Cono Sur Shortlist (OMCS SHL) 56p

geometry shortlists from Cono Sur Olympiad (Cone Sul) with aops links in the names

OLIMPÍADA DE MATEMÁTICA DO CONE SUL

 geometry shortlists collected inside aops: here

from years: 1993, 2003, 2005, 2009,2012, 2018, 2020

1993, 2003, 2005, 2009, 2012, 2018, 2020-21

1993 Shortlist

1993 Cono Sur Shortlist G1

Let $C_1$ and $C_2$ be two concentric circles and $C_3$ an outer circle to $C_1$ inner to $C_2$ and tangent to both. If the radius of $C_2$ is equal to $1$, how much must the radius of $C_1$ be worth, so that the area of is twice that of $C_3$?

1993 Cono Sur Shortlist G2

Let $ABCD$ be a quadrilateral and let $O$ be the point of intersection of diagonals $AC$ and $BD$. Knowing that the area of triangle $AOB$ is equal to $1$, the area of triangle $BOC$ is equal to $2$, and the area of triangle $COD$ is equal to $4$, calculate the area of triangle $AOD$ and prove that $ABCD$ is a trapezoid.

1993 Cono Sur Shortlist G3

Justify the following construction of the bisector of an angle with an inaccessible vertex:

$M \in a$ and $N \in  b$ are taken, the $4$ bisectors of the $4$ internal angles formed by $MN$ are traced with $a$ and $b$. Said bisectors intersect at $P$ and $Q$, then $PQ$ is the bisector sought.

1993 Cono Sur Shortlist G4

Is it possible to locate in a rectangle of $5$ cm by $8$ cm, $51$ circles of diameter $1$ cm, so that they don't overlap? Could it be possible for more than $40$ circles ?

1993 Cono Sur Shortlist G5

A block of houses is a square. There is a courtyard there in which a gold medal has fallen. Whoever calculates how long the side of said apple is, knowing that the distances from the medal to three consecutive corners of the apple are, respectively, $40$ m, $60$ m and $80$ m, will win the medal.

1993 Cono Sur Shortlist G6

Consider in the interior of an equilateral triangle $ABC$ points $D, E$ and $F$ such that$D$ belongs to segment $BE$, $E$ belongs to segment $CF$ and$F$ to segment $AD$. If $AD=BE = CF$ then $DEF$ is equilateral.

1993 Cono Sur Shortlist G7

Let $ABCD$ be a convex quadrilateral, where $M$ is the midpoint of $DC$, $N$ is the midpoint of $BC$, and $O$ is the intersection of diagonals $AC$ and $BD$. Prove that $O$ is the centroid of the triangle $AMN$ if and only if $ABCD$ is a parallelogram.

1993 Cono Sur Shortlist G8

In a triangle $ABC$, let $D$, $E$ and $F$ be the touchpoints of the inscribed circle and the sides $AB$, $BC$ and $CA$. Show that the triangles $DEF$ and $ABC$ are similar if and only if $ABC$ is equilateral.

1993 Cono Sur Shortlist G9

Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.

1993 Cono Sur Shortlist G10

Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$.

1993 Cono Sur Shortlist G11

Let $\Gamma$ be a semicircle with center $O$ and diameter $AB$. $D$ is the midpoint of arc $AB$. On the ray $OD$, we take $E$ such that $OE = BD$. $BE$ intersects the semicircle at $F$ and $P$ is the point on $AB$ such that $FP$ is perpendicular to $AB$. Prove that $BP=\frac13 AB$.

1993 Cono Sur Shortlist G12

Given $4$ lines in the plane such that there are not $2$ parallel to each other or no $3$ concurrent, we consider the following $8$ segments: in each line we have $2$ consecutive segments determined by the intersections with the other three lines.

Prove that:

a) The lengths of the $8$ segments cannot be the numbers $1, 2, 3,4, 5, 6, 7, 8$ in some order.

b) The lengths of the $8$ segments can be $8$ different integers.

1993 Cono Sur Shortlist G13

Determine the real values of $x$ such that the triangle with sides $5$, $8$, and $x$ is obtuse.

1993 Cono Sur Shortlist G14

Prove that the sum of the squares of the distances from a point $P$ to the vertices of a triangle $ABC$ is minimum when $P$ is the centroid of the triangle.

2003 Shortlist

2003 Cono Sur Shortlist G1
Let $O$ be the circumcenter of the isosceles triangle $ABC$ ($AB = AC$). Let $P$ be a point of the segment $AO$ and $Q$ the symmetric of $P$ with respect to the midpoint of $AB$. If $OQ$ cuts $AB$ at $K$ and the circle that passes through $A, K$ and $O$ cuts $AC$ in $L$, show that $\angle ALP = \angle  CLO$.

2003 Cono Sur Shortlist G2
The circles $C_1, C_2$ and $C_3$ are externally tangent in pairs (each tangent to other two externally). Let $M$ the common point of $C_1$ and $C_2, N$  the common point of $C_2$ and $C_3$ and $P$ the common point of $C_3$ and $C_1$.  Let $A$ be an arbitrary point of $C_1$. Line $AM$ cuts $C_2$ in $B$, line $BN$ cuts $C_3$ in $C$ and line $CP$ cuts $C_1$ in $D$. Prove that $AD$ is diameter of $C_1$.

2003 Cono Sur Shortlist G3
An interior $P$ point to a square $ABCD$ is such that $PA = a, PB = b$ and $PC = b + c$, where the numbers $a, b$ and $c$ satisfy the relationship $a^2 = b^2 + c^2$. Prove that the angle $BPC$ is right.

2003 Cono Sur Shortlist G4
In a triangle $ABC$ , let $P$ be a point on its circumscribed circle (on the arc $AC$ that does not contain $B$). Let $H,H_1,H_2$ and $H_3$ be the orthocenters of  triangles $ABC, BCP, ACP$ and $ABP$, respectively. Let $L = PB \cap  AC$ and $J = HH_2 \cap  H_1H_3$. If $M$ and $N$ are the midpoints of $JH$ and $LP$, respectively, prove that $MN$ and $JL$ intersect at their midpoint.

2003 Cono Sur Shortlist G5 problem 4

In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$.

2003 Cono Sur Shortlist G6
Let $L_1$ and $L_2$ be two parallel lines and $L_3$ a line perpendicular to $L_1$ and $L_2$ at $H$ and $P$, respectively. Points $Q$ and $R$ lie on $L_1$ such that $QR = PR$ ($Q \ne  H$). Let  $d$ be the diameter of the circle inscribed in the triangle $PQR$. Point $T$ lies $L_2$ in the same semiplane as $Q$  with respect to line $L_3$ such that $\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH}$ . Let $X$ be the intersection point of $PQ$ and $TH$. Find the locus of the points $X$ as $Q$ varies on $L_1$.

2003 Cono Sur Shortlist G7 problem 3
Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.

2005 Shortlist

2005 Cono Sur Shortlist G1
Given the heights $h_a, h_b$ and $h_c$ of a triangle, construct (explain how) the triangle ABC.

2005 Cono Sur Shortlist G2
Find the ratio between the sum of the areas of the circles and the area of the fourth circle that are shown in the figure. Each circle passes through the center of the previous one and they are internally tangent.

2005 Cono Sur Shortlist G3 problem 4
Let $ABC$ be a isosceles triangle, with $AB=AC$.  A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$

2005 Cono Sur Shortlist G4 problem 2
Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively.
(a) Show that $R$, $Q$, $W$, $S$ are collinear.
(b) Show that $MP=RS-QW$.

2005 Cono Sur Shortlist G5
Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ a point of the minor arc $BC$ of the circle $ABC$ . Let A_2 and $A_3$ be points on sides $AB$ and $AC$ respectively such that $\angle BA_1A_2=\angle OAC$ and $\angle CA_1A_3=\angle OAB$ . Points $B_2, B_3, C_2$ and $C_3$ are similarly constructed, with $B_2$ in $BC, B_3$ in $AB, C_2$ in $AC$ and $C_3$ in $BC$. Prove that lines $A_2A_3, B_2B_3$ and $C_2C_3$ are concurrent.

2005 Cono Sur Shortlist G6
Let $AM$ and $AN$ be the tangents to a circle $\Gamma$ drawn from a point $A$ ($M$ and $N$ lie on the circle). A line passing through $A$ cuts $\Gamma$ at $B$ and $C$, with B between $A$ and $C$ such that $AB: BC = 2: 3$. If $P$ is the intersection point of $AB$ and $MN$, calculate the ratio $AP: CP$ .

2009 Shortlist

2009 Cono Sur Shortlist G1 problem 6
Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square.

Note: The rectangles can overlap and they can protrude over the sides of the square

2009 Cono Sur Shortlist G2
The trapezoid $ABCD$,  of bases $AB$ and $CD$, is inscribed in a circumference $\Gamma$. Let $X$ a variable point of the arc $AB$ of $\Gamma$ that does not contain $C$ or $D$. We denote $Y$ to the point of intersection of $AB$ and $DX$, and let Z be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$ . Prove that the measure of $\angle AZX$ does not depend on the choice of $X.$

2009 Cono Sur Shortlist G3
We have a convex polygon $P$ in the plane and two points $S,T$ in the boundary of $P$, dividing the perimeter in a proportion $1:2$. Three distinct points in the boundary, denoted by $A,B,C$ start to move simultaneously along the boundary, in the same direction and with the same speed. Prove that there will be a moment in which one of the segments $AB, BC, CA$ will have a length smaller or equal than $ST$.

2009 Cono Sur Shortlist G4
Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$  respectively. The $C_1J$ and $A_1 I$ lines are cut into $T$. Prove that lines  $AT$ and $TC$ are perpendicular.

2009 Cono Sur Shortlist G5 problem 3
Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.

2012 Shortlist

2012 Cono Sur Shortlist G1

Let $ABCD$ be a cyclic quadrilateral. Let $P$ be the intersection of $BC$ and $AD$. Line $AC$ intersects the circumcircle of triangle $BDP$ in points $S$ and $T$, with $S$ between $A$ and $C$. Line $BD$ intersects the circumcircle of triangle $ACP$ in points $U$ and $V$, with $U$ between $B$ and $D$. Prove that $PS$ = $PT$ = $PU$ = $PV$.

2012 Cono Sur Shortlist G2

Let $ABC$ be a triangle, and $M$ and $N$ variable points on $AB$ and $AC$ respectively, such that both $M$ and $N$ do not lie on the vertices, and also, $AM \times MB = AN \times NC$. Prove that the perpendicular bisector of $MN$ passes through a fixed point.

2012 Cono Sur Shortlist G3

Let $ABC$ be a triangle, and $M$, $N$, and $P$ be the midpoints of $AB$, $BC$, and $CA$ respectively, such that $MBNP$ is a parallelogram. Let $R$ and $S$ be the points in which the line $MN$ intersects the circumcircle of $ABC$. Prove that $AC$ is tangent to the circumcircle of triangle $RPS$.

2012 Cono Sur Shortlist G4 problem 2

In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

2012 Cono Sur Shortlist G5

Let $ABC$ be an acute triangle, and let $H_A$, $H_B$, and $H_C$ be the feet of the altitudes relative to vertices $A$, $B$, and $C$, respectively. Define $I_A$, $I_B$, and $I_C$ as the incenters of triangles $AH_B H_C$, $BH_C H_A$, and $CH_A H_B$, respectively. Let $T_A$, $T_B$, and $T_C$ be the intersection of the incircle of triangle $ABC$ with $BC$, $CA$, and $AB$, respectively. Prove that the triangles $I_A I_B I_C$ and $T_A T_B T_C$ are congruent.

2012 Cono Sur Shortlist G6 problem 6

Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.

2018 Shortlist

2018 Cono Sur Shortlist G1 problem1

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.

2018 Cono Sur Shortlist G2 problem 5

Let $ABC$ be a acute-angled triangle with $\angle BAC = 60$ and with incenter $I$ and circumcenter $O$. Let $H$ be the diametrically opposite(antipode) to $O$ in the circumcircle of $BOC$. Prove that $IH = BI + IC$

2018 Cono Sur Shortlist G3

Consider the pentagon $ABCDE$ such that $AB = AE = x$, $AC = AD = y$, $\angle BAE = 90^o$ and $\angle ACB = \angle ADE = 135^o$. It is known that $C$ and $D$ are inside the triangle $BAE$. Determine the length of $CD$ in terms of $x$ and $y$.

2018 Cono Sur Shortlist G4

Let $ABC$ be an acute triangle with $AC > AB$. Let $\Gamma$ be the circle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smaller arc $BC$ of this circle. Let $I$ be the incenter of $ABC$ and let $E$ and $F$ be points on sides $AB$ and $AC$, respectively, such that $AE = AF$ and $I$ lies on the segment $EF$. Let $P$ be the second intersection point of the circumcircle of the triangle $AEF$ with $\Gamma$ with $P \ne A$. Let $G$ and $H$ be the intersection points of the lines $PE$ and $PF$ with $\Gamma$ different from $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines AB and $AC$, respectively. Show that the line $JK$ passes through the midpoint of $BC$.

2018 Cono Sur Shortlist G5

We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .

2018 Cono Sur Shortlist G6

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_A$ passes through the points $A$ and $H$ and is tangent to the circumcircle of the triangle $ABC$. Similarly, define the points $X_B$ and $X_C$. Let $O_A$, $O_B$ and $O_C$ be the reflections of $O$ with respect to sides $BC$, $CA$ and $AB$, respectively. Prove that the lines $O_AX_A$, $O_BX_B$ and $O_CX_C$ are concurrent.

2020 Shortlist

2020 Cono Sur Shortlist G1 problem 4

Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.

2020 Cono Sur Shortlist G2

Let $ABC$ be a triangle whose inscribed circle is $\omega$. Let $r_1$ be the line parallel to $BC$ and tangent to $\omega$, with $r_1 \ne BC$ and let $r_2$ be the line parallel to $AB$ and tangent to $\omega$ with $r_2 \ne AB$. Suppose that the intersection point of $r_1$ and $r_2$ lies on the circumscribed circle of triangle $ABC$. Prove that the sidelengths of triangle $ABC$ form an arithmetic progression.

2020 Cono Sur Shortlist G3 problem 3

Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.

2020 Cono Sur Shortlist G4

Let $ABC$ be a triangle with circumcircle $\omega$. The bisector of $\angle BAC$ intersects $\omega$ at point $A_1$. Let $A_2$ be a point on the segment $AA_1$, $CA_2$ cuts $AB$ and $\omega$ at points $C_1$ and $C_2$, respectively. Similarly, $BA_2$ cuts $AC$ and $\omega$ at points $B_1$ and $B_2$, respectively. Let $M$ be the intersection point of $B_1C_2$ and $B_2C_1$. Prove that $MA_2$ passes the midpoint of $BC$.

2020 Cono Sur Shortlist G5 (2021 problem 6)

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $P,Q,R,$ and $S$ be distinct points on the side $BC$ in that order such that $\angle BAP=\angle CAS$ and $\angle BAQ=\angle CAR$. Let $U,V,W$ and $Z$ be the intersections, distinct from $A$, $AP$, $AQ$,$AR$, and $AS$ with $\Gamma$, respectively. Let $X=UQ\cap SW$ , $Y=PV\cap RZ$ , $T=UR\cap VS$, and $K=PW\cap ZQ$ . Also, let $M=KX\cap TY$ and $N=TX\cap KY$. Show that points $M,N$ and $A$ are collinear.

2021 Shortlist

2021 Cono Sur Shortlist G1 problem 2

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

2021 Cono Sur Shortlist G2

Let $ABC$ be an acute triangle. Define $A_1$ the midpoint of the largest arc $BC$ of the circumcircle of $ABC$ . Let $A_2$ and $A_3$ be the feet of the perpendiculars from $A_1$ on the lines $AB$ and $AC$ , respectively. Define $B_1$, $B_2$, $B_3$, $C_1$, $C_2$, and $C_3$ analogously. Show that the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ are concurrent.

2021 Cono Sur Shortlist G3

Let $ABCD$ be a parallelogram with vertices in order clockwise and let $E$ be the intersection of its diagonals. The circle of diameter $DE$ intersects the segment $AD$ at $L$ and $EC$ at $H$. The circumscribed circle of $LEB$ intersects the segment $BC$ at $O$. Prove that the lines $HD$ , $LE$ and $BC$ are concurrent if and only if $EO = EC$.

2021 Cono Sur Shortlist G4

Let $ABC$ be a triangle and $\Gamma$ the $A$- exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$, respectively. Let $S$ be the area of the quadrilateral $ADJE$, Find the maximum value that $$\frac{S}{AJ^2}$$ has and when equality holds.

2021 Cono Sur Shortlist G5

Let $\vartriangle ABC$ be a triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\omega$. $AA'$, $BB'$ and $CC'$ are altitudes of $\vartriangle ABC$ with $A'$ in $BC$, $B'$ in $AC$ and $C'$ in $AB$. $P$ is a point on the segment $AA'$. The perpenicular line to $B'C'$ from $P$ intersects $BC$ at $K$. $AA'$ intersects $\omega$ at $M \ne A$. The lines $MK$ and $AO$ intersect at $Q$. Prove that $\angle CBQ = \angle PBA$.

2021 Cono Sur Shortlist G6 problem 6

Let $ABC$ be a scalene triangle with circle $\Gamma$. Let $P,Q,R,S$ distinct points on the $BC$ side, in that order, such that $\angle BAP = \angle CAS$ and $\angle BAQ = \angle CAR$. Let $U, V, W, Z$ be the intersections, distinct from $A$, of the $AP, AQ, AR$ and $AS$ with $\Gamma$, respectively. Let $X = UQ \cap SW$, $Y = PV \cap ZR$, $T = UR \cap VS$ and $K = PW \cap ZQ$. Suppose that the points $M$ and $N$ are well determined, such that $M = KX \cap TY$ and $N = TX \cap KY$. Show that $M, N, A$ are collinear.

2021 Cono Sur Shortlist G7

Given an triangle $ABC$ isosceles at the vertex $A$, let $P$ and $Q$ be the touchpoints with $AB$ and $AC$, respectively with the circle $T$, which is tangent to both and is internally tangent to the circumcircle of $ABC$. Let $R$ and $S$ be the points of the circumscribed circle of $ABC$ such that $AP = AR = AS$ . Prove that $RS$ is tangent to $T$ .