Brazil Training Lists OMCS 1999 - 2020 219p

 geometry problems from Brazilian Training Lists for Cono Sur,  IberoAmerican and IMO with aops links in the names

collected inside aops: here

Cono Sur Training Lists

1999 - 2020

no problems from 2019

1999 Brazil Cono Sur Training List 2.3

Let $ABCD$ be a convex quadrilateral, $M$ the midpoint of $BC$ and $N$ the midpoint of $AD$. Prove that

$$[ABMN] = [NMCD] \Leftrightarrow BC \parallel AD$$ where $[ \,]$ denotes area.

1999 Brazil Cono Sur Training List 2.6

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle DEA = 90^o$, $AB = CD = EA = 1$ and $BC + DE = 1$. Calculate the area of the pentagon.

1999 Brazil Cono Sur Training List 2.9

Given two circles each external to the other one, we draw a common inner and outer tangent. The resulting tangency points define a chord in each circle. Prove that the point of intersection of the lines of these two chords is collinear with the centers of the circles.

1999 Brazil Cono Sur Training List 3.2

Let $ABC$ be a triangle right in $A$, $I$ the incenter of $ABC$ and $O$ the midpoint of hypotenuse $BC$. Prove that $OI < IA$.

1999 Brazil Cono Sur Training List 3.3

$A,B,C$ are three points on the circumference of a circle of radius $r$, with A$B = BC$. $D$ is a point interior of the circle such that the triangle $BCD$ is equilateral. The line through $A$ and $D$ intersects the circle again at $E$. Prove that $DE = r$.

2000 Brazil Cono Sur Training List 2.1

Let $ABC$ be a triangle right in $A$ and $P$ a point within $ABC$, such that $AB = PB$. If $H$ is the foot of the altitude on $BC$ and $M$ is the midpoint of $BC$, prove that $PM$ is bisector of $\angle HPC$ if and only if $\angle ABC = 60^o$

2000 Brazil Cono Sur Training List 2.2

The angle between the side $AC$ of triangle $ABC$ and the median relative to vertex $A$ is equal to $30^o$. This is also the measure of the angle between side $BC$ and the median relative to vertex $B$. Prove that triangle $ABC$ is equilateral.

2000 Brazil Cono Sur Training List 2.4

Show that it is not possible to cover a convex polygon with $n$ sides using only non convex quadrilaterals.

2000 Brazil Cono Sur Training List 2.5

Let $A,B,C$ be three distinct points on the plane such that $AB = AC$. Find the locus of the points $P$ such that $\angle APB = \angle APC$.

2001 Brazil Cono Sur Training List 2.2

The lengths of the sides of a right triangle are integers. The length of the hypotenuse is not divisible by $5$. Prove that the area of the triangle is a multiple of $10$.

2001 Brazil Cono Sur Training List 2.5

Convex $ABCD$ is inscribed in a semicircle of diameter $AB$. Let $S$ be the intersection point of $AC$ and $BD$, and $T$ a point on A$B$ such that $TS$ is perpendicular to $AB$. Prove that $ST$ is the bisector of the angle $\angle CTS$.

2001 Brazil Cono Sur Training List 2.6

Let $ABC$ be a triangle for which there is an interior point $O$ such that $\angle BAO =\angle CBO  =\angle ACO = 30^o$. Prove that the triangle $ABC$ is equilateral.

2001 Brazil Cono Sur Training List 2.8

Is it possible to cut a circle of paper into pieces and then rearrange the pieces to get a square of the same area?

Note: Only a finite number of cuts are allowed, and each cut is either a straight line segment or an arc of a circle.

2001 Brazil Cono Sur Training List 3.3

The bisector of angle $A$ of $ABC$ intersects side $BC$ at point $U$. Prove that the perpendicular bisector of segment $AU$, the perpendicular on $BC$ from $U$ and the diameter of the circumscribed circle through $A$ are concurrent.

2001 Brazil Cono Sur Training List 3.4

An quadrilateral $ABCD$ is special if it is inscribed and has perpendicular diagonals. Let $ABCD$ be a special quadrilateral , $T$ is the point intersection of the diagonals, $O$ is the center of the circumscribed circumference and $P$ is the midpoint of $BC$. Prove that:

(a) the line perpendicular on the side $AD$ from $T$ contains the point $P$.

(b) $OP =\frac12  AD$.

2001 Brazil Cono Sur Training List 4.1

In the parallelogram $ABCD$ , let $O$ be the point of intersection of the diagonals $AC$ and $BD$. The angles $\angle CAB$ and $\angle DBC$ measures twice the angle $\angle DBA$ , and the angle $\angle ACB$ is $r$ times the angle $\angle AOB$ . Determine $r$.

2001 Brazil Cono Sur Training List 4.2

Let $P$ be a point inside an angle that does not belong to its bisector. Two segments are considered through $Q$, $AB$ and $CD$, with $A$ and $C$ on one side of the angle, $B$ and $D$ on the other side of the angle, and such that $P$ is the midpoint of $AB$ and $CD$ is perpendicular to the angle bisector. Demonstrate that $AB > CD$.

2002 Brazil Cono Sur Training List 1.1

Show how to construct with ruler and compass the triangle $ABC$ knowing the length$a$ of the side $BC$, the measure of the angle $\angle BAC$ and the sum $d$ of the lengths of the other two sides of $ABC$.

2002 Brazil Cono Sur Training List 1.4

The point $D$ on the side $BC$ of the triangle with acute angle $ABC$ is such that $AB = AD$. Let $E$ be the point on the altitude of $ABC$ drawn from $C$ such that the circle $a_1$, with center $E$ tangent to the line $AD$ at point $D$. Let $a_2$ be a circle passing through $C$ and tangent to line $AB$ at $B$. Prove that $A$ is on the line joining the intersection points of $a_1$ and $a_2$.

2002 Brazil Cono Sur Training List 1.10

For a point $P$ inside a triangle of area $S$ we draw three lines, none of them passing through any of the vertices of the triangle, in such a way that each side of the triangle exactly intersects two lines. In this way the triangle is partitioned into six smaller polygons, three of which are triangles. Let $S_1, S_2, S_3$ be the areas of such smaller triangles, prove that $$\frac{1}{S_1}+\frac{1}{S_2}+\frac{1}{S_3} \ge \frac{18}{S}$$

2002 Brazil Cono Sur Training List 2.9

The point $O$ is the center of the circle inscribed in a isosceles trapezoid $ABCD$, of bases $AB > CD$. Let $M$ be the midpoint of $AB$, $E$ be the point where the inscribed circle touches the side $CD$ and $F$ the intersection point of the line $OM$ with the base $CD$. Prove that $DE = CF \Leftrightarrow AB = 2CD$.

2002 Brazil Cono Sur Training List 2.10

Determine the locus of the orthocenters of the inscribed triangles in a given circle.

2002 Brazil Cono Sur Training List 3.8

In a triangle $ABC$ let $O$ be the circumcenter and $P, H$ respectively the feet of the angle bisector and altitude relative to BC. Construct $\vartriangle ABC$ with ruler and compass, knowning the positions of $O,P$ and $H$.

2002 Brazil Cono Sur Training List 3.9

From the vertices of an $ABCD$ quadrilateral, we draw perpendicular to the diagonals $AC$ and $BD$. Prove that the feet of these perpendiculars are the vertices of a quadrilateral similar to the original quadrilateral.

Note: Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are similar with corresponding vertices $A \leftrightarrow A_1$, $B \leftrightarrow B_1$,$C \leftrightarrow C_1$ and $D \leftrightarrow D_1$ if the following two conditions are met:

$$\bullet \frac{AB}{A_1B_1}=\frac{BC}{B_1C_1}=\frac{CD}{C_1D_1}=\frac{DA}{D_1A_1}$$ $$\bullet  \angle A = \angle A_1, \angle B = \angle B_1, \angle C = \angle C_1, \angle D = \angle D_1$$

2002 Brazil Cono Sur Training List 3.10

$AC'BA'CB'$ is a convex hexagon such that $AB'= AC$',$BC'= BA'$, $CA' = CB'$ and $\angle A + \angle B +\angle C = \angle A' + \angle  B' + \angle  C'$. Prove that the area of the triangle $ABC$ is half the area of the hexagon.

PS. Hexagon had a typo as $AC'BACB'$ . I hope it is correct now.

2004 Brazil Cono Sur Training List 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.

2004 Brazil Cono Sur Training List 1.8

Let $ABCDEF$ be a convex hexagon and $A_1$,$B_1$, $C_1$, $D_1$, $E_1$, $F_1$ the midpoints of the sides are $AB$, $BC$, $CD$, $DE$, $EF$, $FA$, respectively. Find the area of the hexagon depending on the areas of the triangles $ABC_1$, $BCD_1$, $CDE_1$, $DEF_1$, $EFA_1$ and $FAB_1$.

2004 Brazil Cono Sur Training List 1.11

A semicircle with center $O$ and diameter $AB$ is given. A straight line intersects $AB$ at $M$ and the semicircle at $C$ and $D$, such that $MB < MA$ and $MD < MC$. The circumcircles of the triangles $AOC$ and $DOB$ intersect again at $K$. Show that the lines $MK$ and $KO$ are perpendicular.

2004 Brazil Cono Sur Training List 2.4

Let $D$ be the midpoint of the base $AB$ of the acute isosceles triangle $ABC$. Pick a point $E$ on $AB$ and let $O$ be the circumcenter of the triangle $ACE$. Prove that the line that passes through $D$ and is perpendicular on $DO$, the line that passes through $E$ and is perpendicular on $BC$ and the line that passes through $B$ and is parallel to $AC$ are concurrent.

2004 Brazil Cono Sur Training List 2.6

In the convex quadrilateral $ABCD$, $AB = BD$ and the angles $\angle BAC$ and $\angle ADC$ are $30^o$ and $150^o$, respectively. Prove that the angles $\angle BCA$ and $\angle ACD$ are equal.

2004 Brazil Cono Sur Training List 2.9

Let $O$ be the circumcenter of an isosceles triangle $ABC$ with $AB = BC$. $M$ is any point of $BO$, $M'$ is symmetric of $M$ wrt the midpoint of $AB$, $K$ is the intersection of $M'O$ with $AB$ and $L$ is the point on the side $BC$ such that $\angle CLO = \angle BLM$ . Show that points $O, K, B$ and $L$ lie on the same circle.

2004 Brazil Cono Sur Training List 3.2

Let $ABC$ be a triangle with $AC < BC$. Let $F$ be the foot of the altitude relative to the side $AB$, $H$ be the orthocenter of $ABC$ and $O$ be the circumcenter of $ABC$. Let also $P$ be the intersection of $AC$ with the perpendicular on $OF$ passing through $F$. Prove that $\angle PHF=\angle BAC$ .

2004 Brazil Cono Sur Training List 3.8

Given a circle $\Gamma$, a line $d$ does not intersect. $M$ and $N$ are variable points on the line $d$ such that the circle with diameter $MN$ is externally tangent to $\Gamma$. Prove that there is a point $P$ on the plane such that, to any such segment $MN$, the angle $\angle MPN$ is fixed.

2004 Brazil Cono Sur Training List 3.10

Let $M$ and $N$ be points on the side $BC$ of a triangle $ABC$ such that $BM = CN$, with $M$ between $B$ and $N$. $P$ and $Q$ lie, respectively, on $AN$ and $AM$ such that $\angle PMC= \angle  MAB$ and $\angle QNB  = \angle NAC$ . Prove that $\angle QBC=\angle  PCB$.

2004 Brazil Cono Sur Training List 3.13

Let $ABCD$ be a convex quadrilateral such that $\angle ABC =\angle ADC = 135^o$ and $AC^2 \cdot BD^2 = 2 \cdot AB \cdot BC \cdot CD \cdot DA$. Prove that the diagonals of the quadrilateral are perpendicular .

2005 Brazil Cono Sur Training List 1.2

Let $O$ be an interior point of the triangle cut-angle $ABC$. the circles with centers at the midpoints of the sides and passing through $O$ intersect again at points $K, L$, and $M$. Show that $O$ is the incenter of $KLM$ if and only if $O$ is the circumcenter of $ABC$.

2005 Brazil Cono Sur Training List 1.7

Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $A'$ be the symmetric of $A$ wrt$BC$, $D$ the point on $AC$ such that $AB = AD$ and $H$ the orthocenter of $ABC$. If l is the external bisector of $\angle BAC$, $M = A'D \cap l$ and $N = CH \cap l$, prove that that $AM = AN$

2005 Brazil Cono Sur Training List 1.10

An acute triangle $ABC$ has circumradius $R$, inradius r and its largest Angle is $\angle BAC$. Let $M$ be the midpoint of $BC$ and $X$ the intersection of two lines tangent to the circumcircle of $ABC$ passing through $B$ and $C$. Prove that $\frac{r}{R}\ge \frac{AM}{AX}.$

2005 Brazil Cono Sur Training List 2.8

Consider the circle $\Gamma$ with center $O$ and radius $R$. Let A be a point outside the $\Gamma$ . Draw a line $\ell$ by $A$, other than $AO$, cutting $\Gamma$ at $B$ and $F$, with $B$ between $A$ and$F$. Let $\ell'$ be the symmetric line of $\ell$ with respect to $AO$. This line cuts $\Gamma$ at $C$ and $D$, with $C$ between $A$ and $D$. Show that the intersection of lines$BD$ and $CF$ is independent of the choice of line $\ell$.

2005 Brazil Cono Sur Training List 2.9

Let $\Gamma$ be a circle and $\ell$ a line that does not intersect it. be AB the diameter of $\Gamma$ perpendicular on $\ell$ , with $B$ closest to $\ell$ . A point $C$ other than $A$ ,$B$ is chosen over $\Gamma$. line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ in $E$, with $B$ and $E$ on the same side of $AC$. Let $F$ be the intersection of $BE$ with $\ell$ and $G$ the second intersection of $AF$ with $\Gamma$. Let $H$ also be the second intersection of $CF$ and $\Gamma$ . Show that $AB$ is perpendicular to $GH$.

2005 Brazil Cono Sur Training List 3.4

Let $\omega$ be the incircle of the triangle $ABC$. Let $M$ be the midpoint of $BC$ and $D$ the intersection of $\omega$ with $BC$. Let $\omega_1$ be the circle of center M passing through $D.$ The circles $\omega_2$ and $\omega_3$ are constructed analogously. show that if $\omega_1$ is tangent to the circle $\Gamma$ of $ABC$, then $\omega_2$ or $\omega_3$ is also tangent to $\Gamma$ .

2005 Brazil Cono Sur Training List 3.7

Let $H$ be the orthocenter of a triangle with an acute angle $ABC$ and $B', C'$ are projections of $B$,$C$ on $AC$ and $AB$, respectively. a straight line passing through $H$ intersects the segments $BC'$ and $CB'$ in $M$ and $N$, respectively. Lines perpendicular on $\ell$ passing through $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$, respectively. Find the locus of the midpoints of $PQ$ when line $\ell$ varies.

2006 Brazil Cono Sur Training List 1.1

All diagonals of a regular $25$-sided polygon are drawn. Prove that there are no $9$ diagonals passing through a point inside the polygon.

2006 Brazil Cono Sur Training List 1.2

Let $M$ be the intersection point of the diagonals of an cyclic quadrilateral $ABCD$, where $\angle AMB$ is acute. The isosceles triangle $BCK$ is constructed outside the quadrilateral, with the base being $BC$, such that $\angle KBC + \angle AMB = 90^o$. Prove that $KM$ is perpendicular to $AD$.

2006 Brazil Cono Sur Training List 1.7

Let $ABC$ be a triangle with circumradius $R = 1$. Let $r$ be the inradius of the triangle $ABC$ and $p$ be the inradius of the orthic triangle $A_1B_1C_1$. Show that $p \le 1- \frac{1}{3}(1+r)^2$,

2006 Brazil Cono Sur Training List 2.2

Let $A_1$, $B_1$ and $C_1$ be the intersections of the internal bisectors of a triangle $ABC$ with sides $BC$, $CA$ and $AB$, respectively, and let $A_2$, $B_2$ and $C_2$ be the midpoints of segments $B_1C_1$, $C_1A_1$ and $A_1B_1$, respectively. Prove that $AA_2$, $BB_2$ and $CC_2$ are concurrent .

2006 Brazil Cono Sur Training List 2.4

Let $ABC$ and $A'BC$ be two equilateral triangles ($A\ne A'$) and $D$ a variable point on the side $AC$ . Let $A'D$ intersects side $AB$ at $E$ and $BD$ intersects $CE$ at $P$. Show that $P$ lies on the circle circumscribed around the triangle $ABC$.

2006 Brazil Cono Sur Training List 2.5

Segment $AB$ intersects two circles of the same radius, and is parallel to the line is connects the centers of these circles and all the intersection points of that segment with the circles is between $A$ and $B$. From $A$, we draw tangents to the circle closest to $A$, and from $B$, we draw tangents to the circle closest to $B$. It is verified that the quadrilateral formed by the four tangents contains both the circles. Prove that this quadrilateral is tangential.

2006 Brazil Cono Sur Training List 2.8

Prove that if all the vertices of a polygon in the Euclidean plane are distant in the maximum $1$ of each other, then the area of this polygon is less than $\frac{\sqrt3}{2}.$

2006 Brazil Cono Sur Training List 3.5

Let $ABCD$ be a convex quadrilateral. Consider the points $E =\overline{AD} \cap \overline{BC}$ and $I = \overline{AC} \cap  \overline{BD}$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB\parallel CD$ and $IC^2 = IA \cdot AC$.

2006 Brazil Cono Sur Training List 3.7

Let $ABCD$ be a convex quadrilateral. Consider the points $P = \overline{AB} \cap  \overline{CD}$ and $Q =  \overline{AD} \cap  \overline{BC}$. Let $O$ be an interior point of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that $\angle AOB +\angle COD = 180^o$.

2006 Brazil Cono Sur Training List 4.4

A point $P$ is inside a triangle $ABC$. The side $AC$ meets the line $BP$ at $Q$ and $AB$ meets $CP$ at $R$. Suppose that $AR = RB = CP$ and $CQ = PQ$. Find $\angle BRC$.

2006 Brazil Cono Sur Training List 4.5

An acute triangle has all its sides of different lengths . Show that the straight line passing through its circumcenter and its incenter intersects the longest and shortest sides of the triangle.

2007 Brazil Cono Sur Training List 1.3

The altitudes $AD$ and $BE$ of triangle $ABC$ meet at orthocenter$H$. The midpoints of $AB$ and $CH$ are $X$ and $Y$ , respectively. Prove that $XY$ is perpendicular to $DE$

2007 Brazil Cono Sur Training List 1.6

Consider the triangle $ABC$ whose lengths of the sides are $a, b, c$, where $a$ is the longest side. Prove that $ABC$ is right if, and only if, $$(\sqrt{a+b}+\sqrt{a-b})(\sqrt{a+c}+\sqrt{a-c})=(a+b+c)\sqrt2$$

2007 Brazil Cono Sur Training List 1.9

A circle with center $I$ is inside another circle. $AB$ is a variable chord of the large circle that is tangent to the small circle. Determine the locus of the circumcenter of the triangle $ABI$.

2007 Brazil Cono Sur Training List 1.11

Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$ and such that $AC \ne BC$. The lines $CH$ and $CI$ cut the circumcircle of $ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^o$ if, and only if, $\angle IDL = 90^o$.

2007 Brazil Cono Sur Training List 2.1

Within a unit square, three points are chosen. Show that the distance between two of them is not greater than $\sqrt6-\sqrt2$.

2007 Brazil Cono Sur Training List 2.2

Let $ABCD$ be a parallelogram. the bisector of $\angle BAD$ cuts $BC$ at $M$ and cuts the extension of $CD$ at $N$. The circumcenter of the triangle $MCN$ is $O$. Prove that $B, O, C, D$ they are concentric.

2007 Brazil Cono Sur Training List 2.7

Let $D$ be a point on the side $BC$ of the acute triangle $ABC$ ($D \ne B$ and $D \ne C$), $O_1$ the circumcenter of the triangle $ABD$, $O_2$ is the circumcenter of triangle $ACD$ and $O$ is the circumcenter of triangle $AO_1O_2$. Determine the locus of point $O$.

2007 Brazil Cono Sur Training List 2.11

Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$ and $\angle PAC = \angle PCB$. the point $Q$ is on the straight line $BC$ satisfying $QA = QP$. Prove that $\angle AQP =2\angle OQB$.

2007 Brazil Cono Sur Training List 3.4

A game is played by two players on an infinite plane containing $51$ pieces, $50$ sheep and $1$ wolf. The first player starts the game by moving the wolf. Then the second player moves some sheep, and so on. On each move, a piece can move up to a distance of $1$ meter. Is it true that, regardless of the initial position of the pieces, the wolf always manages to capture at least one sheep?

2007 Brazil Cono Sur Training List 3.5

Let $ABC$ be a triangle with circumcircle $\omega$ such that $\angle BAC = 60^o$. Given a point $B$ on the arc $BC$ of $\omega$, let $O_1$, $O_2$ the circumcenters of the triangles $ABD$ and $ACD$, respectively, $M$ the intersection of $BO_1$ and $CO_2$ and $N$ the circumcircle of $DO_1O_2$. Show that, when varying $D$, $MN$ passes through a fixed point .

2007 Brazil Cono Sur Training List 3.7

Let $\omega$ be the incircle of a triangle ABC. The median $AM$ of $ABC$ intersects $\omega$ at $K$ and $L$. The lines parallel to $BC$ passing through $K$ and $L$ intersect $\omega$ again at $X$ and $Y$. Let $P, Q$ be the intersections of $BC$ with the lines $AX$ and $AY$ . Show that $BP = CQ$.

2007 Brazil Cono Sur Training List 3.8

Let $ABC$ be a triangle with circumcircle $\omega_1$, $O$ the circumcenter of $ABC$ and $\omega_2$ the $A$-excircle . If $M, N, L$ are the touchpoints of $\omega_2$ with the lines $BC, AC, AB$ repsectively and the radii of $\omega_1$ and $\omega_2$ are equal, show that $O$ is the orthocenter of the triangle $MNL$ .

2007 Brazil Cono Sur Training List 4.2

Let $ABCDE$ be a pentagon with $AE = ED$, $AB + CD = BC$ and $\angle BAE + \angle CDE = 180^o$. prove that $\angle AED = 2\angle BEC$.

2007 Brazil Cono Sur Training List 4.7

Let $ABC$ be a triangle and $D$ the point on the extension of side $BC$ beyond $B$ such that $BD = BA$, and let $M$ be the midpoint of $AC$. The angle bisector of $\angle ABC$ cuts $DM$ to $P$ . Prove that $\angle BAP = \angle ACB$.

2008 Brazil Cono Sur Training List 1.3

In the forest where the Smurfs live, Gargamel planted $1280$ pine trees, each $1$ meter in diameter. The forest is a rectangular field of dimensions $1001\times  945$ meters. Grandpa Smurf would like to build seven tennis courts , each measuring $20\times  34$ meters. Grandpa Smurf will be able to do the construction without knocking down no trees, no matter how Gargamel has planted the trees?

2008 Brazil Cono Sur Training List 1.5

Consider five segments in the plane so that any three of them form a triangle. Prove that at least one of these triangles is acute.

2008 Brazil Cono Sur Training List 1.10

Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$ such that $\angle BAC = 60^o$ and $AB > AC$. Let $BE, CF$ be the the altitudes of $ABC$, and $M,N$ points on the segments $BH, HF$ respectively, such $BM = CN$. Determine the value of the expression $\frac{HM + NH}{OH}.$

2008 Brazil Cono Sur Training List 1.11

Let $ABCD$ be a convex quadrilateral, with $AB$ not parallel to $CD$, and let $X$ be a point within $ABCD$ such that $\angle ADX = \angle  BCX < 90^o$ and $\angle DAX = \angle CBX < 90^o$. If the perpendicular bisectors of segments $AB$ and $CD$ intersect in $Y$ , prove that $\angle AYB = 2\angle ADX$

2008 Brazil Cono Sur Training List 2.1

Consider a point $P$ on the outside of a circle $\omega$. The two tangent lines to $\omega$ starting from $P$ intersect $\omega$ at points $A$ and $B$. Let $M$ be the midpoint of $AP$ and $N$ the intersection of $BM$ with $\omega$, prove that $PN = 2MN$.

2008 Brazil Cono Sur Training List 2.5

Let $ABC$ be an acute triangle such that $AB > AC$ and $\angle BAC = 60^o$. Let $O, H$ be the circumcenter, orthocenter of $ABC$, respectively. Knowing that $OH$ intersects $AB$ at $P$ and $AC$ at $Q$, prove that $PO = HQ$.

2008 Brazil Cono Sur Training List 2.9

On the rhombus $ABCD$ with $\angle A = 60^o$, we take points $F,H$ and $G$ on sides $AD$, $DC$ and diagonal $AC$ so that $DFGH$ is a parallelogram. Prove that the triangle $FBH$ is equilateral .

2008 Brazil Cono Sur Training List 3.2

Given a triangle $ABC$ such that $\angle BCA = 60^o$ with circumcenter $O$ and incenter $I$. Let $A_1$ and $B_1$ be the feet of the internal bisectors of angles $\angle CAB$ and $\angle ABC$, respectively. The line $A_1B_1$ intersects the circumcircle of $ABC$ at $A_2$ and $B_2$.

(a) Prove that $OI$ is parallel to $A_1B_1$.

(b) If $R$ is the midpoint of the arc $AB$ not containing $C$ and $P , Q$ are the midpoints of $A_1B_1$, $A_2B_2$, respectively, prove that $RP = RQ$.

2008 Brazil Cono Sur Training List 3.5

Given a circle $\omega$ of center $O$ and a point $P$ outside the $\omega$. Let $\ell_1$, $\ell_2$ be lines through $P$ such that $\ell_1$ touches $\omega$ in $A_1$ and $\ell_2$ interserts $\omega$ in $B$ and $C$. If the tangent lines of $\omega$ passing through $B$ and $C$ intersect at $X$, prove that the segments $AX$ and $PO$ are perpendicular.

2008 Brazil Cono Sur Training List 3.11

Let $ABC$ be an acute triangle with alttiudes $AD$ $BE$ and $CF$ . Let $M$ be the midpoint of the segment $BC$. The circle circumscribed around the triangle $AEF$ cuts the line $AM$ at $A$ and $X$. Line $AM$ cuts line $CF$ at $Y$ . be $Z$ the intersection point of lines $AD$ and $BX$. Show that the lines $YZ$ and $BC$ are parallel.

2008 Brazil Cono Sur Training List 4.6

In a triangle $ABC$, we take points $X, Y$ on sides $AB$, $BC$, respectively. If $AY$ and $CX$ intersect at $Z$ and $AY = YC$ and $AB = ZC$, prove that points $B, X, Z, Y$ are concyclic,

2008 Brazil Cono Sur Training List 4.7

The incircle of the acute triangle $ABC$ is tangent to the sides $AB, BC, CA$ at points $P,Q,R$, respectively. Suppose the orthocenter $H$ of $ABC$ belongs to the segment $QR$ .

(a) Prove that $PH \perp QR$.

(b) Let $I, O$ be the incenter, circumcenter of $ABC$, respectively, and $N$ is the touch point of $AB$ with the $C$-excircle , prove that $I, O, N$ are collinear.

2008 Brazil Cono Sur Training List 4.8

Construct a triangle $ABC$, knowing the positions of:

$\bullet$ $O$, the circumcenter of $ABC$

$\bullet$ $D$, the feet of altitude from vertex $A$

$\bullet$ $P$, the feet of the internal bisector of angle $\angle BAC$

2009 Brazil Cono Sur Training List 1.1

Let $K$ and $L$ be points on sides $AB$ and $AC$, respectively, of the triangle $ABC$ such that $BK = CL$. Let P be the intersection point of the segments $BL$ and $CK$, $M$ a point inside the segment $AC$ such that $MP$ is parallel to the bisector of the angle $\angle BAC$. Prove that $CM = AB$.

2009 Brazil Cono Sur Training List 1.7

Let $ABC$ be a triangle, and let $D, E$ points on the side $BC$ such that $\angle BAD = \angle  CAE$. If $M$ and $N$ are respectively , the touchpoints with $BC$ of the incircles of the triangles $ABD$ and $ACE$, prove that $$\frac{1}{BM}+ \frac{1}{MD}= \frac{1}{NC} +\frac{1}{NE}.$$

2009 Brazil Cono Sur Training List 1.12

Let $ABC$ be an acute triangle with circumcenter $O$. $T$ is the circumcenter of the triangle $AOC$ and $M$ is the midpoint of the side $AC$ . On sides $AB$ and $BC$, points $D$ and $E$ are taken, respectively, such that $\angle BDM = \angle BEM=\angle ABC$. Show that the lines $BT$ and $DE$ are perpendicular.

2009 Brazil Cono Sur Training List 2.1

Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $ABC$, respectively, so that $\angle AXY =2\angle ACB$ and $\angle CYX = 2\angle BAC$. Prove that $$\frac{S(AXYC)}{S(ABC)} \le \frac{AX^2 + XY^2 + YC^2}{AC^2}$$ where $S$ denotes the area of the corresponding figure.

2009 Brazil Cono Sur Training List 2.2

All sides of a convex pentagon have the same length and each of the angles is less than $120^o$. Prove that all angles are not obtuse.

2009 Brazil Cono Sur Training List 2.3

Is it possible for us to cover a cube with three paper triangles (without overlapping)?

2009 Brazil Cono Sur Training List 2.4

A right triangle $ABC$, with hypotenuse $AB$, is inscribed in a circle $\Gamma$. Let $K$ be the midpoint of the arc $BC$ that does not contain $A$, $N$ be the midpoint of the side $AC$ and $M$ be the point of intersection of $KN$ with the circle $\Gamma$. Let $E$ be the intersection point of the tangents to the circle at A and C. Prove that $\angle EMK = 90^o.$

2009 Brazil Cono Sur Training List 2.9

Show that we cannot draw two triangles of area $1$ within a circle of radius $1$ so that the triangles don't have a common point.

2009 Brazil Cono Sur Training List 3.3

An integer coordinate point is a point $(x, y)$ with both coordinates integer . Find the smallest $n$ such that every set with $n$ integer coordinate points has three points that are vertices of a triangle with integer area . (The triangle can be degenerate, in other words, the three points can be on the same line forming a triangle of area zero).

2009 Brazil Cono Sur Training List 3.5

The circles $G_1$ and $G_2$ intersect at two points, $A$ and $B$. A line through $B$ intersects $G_1$ at $C$ and $G_2$ at $D$. A tangent line to $G_1$ that passes through $C$ cuts the tangent line from $G_2$ that passes through $D$ in $E$. The symmetrical line of $AE$, wrt line $AC$, cuts $G_1$ at $F$ (besides $A$). . Prove that $BF$ is tangent to $G_2$.

Note: Consider only the case where $C$ does not belong to the interior of $G_2$ and $D$ does not belong to the interior of $G_1$.

2009 Brazil Cono Sur Training List 3.8

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ intersect at $A$ and $B$. Let $S_3$ be a circle passing through $O_1$, $O_2$ and $A$. The circle $S_3$ intersects $S_1$ at $D \ne A$, $S_2$ at $E$ and the line $AB$ in $C$. Prove that $CD = CB = CE$.

2009 Brazil Cono Sur Training List 3.12

A disc of radius $1$ is covered by $7$ identical discs of radius $r$. Prove that $r \ge \frac12$.

2010 Brazil Cono Sur Training List 1.3

Let $ABCD$ be a cyclic quadrilateral. Prove that $$|AC - BD| \le |AB  -  CD|.$$ When does equality occur?

2010 Brazil Cono Sur Training List 1.5

Let $ABC$ be a non-equilateral triangle whose incircle touches $BC$, $CA$ and $AB$ at $A_1$,$B_1$ and $C_1$, respectively. Let $H_1$ be the orthocenter of the triangle $A_1B_1C_1$ . Prove that $H_1$ lies on the straight line passing through the incenter and circumcenter of the triangle $ABC$.

2010 Brazil Cono Sur Training List 1.9

Points $A_1$,$B_1$ and $C_1$ lie on sides $BC$; $CA$ and $AB$ of the isosceles triangle $ABC$ ($AB = BC$). Knowing that $\angle BC_1A_1 = \angle CA_1B_1 = \angle BAC$ and $BB_1 \cap CC_1 = P$, prove that $AB_1PC_1$ is cyclic.

2010 Brazil Cono Sur Training List 1.12

The quadrilateral $ABCD$ is inscribed in a circle $\omega$ with center $O$. The bisector of $\angle ABD$ meets $AD$ and $\omega$ at points $K$ and $M$, respectively. The bisector of $\angle CBD$ meets $CD$ and $\omega$ at points $L$ and $N$, respectively. Suppose $KL\parallel MN$. Prove that the circumcircle of the triangle $MON$ passes through midpoint of $BD$.

2010 Brazil Cono Sur Training List 2.2

Show that for any convex polygon of area $1$ there is a parallelogram of area $2$ that contains it.

2010 Brazil Cono Sur Training List 2.3

Let $ABC$ ($AC \ne  BC$) be a triangle with the angle $\angle ACB$ acute and $M$ the midpoint of $AB$. Consider the point $P$ on the segment $CM$ so that the bisectors of the angles $\angle PAC$ and $\angle PBC$ intersect at the point $Q$ of $CM$. Find the angles $\angle APB$ and $\angle AQB$.

2010 Brazil Cono Sur Training List 2.4

The circle inscribed with the triangle $ABC$ is tangent to sides $AB$, $BC$ and $CA$ at points $F, D$ and $E$, respectively. The segment $AD$ is bisected at $X$ by the inscribed circle, i.e., $AX = XD$. If $XB$ and $XC$ cut the circle at $Y$ and $Z$, respectively, prove that $EY = FZ$.

2010 Brazil Cono Sur Training List 2.9

Let $ABC$ be an acute triangle and $D$ a point on the side $AB$. The circumcircle of the triangle $BCD$ intersects the side $AC$ at $E$. The circumcircle of the triangle $ADC$ intersects the side $BC$ at $F$. Let $O$ be the circumcenter of the triangle of the triangle $CEF$. Prove that the points $D$ and $O$ and the circumcenters of the triangles $ADE$, $ADC$, $DBF$ and $DBC$ are concyclic and the line $OD$ is perpendicular to $AB$

2010 Brazil Cono Sur Training List 2.10

The convex set $F$ does not cover a semicircle of radius $R$. It is possible that two sets congruent to $F$ cover a circle of radius $R$? What if $F$ is not convex? ´

2010 Brazil Cono Sur Training List 3.5

Let $ABCD$ be a cyclic quadrilateral and let $U$ be the intersection point of the sides $AB$ and $CD,$ and $V$ the intersection point of the sides $BC$ and $DA$ . The straight line starting from $V$ that is perpendicular on the bisector of the $\angle AUD,$ cuts the segments $UA$ and $UD$ at $X$ and $Y$ , respectively. Prove that $AX \cdot DY = BX \cdot CY$.

2010 Brazil Cono Sur Training List 3.6

Consider a triangle $ABC$. Let $D$ be the foot of the altitude from the vertex $A$ and let $E ,F$ be points on the sides $AB$, $AC$ respectively such that $\angle ADE = \angle ADF$. Prove that the lines $AD$, $BF$, and $CE$ are concurrent.

2010 Brazil Cono Sur Training List 3.7

Let $BE$ and $CF$ be the altitudes of the triangle $ABC$. Prove that $AB = AC$ if and only if $AB + BE = AC + CF$.

2010 Brazil Cono Sur Training List 3.8

Let $ABC$ be a scalene triangle. Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ respectively, such that $$\frac{BA_1}{BC} =\frac{CB_1}{CA} =\frac{AC_1}{AB}.$$ Prove that if the triangles $AB_1C_1$, $BC_1A_1$, $CA_1B_1$ have the same circumradius, then $A_1, B_1, C_1$ are the midpoints of triangle $ABC$

2010 Brazil Cono Sur Training List 4.1

In an acute triangle $ABC$, $CF$ is alttiude and $BM$ is median. If $BM = CF$ and $\angle MBC = \angle FCA$, prove that $ABC$ is equilateral.

2010 Brazil Cono Sur Training List 4.5

Consider two circles $\omega_1$ and $\omega_2$ that intersect at two points $A$ and $B$. Let $\ell$be a straight line passing through $B$ that intersects $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively. The tangent to $\omega_1$ passing through $C$ intersects the tangent to $\omega_2$ passing through $D$ at $E$. If the symmetric line of $AE$ wrt $AC$ intersects $\omega_1$ at $F$, $F \ne A$, show that $BF$ is tangent to $\omega_2$.

2010 Brazil Cono Sur Training List 4.10

Given a triangle $ABC$, let $D$ and $E$ be interior points of sides $AB$ and $AC$, respectively, such that $B, D, E, C$ are concyclic. Let $F$ be the intersection of the lines $BE$ and $CD$. Circles circumscribed to the triangles $ADF$ and $BCD$ intersects at $G$ and $D$. Show that the line $GE$ intersects the segment $AF$ at its midpoint.

2010 Brazil Cono Sur Training List 4.12

Squares $ABDE$ and $ACFG$ are constructed externally to an acute triangle $ABC$. If $H$is the orthocenter of $ABC$, show that the lines $AH$, $BF$ and $CD$ are concurrent.

2011 Brazil Cono Sur Training List 1.2

Let $\vartriangle ABC$ be isosceles and right with hypotenuse $AB = \sqrt2$. Determine the positions of the points $X,Y,Z$ on the sides $BC$, $CA$, $AB$, respectively, such that the triangle $\vartriangle XYZ$ is right and isosceles with minimum area.

2011 Brazil Cono Sur Training List 1.4

The quadrilateral $ABCD$ circumscribes a given circle $S$. Touchpoints of $S$ with sides $AB$, $BC$, $CD$ and $AD$ are points $E$, $F$, $G$ and $H$, respectively. The intersection of $AC$ and $BD$ is $I$. Prove that $$\frac{AI}{IC} = \frac{AH}{CD}.$$

2011 Brazil Cono Sur Training List 1.11

Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$ and $AC \ne BC$. The lines $CH$ and $CI$ meet again the circumcircle of the triangle $\vartriangle ABC$ at the points $D$ and $L$, respectively. Prove that $\angle CIH = 90^o$ if, and only if, $\cos \angle A + \cos  \angle B = 1$.

2011 Brazil Cono Sur Training List 2.1

In the triangle $\vartriangle ABC$, satisfying $AB + BC = 3AC$, the incircle has center $I$ and is tangent to sides $AB$ and $BC$ at $D$ and $E$, respectively. Let $K$ and $L$ be the symmetric points of $D$ and $E$ wrt $I$. Prove that the quadrilateral $ACKL$ is cyclic.

2011 Brazil Cono Sur Training List 2.9

Let $ABCD$ be a parallelogram. A variable line $\ell$ passing through $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$ , respectively. Let $K$ and $L$ be the centers of the excircles of the triangles $ABX$ and $ADY$ , which touch the sides $BX$ and $DY$ , respectively. Prove that the size of the angle $\angle KCL$ does not depend on the choice of the line $\ell$.

2011 Brazil Cono Sur Training List 2.12

The median $AM$ of the triangle $\vartriangle ABC$ intersects its incircle $\omega$ at $K$ and $L$. The lines through $K$ and $L$ parallel to $BC$ intersect $\omega$ again at $X$ and $Y$ . The lines $AX$ and $AY$ intersects $BC$ at $P$ and $Q$. Prove that $PB = CQ$.

2011 Brazil Cono Sur Training List 3.4

Let $ABC$ be a triangle such that $\angle A = 90^o$ and $\angle B <  \angle C$. Let $D$ be the intersection of the tangent by $A$ to the circumcircle of $ABC$ and the straight $BC$. Let $E$ be the symmetric of $A$ wrt $BC$ and $X$ be the foot of the perpendicular from $A$ on $BE$. Let $Y$ be the midpoint of $AX$ and $Z$ be the second intersection of $BY$ and the circumcircle of $ABC$. Prove that $BC$ is tangent to the circumcircle of $ADZ$.

2011 Brazil Cono Sur Training List 3.7

Let $I$ be the incenter of a triangle $ABC$ with $AB \ne AC$. The lines $BI$ and $CI$ intersect sides $AC$ and AB at points $D$ and $E$, respectively. Find $\angle BAC$, knowing that $DI = EI$.

2011 Brazil Cono Sur Training List 4.1

Let $P$ be a point inside the triangle $ABC$. the lines $BP$, $CP$ intersect $AC$, $AB$ at $Q$, $R$, respectively. Knowing that $AR = RB = CP$ and $CQ = PQ$, find the measure of the angle $\angle BRC$.

2011 Brazil Cono Sur Training List 4.2

(a) Five identical paper triangles are given over a table. Each of them can move in any direction parallel to itself (i.e. without rotation). Prove or disprove: each triangle can be covered by the other four.

(b) Suppose the triangles in (a) are equilateral. Show that each triangle can be covered by the other four.

2011 Brazil Cono Sur Training List 4.3

Let $ABC$ be a triangle and $P$ be a point on the plane such that the triangles $PAB$, $PBC$ and $PCA$ have the same area and the same perimeter. Prove the statements below:

(a) If $P$ lies inside $ABC$, then $ABC$ is equilateral. ´

(b) If $P$ lies outside of $ABC$, then $ABC$ is right.

2011 Brazil Cono Sur Training List 4.5

Let $ABC$ be a triangle, right at $C$. The internal bisectors $AA_1$ and $BB_1$ intersect at incenter $I$. If $O$ is the circumcenter of the triangle $A_1B_1C$, prove that $OI$ and $AB$ are perpendicular.

2012 Brazil Cono Sur Training List 1.4

Let $M$, $N$ and $P$ be the intersection points of the incircle of the $\vartriangle ABC$ with the sides $AB$, $AC$ and $BC$, respectively. Prove that the intersection points of the altitudes of the $\vartriangle MNP$, the circumcenter of the $\vartriangle ABC$, and the incenter of the $\vartriangle ABC$ are collinear.

2012 Brazil Cono Sur Training List 1.5

In the triangle $ABC$, let $J$ be the center of the excircle that is tangent to side $BC$ in $A_1$ and to the extensions of sides $AC$ and $AB$ in $B_1$ and $C_1$, respectively. Suppose lines $A_1B_1$ and AB are perpendicular and intersect at ̃ $D$. Let $E$ be the foot of the perpendicular of ́ $C_1$ n the line $DJ$. Determine the angles $\angle BEA_1$ and $\angle AEB_1$.

2012 Brazil Cono Sur Training List 2.4

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be points in $ABCD$ such that $PQDA$ and $QPBC$ are cyclic quadrilaterals. Suppose there is a point $E$ on the segment $PQ$ such that $\angle PAE = \angle QDE$ and $\angle PBE = \angle QCE$. Show that the quadrilateral $ABCD$ is cyclic.

2012 Brazil Cono Sur Training List 3.1

The triangles $\vartriangle ABC$ and $\vartriangle A'B'C'$ are the same with $AB = A'B'$ , $AC = A'C'$ and $BC$ = $B'C'$ but they have opposite orientations. Prove that the midpoints of ́ $AA'$ , $BB'$ and $CC'$ are collinear.

2012 Brazil Cono Sur Training List 3.4

Let $ABC$ be an acute triangle and $M$ be the midpoint of ́ $BC$. There is only one interior point $N$ such that $\angle ABN = \angle BAM$ and $\angle ACN = \angle CAM$. Prove that $\angle BAN = \angle CAM$.

2012 Brazil Cono Sur Training List 4.3

Let $ABCDE$ be a convex pentagon in the coordinate plane. Each of its vertices is an lattice point. The five diagonals of $ABCDE$ form a convex pentagon $A_1B_1C_1D_1E_1$ within ABCDE. Prove that this smaller pentagon contains a lattice point on its edge or interior.

2012 Brazil Cono Sur Training List 4.5

A circle is inscribed in the isosceles trapezoid $ABCD$. Let $K$ and $L$ be the intersection points of the circle with $AC$ (with $K$ between $A$ and $L$). Find the value of $$\frac{AL \cdot  KC}{AK \cdot  LC}.$$

2012 Brazil Cono Sur Training List 5.2

Let $a, b$, and $c$ be the lengths of the sides of a triangle and let $p$ the semiperimeter and $r$ the inradius of the triangle. Prove that

$$\frac{1}{(p-a)^2}+\frac{1}{(p - b)^2}+\frac{1}{(p-c)^2}\ge \frac{1}{r^2}$$

2012 Brazil Cono Sur Training List 5.4

The triangle $ABC$ has orthocenter $H$. The projections of $H$ on the internal and external bisectors of the angle $\angle BAC < 90^o$ are $P$ and $Q$, respectively. Prove that $PQ$ passes through the midpoint ́of $BC$.

2012 Brazil Cono Sur Training List 6.2

Let $ABCDE$ be a convex pentagon such that: ́ $\angle BAC = \angle CAD = \angle DAE$ and $\angle ABC =\angle ACD = \angle ADE$. Diagonals $BD$ and $CE$ intersect at $P$. Prove that $AP$ bisects the side $CD$.

2012 Brazil Cono Sur Training List 7.1

Let $ABC$ be a triangle inscribed in a circle of radius $R$ and let $P$ be a point inside it. Prove that $$\frac{PA}{BC^2} +\frac{PB}{CA^2}+\frac{PC}{AB^2} \ge \frac{1}{R}.$$

2012 Brazil Cono Sur Training List 7.2

Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. On sides $AB$ and $AC$, consider the points $M$ and $N$, respectively, both different from the midpoints on the sides, such that $AM^2 + AN^2 = BM^2 + CN^2$ and $\angle MDN = \angle BAC$. Prove that $\angle BAC = 90^o$.

2012 Brazil Cono Sur Training List 8.2

The acute triangle $ABC$ is given with $\angle BAC = 30^o$. The altitudes $BB_1$ and $CC_1$ are drawn. Let $B_2$ and $C_2$ be the midpoints of$AC$ and $AB$, respectively. Prove that the segments $B_1C_2$ and $B_2C_1$ are perpendicular.

2013 Brazil Cono Sur Training List 1.3

Given a scalene triangle $ABC$, be $A'$, $B'$, $C'$ the intersection points of the internal angle bisectors of $A$, $B$, $C$ with opposite sides, respectively. Let $A'''$ be the intersection of ̃ $BC$ and the perpendicular bisector of $AA'$. Define $B''$ and $C''$ analogously. Show that $A''$ , $B''$ and $C''$ are collinear.

2013 Brazil Cono Sur Training List 1.5

Let $O$ be the circumcenter of an acute $\vartriangle ABC$ and $A_1$ a point on the smallest arc $BC$ of the circumcircle of $\vartriangle ABC$ . Let $A_2$ and $A_3$ be points on sides $AB$ and $AC$ respectively, such that $\angle BA_1A_2 =  \angle OAC$ and $\angle CA_1A_3 =  \angle OAB$. Demonstrate that line $A_2A_3$ passes through the orthocenter of the $\vartriangle ABC$.

2013 Brazil Cono Sur Training List 2.4

Two circles intersect at $2$ points $A$ and $B$. A straight line that passes through point $A$ cuts the two circles at points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of arcs $BC$ and $BD$ (which do not contain point $A$) on their respective circles. Let $K$ be the midpoint of the segment ́ $CD$. Prove that $\angle MKN = 90^o$

2013 Brazil Cono Sur Training List 2.7

Let $\vartriangle ABC$ be an acute triangle with $AB \ne AC$, let $V$ be the intersection of the internal bisector of the angle $\angle A$ with $BC$ and let $D$ be the foot of the altitude drawn from $A$. If $E$ and $F$ are the intersections of ̃circumcircle of the triangle $\vartriangle ADV$ with the lines $CA$ and $AB$, respectively, prove that the lines $AD$, $BE$ and $CF$ are concurrent.

2013 Brazil Cono Sur Training List 3.6

Consider five points on the plane so that each triangle with vertices in these three points has an area less than or equal to $1$. Prove that the five points can be covered by a trapezoid of area at most $3$.

2013 Brazil Cono Sur Training List 4.1

Let $ABCD$ be a rhombus and $P$ a point on the side $BC$ . The circle passing through $A$, $B$ and $P$ intersects the line $BD$ once more at point $Q$ and the circle passing through $C$, $P$ and $Q$ intersects $BD$ once more at point $R$. Prove that $A$, $R$, and $P$ are collinear points.

2013 Brazil Cono Sur Training List 4.2

In the triangle $\vartriangle ABC$, $\angle ABC = \angle ACB = 40^o$ . Let $D$ be the point on $AC$ such that $BD$ and bisector of $\angle ABC$. Prove that $BD + DA = BC$.

2013 Brazil Cono Sur Training List 4.3

Let $R$ be the radius of the circle circumscribed to the triangle $\vartriangle  ABC$ of sides $a$, $b$, and $c$. If $R=\frac{a\sqrt{bc}}{b + c}$ , find the angles of the triangle.

2014 Brazil Cono Sur Training List 1.6

Let $k$ be the inscribed circle of a triangle ́$ABC$ not isosceles and let $I$ be the center of $k$. The circle $k$ touches sides $BC$, $CA$, and $AB$ at points $P, Q$, and $R$, respectively. Line $QR$ meets $BC$ at point $M$. Consider a circle $k'$ that contains the points $B$ and $C$. Let $N$ be the intersection of $k$ and $k'$. The circumcircle of the triangle $MNP$ intersects the line $AP$ at point $L$, different from $P$. Prove that points $I, L$ and $M$ are collinear.

2014 Brazil Cono Sur Training List 1.7

The circles $k$ and $k'$ with centers $O$ and $O'$ , respectively, are externally tangent at point $D$ and internally tangent to a circle $k''$ at points E and F, respectively. The line $t$ is common tangent of $k$ and $k'$ passing through $D$. Let $AB$ be the diameter of $k''$ perpendicular to $t$, such that $A, E$, and $O$ are on the same side wrt $t$. Prove that the lines $AO$, $BO'$, $EF$ and $t$ are concurrent.

2014 Brazil Cono Sur Training List 1.8

Consider the point $O$ inside the triangle $ABC$ such that $\angle AOB = \angle  BOC =\angle  COA = 120^o$.

Prove that $$\frac{AO^2}{BC} +\frac{BO^2}{CA} +\frac{CO^2}{AB} \ge \frac{AO + BO + CO}{\sqrt3}$$

2014 Brazil Cono Sur Training List 2.2

Given an acute triangle $ABC$, let $D$ be a point on side $BC$. Let $M_1$, $M_2$, $M_3$, $M_4$ and $M_5$ are the midpoints of the segments $AD$, $AB$, $AC$, $BD$ and $CD$, respectively. Let $O_1$, $O_2$, $O_3$ and $O_4$ be the circumcenters of the triangles $ABD$, $ACD$, $M_1M_2M_4$ and $M_1M_3M_5$, respectively. If $S$ and $T$ are the midpoints of segments $AO_1$ and $AO_2$, respectively, prove that $SO_3O_4T$ is an isosceles trapezoid.

2014 Brazil Cono Sur Training List 2.7

The circles $k_0, k_1, k_2, k_3$ and $k_4$ are in a plane such that for $i = 1, 2, 3, 4$ a circle $k_i$ is tangent externally to $k_0$ at point $T_i$ and $k_i$ is tangent externally to $k_{i+1}$ at point $S_i$ ($k_5 = k_1$). Let $O$ be the center of $k_0$. The lines $T_1T_3$ and $T_2T_4$ intersect at the point $T$ and lines $S_1S_3$ and $S_2S_4$ intersect at point S. Prove that points $O, T$ and $S$ are collinear.

2014 Brazil Cono Sur Training List 3.2

Calculate the minimum possible value of the perimeter of the triangle $ABC$, knowing that:

$\bullet$ $\angle A = 2 (  \angle B)$

$\bullet$ $\angle C > 90^o$

$\bullet$ sides $a, b, c$ of the triangle are integers.

2014 Brazil Cono Sur Training List 3.3

Given a triangle $ABC$ and a point $P$ inside it, let $D$ be the other intersection point of the line $BP$ with the circumscribed circle of the triangle $ABC$ and $Q$ the orthogonal projection of $P$ on the side $AC$ . Knowing that $\angle PAD = \angle APQ$ and $\angle PCD = \angle CPQ$ , prove that BQ is bisector of the angle $\angle ABC$.

2014 Brazil Cono Sur Training List 4.5

Let H be the orthocenter of an acute triangle $ABC$. Show that the triangles $ABH$, $BCH$ and $CAH$ has the same perimeter if and only if the triangle $ABC$ is equilateral.

2014 Brazil Cono Sur Training List 4.6

Let $Q$ be a point on the circle of diameter $AB$, where $Q$ is different from $A$ and $B$. Let $QH$ be the straight line perpendicular to $AB$ that passes through $Q$, where $H$ belongs to $AB$. the points of intersection of the circle of diameter $AB$ and the circle of center $Q$ and radius $QH$ are $C$ and $D$. Prove that $CD$ passes through the midpoint of $QH$.

2015 Brazil Cono Sur Training List 1.9

Let $O$ be the circumcenter of the acute triangle ABC. Let $\Gamma$ be the circle that passes through points $A, B$ and $O$. Lines $CA$ and $CB$ intersect $\Gamma$ again at points $D$ and $E$. Prove that $CO$ perpendicular to $DE$.

2015 Brazil Cono Sur Training List 1.10

Let $BC$ be the diameter of a semicircle and $A$ the midpoint of the arc. Let $M$ be a point on $AC$ and let $P$ and $Q$ be the feet of the perpendiculars from $A$ and $C$ on the line $BM$, respectively. Prove that $BP = PQ + QC$.

2015 Brazil Cono Sur Training List 1.11

The medians $AD, BE$ and $CF$ of the $ABC$ triangle intersect at the point $G$. Given that the quadrilaterals $AFGE$ and $BDGF$ are cyclic, prove that the triangle $ABC$ is equilateral.

2015 Brazil Cono Sur Training List 1.12

Given the acute triangle $ABC$, let $M$ and $N$ be points on sides $AB$ and $AC$, respectively. The circles of diameter $BN$ and $CM$ intersect at points $P$ and $Q$. Prove that $P, Q$ and the orthocenter $H$ are collinear.

2015 Brazil Cono Sur Training List 1.17

Consider an acute triangle $ABC$ with $AB \ne AC$. Let $D$ be the foot of the internal bisector starting from $A$ and let $E$ and $F$ be the feet of the altitudes drawn the vertices $B$ and $C$, respectively. The circumcircles of the triangles $DBF$ and $DCE$ intersect again at point $M$, with $M \ne D$. Prove that $ME = MF$.

2015 Brazil Cono Sur Training List 2.9

A rectangle $ABDE$ is drawn over side $AB$ of the acute triangle $ABC$ so that $C$ is over the side $DE$. Similarly, the rectangles $BCFG$ and $CAHI$ are defined, i.e. $A$ is on the side $FG$ and $B$ is on the side $HI$ . The midpoints of $AB$, $BC$ and $CA$ are $J, K$ and $L$, respectively. Prove that the sum of the angles $\angle GJH$, $\angle IKD$ and $ELF$ is $180^o$.

2015 Brazil Cono Sur Training List 2.10

Let $ABC$ be an isosceles triangle such that $CA = CB$, let $O$ its circumcenter and $I$ its incenter. Consider a point $D$ on side $BC$ such that $DO$ is perpendicular to $BI$. Prove that $DI$ is parallel to $AC$.

2015 Brazil Cono Sur Training List 2.11

Let $AH_a$ and $BH_b$ be the altitudes of the acute triangle $ABC$. The points$P$ and $Q$ are the projections of $H_a$ on the sides $AB$ and $AC$. Prove that the line $PQ$ cuts the segment $H_aH_b$ at its midpoint.

2015 Brazil Cono Sur Training List 2.12

Consider a scalene triangle $ABC$. The straight line through $A$ parallel to side $BC$ cuts the circumcircle of the triangle $ABC$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the lines perpendicular on $BC$, $CA$ and $AB$ from $A_1$, $B_1$ and $C_1$, respectively, are concurrent.

2015 Brazil Cono Sur Training List 2.17

The quadrilateral $ABCD$ is inscribed on a circle of diameter $AC$ and such that the straight line $AC$ is the perpendicular bisector of $BD$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Choose points $F$ on the ray $DA$ and $G$ on the ray $BA$ so that $DG$ and $BF$ are parallel. The point $H$ is the orthogonal projection of $C$ on the line $FG$. Prove that points $B, E, F$ and $H$ are concyclic.

2015 Brazil Cono Sur Training List 3.9

In the triangle $ABC$, let $D, E$ and $F$ be the midpoints of the sides and $P, Q$ and $R$ the midpoints of the medians

$AD$, $BE$ and $CF$, respectively. Prove the value of $$\frac{AQ^2 + AR^2 + BP^2 + BR^2 + CP^2 + CQ^2}{AB^2 + BC^2 + CA^2}$$ does not depend on the shape of the triangle and find its value.

2015 Brazil Cono Sur Training List 3.10

Consider a triangle $ABC$ with $\angle B > 90^o$ , such that for a point $H$ on the side AC, we have $AH = BH$ and $\angle HBC = 90^o$. Let $D$ and $E$ be the midpoints of $AB$ and $BC$, respectively. Through $H$, a parallel to $AB$ ́drawn, cutting $DE$ into $F$. Prove that $\angle BCF = \angle ACD$.

2015 Brazil Cono Sur Training List 3.11

Let $\Gamma$ be the circumcircle of a triangle $ABC$. Let $P$ be a point on $\Gamma$. Let $D$ and $E$ be the feet of the perpendiculars from $P$ on the lines $AB$ and $BC$, respectively. Determine the locus of the circumcenter of the triangle $PDE$ while $P$ moves along $\Gamma$.

2015 Brazil Cono Sur Training List 3.12

Let $ABC$ be an acute triangle. Let $AA_1$ and $BB_1$ be altitudes , with $A_1$ on $BC$ and $B_1$ on $AC$. Let $D$ be a point on the arc $AB$ of the circumcircle of $ABC$ that contains $C$. Lines $AA_1$ and $BD$ intersect at $P$ and lines $BB_1$ and $AD$ intersect at $Q$. Prove that the midpoint of $PQ$ is on the line $A_1B_1$.

2015 Brazil Cono Sur Training List 3.17

The circles $\omega_1$, with center $O_1$ and $\omega_2$, with center $O_2$, are tangent externally at point $T$. A third circle $\Omega$, with center $O$, ́is tangent to $\omega_1$ at $A$ and $\omega_2$ at $B$, and such that $O_1$ and $O_2$ are inside $\Omega$. the tangent line to $\omega_1$ and $\omega_2$ at $T$ cuts $\Omega$ at $K$ and $L$. Let $D$ be the midpoint of $KL$. Prove that $\angle O_1OO_2 = \angle ADB$.

2015 Brazil Cono Sur Training List 4.9

In a triangle $ABC$, $I$ is the incenter and $O$ ́is the circumcenter. Choose point $P, Q$ and $R$ in segments $IA$, $IB$ and $IC$, respectively, such that $$IP \cdot IA = IQ \cdot IB = IR \cdot IC.$$ Prove that points $I$ and $O$ lie on the Euler line of the triangle $PQR$.

2015 Brazil Cono Sur Training List 4.10

The incircle of triangle $ABC$ has center $I$ and tangent sides $AB$ and $AC$ at points$P$ and $Q$, respectively. Lines $BI$ and $CI$ intersect line $PQ$ at points $K$ and $L$, respectively. Prove that the circumcircle of the triangle $IKL$ is tangent to the incircle of triangle $ABC$ if and only if $AB + AC = 3BC$

2015 Brazil Cono Sur Training List 4.11

Let $ABC$ be a triangle. Choose a $D$ point inside it. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ a circle that passes through $C$ and $D$ such that the other intersection point of $\omega_1$ and $\omega_2$ is on the line $AD$. Let $E$ and $F$ be the points in $BC$ that cut $\omega_1$ and $\omega_2$, respectively, with $E \ne B$ and $F \ne C$. Let $X$ a intersection of $DF$ and $AB$ and $Y$ the intersection of $DE$ and $AC$. Show that $XY$ is parallel to $BC$.

2015 Brazil Cono Sur Training List 4.12

Let $ABCD$ be a convex quadrilateral that has an inscribed circle with center $I$. Suppose the straight line that passes through $A$, perpendicular on $AB$ cuts the line $BI$ at $M$ and that the line that passes through $A$, perpendicular on $AD$ cuts the line $DI$ at $N$. Prove that $MN$ ́is perpendicular on $AC$.

2015 Brazil Cono Sur Training List 4.18

Let $O$ and $I$ be the circumcenter and incenter of the triangle $ABC$, respectively. Let $D$ be the touchpoint of the incircle of $ABC$ with $BC$. Let $E$ and $F$ be the intersections of the lines $AI$ and $AO$ with the circumcircle of $ABC$, respectively, with $A \ne E$ and $A \ne F$. Let $S$ be the intersection ̧of the lines $IF$ with $DE$, $M$ the intersection ̧of the lines $SC$ with $BE$ and $N$ the intersection of lines $AC$ with $BF$. Prove that the points $M, I$ and $N$ are collinear.

2016 Brazil Cono Sur Training List 1.7

Let $AD$, $BE$ and $CF$ be the altitudes of the acute triangle $ABC$. Let $P, Q$ and $R$ be the feet of the perpendiculars from $A, B$ and $C$ on the lines $EF$, $FD$ and $DE$, respectively. Prove that the lines $AP$, $BQ$ and $CR$ are concurrent.

2016 Brazil Cono Sur Training List 1.8

Let $ABC$ be an equilateral triangle. For a point $M$ inside the triangle $ABC$, let $D$, $E$ and $F$ be the feet of perpendiculars from $M$ on sides $BC$, $CA$ and $AB$, respectively. Find the locus of the points $M$ for which $\angle FDE = 90^o$.

2016 Brazil Cono Sur Training List 1.9

Let ABCD be a circumscribed around a circle $\Gamma$ . Knowing that $\angle A = \angle B = 120^o$, $\angle D = 90^o$, and $BC=1$, determine the length of the side $AD$.

2016 Brazil Cono Sur Training List 1.16

Let $ABC$ be an acute triangle. Denote by $D$ the foot of the perpendicular from $A$ on the side $BC$, by $M$ the midpoint of $BC$ and by $H$ the orthocenter of $ABC$. Let $E$ be the intersection point of the circle $\Gamma$ of the triangle $ABC$ with the ray $MH$ and let $F$ be the intersection (different from $E$) of the line $ED$ with the circle $\Gamma$. Prove that $$\frac{BF}{CF}=\frac{BA}{CA}.$$

2016 Brazil Cono Sur Training List 2.7

Let $ABC$ be a triangle right at $A$ and let $I$ be its incenter. Line $BI$ intersects $AC$ at $D$, and line $CI$ intersects $AB$ at $E$. Let $P$ and $Q$ be the feet of the perpendiculars from $D$ and $E$ on $BC$, respectively. Prove that $IP = IQ$.

2016 Brazil Cono Sur Training List 2.8

Let $ABCD$ be a convex quadrilateral. Suppose that points $M$ and $N$ are on sides $AB$ and $BC$, respectively such that $[AMCD] = [CMB]$ and $[ANCD] = [ANB]$. Demonstrate that the line $MN$ passes through the midpoint of the diagonal $BD$ .

Note: $[P]$ denotes the area of the polygon $P$.

2016 Brazil Cono Sur Training List 2.9

Let $ABC$ be a triangle and $H$ its orthocenter. The circumcircle of $ABC$ intersects the circle of diameter $AH$ at $P \ne A$. Prove that $HP$ passes through the midpoint of $BC$.

2016 Brazil Cono Sur Training List 2.16

Let $ABC$ be a triangle and let $D, E, F$ be the touchpoints of the incircle of $ABC$ with sides $BC$, $CA$, $AB$, respectively. The segment $AD$ intersects the incircle of $ABC$ again at point $P$. A perpendicular on $AD$ from $P$ intersects $EF$ at point $Q$. Lines $DE$ and $DF$ intersect line $AQ$ at $X$ and $Y$ , respectively. Show that $A$ is the midpoint of $XY$.

2016 Brazil Cono Sur Training List 3.7

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose the internal bisector of angle $\angle B$ cuts $AC$ at point $D$ and $BC = BD + AD$. Determine the measure of the angle $\angle A$.

2016 Brazil Cono Sur Training List 3.8

Let $ABC$ be a triangle and let $M, N$ and $P$ be points on the line $BC$ such that $AM$, $AN$ and $AP$ are altitude, angle bisector and median of the triangle, respectively. It is known that $$\frac{[AMP]}{[ABC]}=\frac14 \,\,\,, \,\,\, \frac{[ANP]}{[ABC]}= 1 - \frac{\sqrt3}{2}.$$ Determine the angles of triangle $ABC$.

Note: $[P]$ denotes the area of the polygon $P$.

2016 Brazil Cono Sur Training List 3.9

Let $ABCD$ be a tetrahedron and let $E, F, G, H, K$ and $L$ be points on the segments $AB$, $BC$, $CA$, $DA$, $DB$ and $DC$, respectively, so that $$AE \cdot BE = BF\cdot  CF = CG \cdot  AG = DH \cdot  AH = DK \cdot  BK = DL \cdot  CL.$$ Prove that the six points marked on the sides of the tetrahedron are on the same sphere.

2016 Brazil Cono Sur Training List 3.16

Let $ABC$ be a triangle with all its acute angles, of altitudes $AD$, $BE$, and $CF$ (with $D$ in $BC$, $E$ in $AC$ and $F$ in $AB$). Let $M$ be the midpoint of segment $BC$. The circle circumscribed around the triangle $AEF$ cuts line $AM$ at $A$ and $X$. Line $AM$ cuts line $CF$ at $Y$ . Let $Z$ be the intersection point between lines $AD$ and $BX$. Prove that the lines $YZ$ and $BC$ are parallel.

2016 Brazil Cono Sur Training List 4.7

Let $ABC$ be an acute triangle such that its inscribed circle intersects the sides $AB$ and $AC$ at $D$ and $E$, respectively. Let $X$ and $Y$ be the respective intersection points of the internal angle bisectors of the angles $\angle ACB$ and $\angle ABC$ with the segment $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $XYZ$ is equilateral if, and only if, $\angle BAC = 60^o$.

2016 Brazil Cono Sur Training List 4.8

Let $ABCD$ be a cyclic quadrilateral, and let $P$ be the intersection of the diagonals $AC$ and $BD$. Let $K$ and $L$ be the feet of the perpendiculars from $P$ on sides $AD$ and $BC$, respectively. If $M$ ́is the midpoint of $AB$, prove that $MK = ML$.

2016 Brazil Cono Sur Training List 4.9

Let $ABC$ be an acute and scalene triangle, with $\angle BAC = 30^o$. The internal and external angle bisectors of $\angle ABC$ intersect the line $AC$ at $B_1$ and $B_2$, respectively. The internal and external angle bisectors of $\angle ACB$ intersect the line $AB$ at $C_1$ and $C_2$, respectively. Suppose the circles of diameters $B_1B_2$ and $C_1C_2$ intersect at a point $P$ inside the triangle $ABC$. Prove that $\angle BPC = 90^o$.

2016 Brazil Cono Sur Training List 4.14

Let $ABC$ be an acute triangle, and let $P, Q, R$ be the midpoints of the arcs $BC,CA,AC$ of the circumcircle of $ABC$. Prove that $AP, BQ, CR$ intersected by $BC, CA, AB$ at $L, M, N$, respectively, prove that: $$\frac{AL}{PL}+\frac{BM}{QM} +\frac{LN}{RN} \ge 9$$

2017 Brazil Cono Sur Training List 1.7

Given a triangle isosceles $ABC$ with $AB = BC$. A point $M$ is chosen within $ABC$ such that $\angle AMC = 2\angle ABC$. A point $K$ is on the segment $AM$ so that $\angle BKM = \angle ABC$. Prove that $BK = KM + MC$.

2017 Brazil Cono Sur Training List 1.8

Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2\angle ADB$, $\angle ABD = 2\angle CDB$ and $AB = CB$. Prove that $AD = CD$.

2017 Brazil Cono Sur Training List 1.9

Let $M$ be the intersection point of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. The internal bisector of the angle $\angle ACD$ intersects the ray $BA$ at a point $K$. It is known that $MA \cdot MC + MA \cdot CD = MB \cdot MD$. Prove that $\angle BKC = \angle CDB$.

2017 Brazil Cono Sur Training List 1.12

Let $ABCD$ be an ́convex quadrilateral such that $\angle DAB = 90^o$. Let $M$ be the midpoint of $BC$. Knowing that $\angle ADC = \angle BAM$, prove that $\angle ADB = \angle CAM$.

2017 Brazil Cono Sur Training List 2.7

On the circumcircle of triangle $ABC$, let $A_1$ be the point diametrically opposite vertex $A$. Let $A'$ be the intersection point to that of $AA_1$ and $BC$. The perpendicular to the line $AA'$ through $A'$ intersects sides $AB$ and $AC$ into $M$ and $N$, respectively. Prove that the points $A, M, A_1, N$ are on a circle whose center is at the altitude from vertex $A$ of triangle $ABC$.

2017 Brazil Cono Sur Training List 2.8

Let $ABC$ be a triangle, $I$ its incenter and $D$ the foot of the perpendicular of$I$ to the side $BC$. Let $P$ and $Q$ be the orthocenters of the triangles $AIB$ and $AIC$, respectively. Prove that $P, Q, D$ are collinear.

2017 Brazil Cono Sur Training List 2.9

Let $ABC$ be a triangle with orthocenter $H$. The lines $AH$, $BH$, $CH$ intersect the circumcircle of $ABC$ again at $D, E, F$, respectively. Determine the maximum value of $\frac{Area\,(DEF)}{Area \,(ABC)}$.

2017 Brazil Cono Sur Training List 2.13

Let $ABC$ be a sharp-angled triangle of circumcenter $O$ and let $D, E, F$ be the feet of altitudes on the sides $BC$, $CA$, $AB$, respectively. Let $\omega$ be the circumcircle of $ABC$, let $\omega'$ is the circle of $DEF$ and let $P$ be a variable point about $\omega$. Consider a circle tangent internally to $\omega$ at by $P$ and tangent externally the $\omega'$ at $Q$. Prove that the line $PQ$ passes through a fixed point on the line $HO$.

2017 Brazil Cono Sur Training List 3.7

The point $P$ located internally in the triangle $ABC$ , such that $\angle BAP = \angle PAC = 20^o$, $\angle PCA = 10^o$ and $\angle PCB = 30^o$. Calculate the measure of the angle $\angle PBC$.

2017 Brazil Cono Sur Training List 3.8

Let $H$ be the orthocenter and $M$ the midpoint of the side $BC$ in the acute-angled triangle $ABC$. Let $N$ be a point belonging to the extension of the side $HM$ such that $HM = MN$. Prove that $M$ belongs to circumscribed circle of triangle $ABC$.

2017 Brazil Cono Sur Training List 3.9

Two given right triangles are such that the incircle of the first is exactly equal to the circumcircle of the second. Let $S$ and $S'$ be the areas of the first and second triangle, respectively. Prove that $$\frac{S}{S'}\ge 3+2\sqrt2$$

2017 Brazil Cono Sur Training List 3.12

Let $ABCD$ be a convex quadrilateral. Let $n \ge 2$ be an integer. Prove that there are $n$ triangles of the same area with the following properties:

$\bullet$ There are no overlapping triangles

$\bullet$ Each triangle is contained within or within the perimeter of $ABCD$

$\bullet$ The sum of the areas of all triangles ́and at least $\frac{4n}{4n + 1}$ of the area of the quadrilateral $ABCD$.

2017 Brazil Cono Sur Training List 4.7

Determine all angles of a ́convex quadrilateral $ABCD$ such that $\angle  ABD = 29^o$, $\angle  ADB = 41^o$, $\angle ACB =82^o$ and $\angle ACD = 58^o$.

2017 Brazil Cono Sur Training List 4.8

In a convex quadrilateral $ABCD$, the angles $\angle A$ and $\angle C$ have the same measure and the bisector of $\angle B$ passes through the midpoint of the side $CD$ . If $CD = 3AD$, determine the ratio $\frac{AB}{BC}$

2017 Brazil Cono Sur Training List 4.9

Let $ABC$ be a triangle. Construct isosceles triangles $BCD$, $CAE$ and $ABF$ externally to $ABC$, with base $BC$, $CA$ and $AB$, respectively. Prove that the lines that pass through $A, B$ and $C$ and are perpendicular to $EF$, $FD$ and $DE$, respectively, are concurrent.

2017 Brazil Cono Sur Training List 4.13

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$ with $BC > CA$. Let $F$ be the foot of the altitude from $C$ of this triangle. The line perpendicular to $OF$ at point $F$ intersects the line $AC$ at point $P$. Prove that $\angle FHP = \angle BAC$.

2018 Brazil Cono Sur Training List 1.7

The $ABCD$ quadrilateral is inscribed in a circle. Let $M$ be the intersection point of their diagonals and $L$ the midpoint of the arc $AD$ that does not contain other vertices of the quadrilateral. Prove that the distances from $L$ to centers of the triangles inscribed in the triangles $\vartriangle ABM$ and $\vartriangle CDM$ are equal.

2018 Brazil Cono Sur Training List 1.8

In the figure below, $ABC$ is an isosceles triangle with $BA = BC$. Point D is inside it so that $\angle ABD = 13^o$, $\angle ADB = 150^o$ and $\angle ACD = 30^o$. Furthermore, $ADE$ is an equilateral triangle. Determine the value of the angle $\angle DBC$.

2018 Brazil Cono Sur Training List 1.9

Squares $BAXX'$ and $CAYY'$ are built externally on the sides of the isosceles triangle $ABC$ with equal sides $AB = AC$. Let $E$ and $F$ be the feet of the perpendiculars of an arbitrary point $K$ of the segment $BC$ on segments B$Y$ and $CX$ , respectively. Let $D$ be the midpoint of $BC$.

a) Prove that $DE = DF$.

b) Find the locus of the midpoint of $EF$

2018 Brazil Cono Sur Training List 1.10

Clipping a convex $n$-gon means choosing a pair of consecutive sides $AB$, $BC$ and replacing them by three segments $AM$, $MN$ and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, we cut the $MBN$ triangle to obtain an convex $(n + 1)$-gon. A regular hexagon $P_6$ of Area $1$ is clipped to get a heptagon $P_7$ . So $P_7$ is clipped in one of the $7$ possible ways to get an octagon $P_8$, and so on. Prove that no matter how clippings happen, the area of $P_n$ is greater than $\frac13$ for all $n \ge 6$.

2018 Brazil Cono Sur Training List 2.7

The equilateral triangle $DCE$ is constructed externally on the side $DC$ of the parallelogram $ABCD$ . Let $X$ be an arbitrary point in the plane. Prove that $XA + XB + AD \ge XE$.

2018 Brazil Cono Sur Training List 2.8

Let $P$ be the intersection point of the diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $AB = AC = BD$. Let $O$ and $I$ be the circumcenter and incenter of the triangle $ABP$. Prove that if $O\ne I$, then it is $OI \perp CD$.

2018 Brazil Cono Sur Training List 2.9

Let $ABC$ be a triangle with $AC = BC$. A point $P$ internal to $ABC$ is such that $\angle PAB =  \angle PBC$. Le $M$ be the midpoint of $AB$, prove that $\angle APM + \angle BPC = 180^o$.

2018 Brazil Cono Sur Training List 2.10

Let $P =\{P_1, P_2, ..., P_{1997}\}$ be a set of 1997 points within a circle of radius $1$, with $P_1$ being the center of the circle. For $k = 1, 2, ..., 1997$ let $x_k$ be the distance from $P_k$ to the point of $P$ closest to $P_k$. Prove that $x_1^2+x_2^2+...+x_{1997}^2 \le 9$.

2018 Brazil Cono Sur Training List 3.7

Let $ABCDE$ be a convex pentagon. Suppose $BD \cap CE = A'$, $CE \cap DA = B'$, $DA \cap EB = C'$, $EB \cap AC = D'$ and $AC \cap BD = E'$. Also suppose that $(ABD') \cap (AC'E) = A''$, $(ECB') \cap (BD'A) = B''$, $(CDA') \cap (CE'B) = C''$, $(DEB') \cap DA'C = D''$ and $(EAC') \cap (EB'D) = E''$. Prove that $AA''$, $BB''$ , $CC''$, $DD''$ and $EE''$ are concurrent.

2018 Brazil Cono Sur Training List 3.8

Let $D$ be a point on the side $BC$ of the triangle \vartriangle ABC such that $AD =\frac{BD^2}{AB + AD} =\frac{CD^2}{AC+AD}$. Let E be a point such that $D$ lies on $[AE]$ and $CD =\frac{DE^2}{CD+CE}$. Prove that $AE = AB + AC$.

2018 Brazil Cono Sur Training List 3.9

Given the $\vartriangle ABC$ isosceles with $AB = AC > BC$. The perpendicular bisector of $AB$ meets the external bisector of $\angle ADB$ at point $P$, The perpendicular bisector of $AC$ meets the external bisector of $\angle ADC$ at point $Q$. Prove that $B, C, P$ and $Q$ are concyclic.

2018 Brazil Cono Sur Training List 3.10

Given a convex polygon $A_1A_2...A_n$ of n sides. Prove or disprove that the largest circles circumscribed to the triangles $A_iA_jA_k,$ with $i, j, k \in \{1, 2, ..., n\}$ and $i\ne k \ne j$, are of the form $A_{\ell}A_{\ell+1}A_{\ell+2}$.

2020 Brazil Cono Sur Training List 1.2

Let $D$ be a point on side $AB$ of a triangle $ABC$. A point $L$ lies inside triangle $ABC$ such that $BD = LD$ and $\angle LAB = \angle LCA = \angle DCB$. We know that $\angle ALD + \angle ABC = 180^o$. Prove that $\angle BLC = 90^o$.

2020 Brazil Cono Sur Training List 1.7

Let $I$ be the incenter and $AB$ the shortest side of a triangle $ABC$. The circle with center $I$ passing through $C$ intersects the ray $AB$ into $P$ and the ray $BA$ into $Q$. Let $D$ be the touchpoint between the circle exscribed to $ABC$, relative to point $A$ and side $BC$. Let $E$ be the the reflection point of $C$ wrt to point $D$. Prove that $PE \perp CQ$.

2020 Brazil Cono Sur Training List 1.9

Let $G$ be the centroid of a right triangle $ABC$ with $\angle BCA = 90^o$. Let P be a point on the ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on the ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circles circumscribed around the triangles $AQG$ and $BPG$ intersect at a point on the side $AB$.

2020 Brazil Cono Sur Training List 2.3

Let $ABC$ be an acute triangle. Points $M$ and $N$ are the midpoints of sides $AB$ and $BC$, respectively, and $BH$ is the altitude relative to $AC$, with $H$ over $AC$. The circles circumscribed around the triangles $AHN$ and $CHM$ intersect again at point $P$, $P \ne H$. Prove that PH passes through the midpoint of $MN$.

2020 Brazil Cono Sur Training List 2.6

Let $ABC$ be an acute triangle and $O$ its circumcenter. Point $H$ is the foot of the altitude relative to vertex $A$ and the points $P$ and $Q$ are the feet of the perpendiculars of $H$ to sides $AB$ and $AC$, respectively. Given that $AH^2 = 2AO^2$, prove that $O, P$, and $Q$ are collinear.

2020 Brazil Cono Sur Training List 2.11

A parallelogram $ABCD$ is given, with AB < AC < BC. Points $E$ and $F$ are selected on the circle $\omega$ of $ABC$ such that the tangent to $\omega$ at these points pass through $D$ and segments $AD$ and $CE$ intersect. Knowing that $\angle ABF = \angle DCE$, determine the angle $\angle ABC$.

2020 Brazil Cono Sur Training List 3.3

Let $ABC$ be a triangle with $\angle BAC =\frac{\pi}{6}$ and circumradius $1$. For a point $X$ inside or on the edge of the triangle, let $m(X) = \min \, (AX, BX, CX)$. determine the angles of ABC if $\max \, (m(X)) = \frac{\sqrt3}{3}$.

2020 Brazil Cono Sur Training List 3.4

Let $ABCD$ be a trapezoid with $AB \parallel CD$. The sides $DA$, $AB$, $BC$ are tangent to a circle touching $AB$ in $P$. The sides $BC$, $CD$, $DA$ are tangent to a circle that plays $CD$ in $Q$. Prove that the lines $AC$, $BD$ and $PQ$ are concurrent.

2020 Brazil Cono Sur Training List 3.10

The convex quadrilateral $ABCD$ satisfies $\angle B = \angle C$ and $\angle D = 90^o$. Suppose $|AB| = 2|CD|$. Prove that the bisector of $\angle ACB$ is perpendicular to $CD$.

2020 Brazil Cono Sur Training List 4.8

Let $\vartriangle ABC$ be a triangle such that $AC = BC$. $P$ is a point on arc $AB$ of the circumcircle of $ABC$, that does not contain $C$. Let $D$ be the foot of the perpendicular from $C$ to $PB$. Prove that $PA + PB = 2PD$.

2020 Brazil Cono Sur Training List 4.9

Let $\vartriangle ABC$ be an obtuse-angled triangle in $C$ such that $2\angle BAC = \angle ABC$. Let $P$ be a point on the side$AB$ such that $BP = 2BC$. Let $M$ be the midpoint of $AB$ ($M$ is between $P$ and $B$). Show that the perpendicular on $AC$ through of $M$ intersects $PC$ at its midpoint.

2020 Brazil Cono Sur Training List 4.12

In a given triangle $\vartriangle ABC$, $\angle A = 90^o$ and $M$ is the midpoint of $BC$. Choose $D$ in segment $AC$ such that $AM = AD$ and let $P$ be the other point of intersection between the circumcircles of triangles $\vartriangle  AMC$ and $\vartriangle BDC$. Prove that $P$ lies on the bisector of angle $\angle  ACB$.