF=\angle FAE$. Show that if $DF+BE=AE$, then $ABCD$ is a square.
Let $AB$ and $AC$ be tangent to the circle $\Gamma$ at $B$ and $C$ respectively. Let $D$ be a point on the extension of $AB$, closer to $B$, and let $P$ the second intersection point of of ̃ $\Gamma$ with the circumscribed circle of triangle $ACD$. Let $Q$ be the projection of $B$ on $CD$. Prove that $\angle DP Q = 2\angle ADC$.
On the convex quadrilateral $ABCD$, sides $AB$ and $CD$ are not parallel and have the same length. Let $P$ be any point on the segment that joins the midpoint of diagonals $AC$ and $BD$. Prove that the sum of the distances from $P$ to lines $AB$ and $CD$ do not depend on the choice of point $P$.
A point $D$ was chosen inside the side $BC$ of an acute triangle $\vartriangle ABC$ and another point $P$ was chosen inside the segment $AD$ but which doesn't lie on the median of vertex $C$. The line containing this median intersects the circumcircle of the triangle $\vartriangle CPD$ at $K$ ($K \ne C$). Show that the circumcircle of the triangle $\vartriangle AKP$ passes, in addition to point $A$, through another fixed point that does not depend on the choice of points $D$ and $P$.
Consider the triangle $ABC$. Inside sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that $AE = BD$. Let $P$ be the intersection point of $AD$ and $BE$ and $Q$ is a point in the plane such that $APBQ$ is a parallelogram. Prove that $Q$ is on the bisector of $\angle ACB$.
Let $ABC$ be a triangle with $\angle ABC = 90^o$. Denote by $\Gamma$ a circle with center on the side $BC$ and tangent to the side $AC$. Let $AE$ be the tangent line to Γ, different of $AC$, with $E$ on $\Gamma$. Let $B'$ be the midpoint of $AC$ and $M$ the intersection of $BB'$ and $AE$. Prove that $MB = ME$.
Prove that there are $2012$ non-similar triangles with angles $A, B$ and $C$ satisfy under the following conditions:
1. $\frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} =\frac{12}{7}$
2. $\sin A \sin B \sin C =\frac{12}{25}$
Let $A, B, C, X, Y$ and $Z$ be points in the plane. Prove that the circumcircles of triangles $AYZ$, $BZX$, $CXY$ are concurrent at a point if and only if the circumcircles of the triangles $XBC$, $YCA$ and $ZAB$ are concurrent at a point.
Let $O$ and $H$ be the circumcenter and orthocenter of $\vartriangle ABC$, respectively. Let $M$ and $N$ be the midpoints of $BH$ and $CH$, respectively. Point $B'$ lies on the circumscribed circle of $\vartriangle ABC$ so that $BB'$ is a diameter. If the quadrilateral $HONM$ is tangential, prove that $B'N =\frac{AC}{2}$.
Let $A$ be a point outside a circle $\omega$. Through A are drawn two straight lines, each an intersects ω at two points. The first intersects $\omega$ at $B$ and $C$, the second intersects at $D$ and $E$ ($D$ is between $A$ and $E$). The line through $D$ parallel to $BC$ cuts $\omega$ again at $F$. The line $AF$ cuts $\omega$ again at $T,$ different from $F$. The lines $BC$ and $ET$ meet at $M$. The point N is such that $M$ is the midpoint of $AN$. Let $K$ be the midpoint of $BC$. Prove that points $D, E, K$ and $N$ line on a circle..
A convex quadrilateral $ABCD$ is inscribed on a circle of center $O$ and circumscribed in a circle of center $I$. Prove that$$\frac{IB^2}{AB \cdot BC}+\frac{IC^2}{BC \cdot CD}+\frac{ID^2}{CD \cdot DA}+ \frac{IA^2}{DA \cdot AB}=2$$
Let $ABC$ be a triangle with $\angle ABC > \angle BCA$ and $\angle BCA \ge 30^o$. The internal bisectors of angles $\angle ABC$ and $\angle BCA$ meet opposite sides of the triangle at points $D$ and $E$, respectively. The point $P$ is the intersection point of the segments $BD$ and $CE$ . Knowing that $PD = PE$ and that the radius of the incircle of the triangle $ABC$ is equal to $1$, determine the maximum value of the sidelength $BC$ .
Let $ABCDE$ be a convex pentagon inscribed in a circle such that $AB = BC$ and $CD = DE$. Segments $AD$ and $BE$ intersect at point $P$ and segment $BD$ intersects the segment $CA$ at $Q$ and the segment $CE$ at $T$. Prove that the triangle $PQT$ ́is isosceles.
The incircle of the triangle $ABC$ is tangent to $AB$ and $AC$ at $D$ and $E$, respectively. A circle passes through points $B$ and $C$ and is tangent to line $DE$ at point $X$. Prove that the angle $\angle BXC$ ́is obtuse.
Consider a triangle $ABC$ where $\angle A$ is the smallest of the three internal angles and let $\Gamma$ be the circle passing through the vertices of triangle $ABC$. Let $D$ be a point on the arc $BC$ of $\Gamma$ that does not contain $A$. Let $E$ be a point on the arc $AC$ of $\Gamma$ that does not contain $B$, such that $CE = BD$ and let $F$ be a point on the arc $AB$ of $\Gamma$ that does not contains $C$ such that $BF = CD$. The $EF$ segment cuts sides $AB$ and $AC$ at points $P$ and $Q$, respectively. It is also known that $DF$ cuts side $AB$ at $X$ and $DE$ cuts side $AC$ at $Y$. Let $R$ be the intersection point of the lines $PY$ and $QX$. Prove that the quadrilateral $APRQ$ is cyclic.
Let $ABC$ be a triangle, with circumcenter $O$, where $AB < AC$. Let $I$ be the midpoint of the arc $BC$ that does not contains $A$. In segment $AC$, let $K$ be a point, different from $C$, such that $IK = IC$. The line $BK$ cuts the circumcircle of triangle $ABC$ at point $D$, different from $B$, and intersects $AI$ at point $E$. Line $DI$ intersects line $AC$ at point $F$.
a) Prove that $BC = 2 EF$.
b) In segment $DI$, let M be a point such that $CM \parallel AD$. Lines $KM$ and $BC$ meet at point $N$. The circumcircle of triangle $BKN$ cuts the circumcircle of triangle $ABC$ at point $P$ other than $B$. $Q$ is the midpoint of $AD$. Prove that the point $Q$ lies on the line $PK$.
Let $ABC$ be a triangle. Let $P_1$ and $P_2$ be two points on the side $AB$ such that $P_2$ is in segment $BP_1$ and $AP_1 = BP_2$. Similarly, let $Q_1$ and $Q_2$ be points on the side $BC$ such that $Q_2$ is in segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ intersect at point $R$. The circles of triangles $P_1P_2R$ and $Q_1Q_2R$ intersects again at point $S \ne R$, which is inside the triangle $P_1Q_1R$. Finally, let $M$ be the midpoint of the side $AC$ . Prove that the angles $\angle P_1RS$ and $\angle Q_1RM$ are equal.
Let $I$ be the incenter of the triangle $ABC$. Let $A_1$, $B_1$ and $C_1$ be the points of tangency of the incircle of the triangle $ABC$ with sides $BC$, $AC$ and $AB$ respectively. Let $B$ and $K$ be the intersection ̧points of the line $BC$ with the circumscribed circle of the triangle $BC_1B_1$ . Let $C$ and $L$ be the intersection ̧points of the line $BC$ with a circumscribed circle of the triangle $CB_1C_1$. Prove that the lines $LC_1$, $KB_1$ and $IA_1$ are concurrent at the same point.
Let $ABCD$ be an cyclic quadrilateral such that the rays $\overrightarrow{AB}$ and $\overrightarrow{DC}$ intersect at $K$. Let $M$ and $N$ be the midpoints of segments $AC$ and $KC$, respectively. Suppose points $M, B, N$ and $D$ lie on the same circle. Determine the measure of the angle $\angle ADC$.
Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$, respectively, intersect at points $A$ and $B$. A straight line passing through $B$ cuts $\omega_1$ again at $C$ and $\omega_2$ again at $D$, both different from $B$. The tangent lines to $\omega_1$ at $C$ and $\omega_2$ at $D$ meet at point $E$. Let $\Gamma$ be the circle that passes through points $A$, $O_1$ and $O_2$. Let $F$ be the second intersection point of segment $AE$ with circle $\Gamma$. Prove that the length of $EF$ is equal to the diameter of $\Gamma$.
Let $ABCD$ be a rhombus such that triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the point of intersection of the diagonals $AC$ and $BD$.Prove that $\angle XMN = \angle DAM$ and $\angle XNM = \angle BAN$.
Let $ABC$ be a triangle, with $BC$ being its shortest side. Let $X, Y$ be points on sides $AB, AC$, respectively such that $XB = BC = CY$ . Let $K$ and $L$ be points in the extension of the side $BC$ beyond $B$ and in the extension of the side $BC$ beyond $C$, respectively, such that $KB = BC = CL$. Let $M$ be the intersection point of the lines $KX $and $LY$ .If $G$ is the centroid of triangle $KLM$, prove that $G$ is the incenter of triangle $ABC$.
Let $k$ be the circumcircle of triangle $ABC$ and $D$ a point on the arc $AB$ that does not contain $C$. Let $I_a$ and $I_B$ be the incenters of triangles $ADC$ and $BDC$, respectively. Prove that the circumcircle of the triangle $I_AI_BC$ is tangent to $k$ if, and only if,
$$\frac{AD}{BD}=\frac{AC+CD}{BC+CD}$$
also
An equilateral triangle $ABC$ is inscribed in a circle $\Omega$ and has incircle $\omega$. Points $P$ and $Q$ are in segments $AC$ and $AB$, respectively, such that $PQ$ is tangent to $\omega$. The circle $\Omega_B$ has center $P$ and radius $PB$ and the circle $\Omega_C$ is defined similarly. Prove that $\Omega$, $\Omega_B$ and $\Omega_C$ have a common point.
Let $ABC$ be a triangle with incenter $I$ such that $AB < BC$. If $M$ is the midpoint of $AC$ and $N$ is the midpoint of the arc $AC$ containing $B$ of the circle circumscribed around $ABC$. Prove that $\angle IMA = \angle INB$.
Let $\Gamma$ be a circle with center $K$ and let $M$ be a point on $\Gamma$. Let $\omega$ be a semicircle with diameter $KM$ and let $L$ be a point within the segment $KM$. The line perpendicular to $KM$ through point $L$ intersects $\omega$ at $Q$ and intersects line at points $P_1$ and $P_2$ such that $P_1Q > P_2Q$. Finally, the line $MQ$ intersects $\Gamma$ for the second time at point $R$, distinct from $M$. Let $S_1$ and $S_2$ be the areas of triangles $MP_1Q$ and $P_2RQ$, respectively.
(a) Prove that $MQ = QR$.
(b) Prove that $\frac{S_1}{S_2}<3 + 2\sqrt2$
Let $ABC$ be a triangle and $E$ and $F$ two arbitrary points on sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. The point $D$ is such that $EF$ bisects the segment $MD$ . Finally, $O$ is the circumcenter of triangle $ABC$. Prove that $D$ lies on line $BC$ if and only if $O$ lies on the circumcircle of triangle $AEF$.
Let $ABC$ be a acute triangle and $AA_1$, $BB_1$ ,$CC_1$ be the altitudes . The straight line perpendicular to $AC$ through $A_1$ intersects the line $B_1C_1$ at point $D$. Prove that $\angle ADC= 90^o$.
Let $ABC$ be an acute triangle whose circumcircle is $\omega$. Let $D$ and $E$ be points on the segments $AB$ and $BC$, respectively, such that $AC$ is parallel to $DE$. Let $P$ and $Q$ be points on the minor arc $AC$ of $\omega$ such that $DP$ is parallel to $EQ$. The lines $QA$ and $PC$ intersect the line $DE$ at $X$ and $Y$ , respectively. Prove that $\angle XBY + \angle PBQ = 180^o$.
Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.
Let $ABC$ be a triangle and $D$ is a point inside of $\triangle ABC$. The point $A'$ is the midpoint of the arc $BDC$, in the circle which passes by $B,C,D$. Analogously define $B'$ and $C'$, being the midpoints of the arc $ADC$ and $ADB$ respectively. Prove that the four points $D,A',B',C'$ are concyclic.
Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.
Consider a circle $ \alpha$ tangent to two paralell lines $ l_1$ and $ l_2$ at $ A$ and $ B$, respectively. Call $ C$ any point of $ l_1$, since it's not $ A$. Let $ D$ and $ E$ be two points in the circunference, such that they are not coincident and are at the same side of $ AB$ as $ C$. Let $ CD\cap \alpha = \{F,D\}$ and $ CE\cap \alpha = \{G,E\}$. Let $ AD\cap l_2 = H$ and $ AE\cap l_2 = I$.Let $ AF\cap l_2 = J$ and $ AG\cap l_2 = K$. Show that $ JK = HI$.
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