Brazil TST 1997 - 2020 (IMO - OMCS - OIM) 72p

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 

1997 Brazil IMO 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.

1997 Brazil IMO TST 1.5

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 $S_{A_1B_1C_1}\ge9S_{ABC}$.

1997 Brazil IMO TST 2.1

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 $BD$.

1998 Brazil IMO TST 1.4

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 $BC$.

1998 Brazil IMO TST 2.1

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.

1999 Brazil IMO TST 1.3

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 $DE$.

1999 Brazil IMO TST 2.2

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

$$\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.$$

2000 Brazil IMO TST 1.3

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 $AM$.

2000 Brazil IMO TST 2.1

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 $\angle{BAC}$.

2001 Brazil IMO TST 1.4

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.

2001 Brazil IMO TST 2.3

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 $AF = AG.$

2004 Brazil IMO TST 1.4

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 $\angle BFP=\frac12\angle ABC$.

2004 Brazil IMO TST 2.3

Determine the locus of points $M$ in the plane of a given rhombus $ABCD$ such that $MA\cdot MC+MB\cdot MD=AB^2$.

2008 Brazil IMO TST 3.1

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 $\angle APB = \frac13  \angle AOB$.

2009 Brazil IMO TST 4.1

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 .

2010 Brazil IMO TST 3.1

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 $AB$.

2011 Brazil IMO TST 4.2

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.

2013 Brazil IMO TST 3.1

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.

2013 Brazil IMO TST 4.2

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.

2018 Brazil IMO TST 4.6

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

1999 Brazil Cono Sur TST 1.3

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 .

1999 Brazil Cono Sur TST 2.3

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.

2000 Brazil Cono Sur TST 1.1

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 $ABC$.

2000 Brazil Cono Sur TST 2.3

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 $AD^3 = AB \cdot AC \cdot AP$.

2001 Brazil Cono Sur TST 1.4

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.

2001 Brazil Cono Sur TST 2.2

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.

2002 Brazil Cono Sur TST 1.3

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 $\frac{1}{AC}+\frac{1}{AD}=\frac{2}{AB}$.

2004 Brazil Cono Sur TST 1.2

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.

2004 Brazil Cono Sur TST 2.1

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.

2005 Brazil Cono Sur TST 1.1

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 $\angle PQB$.

2005 Brazil Cono Sur TST 2.3

(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).

2005 Brazil Cono Sur TST 2.5

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 $\angle CIM$.

2005 Brazil Cono Sur TST 3.1

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.

2006 Brazil Cono Sur TST 1.3

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.

2006 Brazil Cono Sur TST 2.4

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.

2006 Brazil Cono Sur TST 3.1

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 $BB_1$.

2007 Brazil Cono Sur TST 1.3

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 $AB = AC$.

2008 Brazil Cono Sur TST 1.2

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.

2008 Brazil Cono Sur TST 2.2

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.

2009 Brazil Cono Sur TST 1.3

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.

2010 Brazil Cono Sur TST 1.4

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 $\angle DP Q = 2\angle  ADC$.

2010 Brazil Cono Sur TST 2.2

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 $P$.

2010 Brazil Cono Sur TST 3.3

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 $P$.

2011 Brazil Cono Sur TST 1.3

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 $\angle ACB$.

2012 Brazil Cono Sur TST 1.1

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 $MB = ME$.

2012 Brazil Cono Sur TST 2.3

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. $\sin A \sin B \sin C =\frac{12}{25}$

2012 Brazil Cono Sur TST 3.2

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.

2013 Brazil Cono Sur TST 1.2

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 $B'N =\frac{AC}{2}$.

2013 Brazil Cono Sur TST 2.4

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..

2013 Brazil Cono Sur TST 3.2

A convex quadrilateral $ABCD$ is inscribed on a circle of center $O$ and circumscribed in a circle of center $I$. Prove that $$\frac{IB^2}{AB \cdot BC}+\frac{IC^2}{BC \cdot CD}+\frac{ID^2}{CD \cdot DA}+ \frac{IA^2}{DA \cdot AB}=2$$

2014 Brazil Cono Sur TST 1.3

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 $BC$ .

2014 Brazil Cono Sur TST 2.1

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.

2014 Brazil Cono Sur TST 3.2

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.

2015 Brazil Cono Sur TST 1.1

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.

2015 Brazil Cono Sur TST 2.3

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 $PK$.

2015 Brazil Cono Sur TST 3.4

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.

2016 Brazil Cono Sur TST 1.3

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.

2016 Brazil Cono Sur TST 2.1

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 $\angle ADC$.

2016 Brazil Cono Sur TST 3.2

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 $\Gamma$.

2017 Brazil Cono Sur TST 1.3

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 $\angle XNM = \angle BAN$.

2017 Brazil Cono Sur TST 2.1

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 $ABC$.

2017 Brazil Cono Sur TST 3.4

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

2018 Brazil Cono Sur TST 1.3

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.

2018 Brazil Cono Sur TST 2.2

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 $\angle IMA = \angle INB$.

2018 Brazil Cono Sur TST 3.1

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 $\frac{S_1}{S_2}<3 + 2\sqrt2$

2019 Brazil Cono Sur TST 1.3

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 $AEF$.

2019 Brazil Cono Sur TST 2.1

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 $\angle ADC= 90^o$.

2019 Brazil Cono Sur TST 3.3

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 $\angle XBY + \angle PBQ = 180^o$.

2020 Brazil Cono Sur TST 1.3

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.

2020 Brazil Cono Sur TST 2.4

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.

2020 Brazil Cono Sur TST 3.1

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 $CEP$.

<=2009  Brazil Cono Sur TST 

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 $JK = HI$.