Cono Sur / Southern Cone 1989 - 2018 (OMCS) 53p

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

OLIMPÍADA DE MATEMÁTICA DO CONE SUL

1989 - 2018

Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ($a \neq b$) have equal areas. Find $x$.

Let $ABCD$ be a square with diagonals $AC$ and $BD$, and $P$ a point in one of the sides of the square. Show that the sum of the distances from P to the diagonals is constant.

Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.

in 1990 this contest did not take place

Let $A, B$ and $C$ be three non-collinear points and $E$ ($\ne B$) an arbitrary point not in the straight line $AC$. Construct the parallelograms $ABCD$ and $AECF$. Prove that $BE \parallel DF$.

Cono Sur 1991 P5
Given a square $ABCD$ with side $1$, and a square inside $ABCD$ with side $x$, find (in terms of $x$) the radio $r$ of the circle tangent to two sides of $ABCD$ and touches the square with side $x$. (See picture).
Cono Sur 1992 P2
Let $P$ be a point outside the circle $C$. Find two points $Q$ and $R$ on the circle, such that $P,Q$ and $R$ are collinear and $Q$ is the midpopint of the segmenet $PR$. (Discuss the number of solutions).

Cono Sur 1992 P5
In a $\triangle {ABC}$, consider a point $E$ in $BC$ such that $AE \perp BC$. Prove that $AE=\frac{bc}{2r}$, where $r$ is the radio of the circle circumscripte, $b=AC$ and $c=AB$.

Cono Sur 1993 P2
Consider a circle with centre $O$, and $3$ points on it, $A,B$ and $C$, such that $\angle {AOB}< \angle {BOC}$. Let $D$ be the midpoint on the arc $AC$ that contains the point $B$. Consider a point $K$ on $BC$ such that $DK \perp BC$. Prove that $AB+BK=KC$.

Cono Sur 1994 P2
Consider a circle $C$ with diameter $AB=1$. A point $P_0$ is chosen on $C$, $P_0 \ne A$, and starting in $P_0$ a sequence of points $P_1, P_2, \dots, P_n, \dots$ is constructed on $C$, in the following way: $Q_n$ is the symmetrical point of $A$ with respect of $P_n$ and the straight line that joins $B$ and $Q_n$ cuts $C$ at $B$ and $P_{n+1}$ (not necessary different). Prove that it is possible to choose $P_0$ such that:
i) $\angle {P_0AB} < 1$.
ii) In the sequence that starts with $P_0$ there are $2$ points, $P_k$ and $P_j$, such that $\triangle {AP_kP_j}$ is equilateral.

Cono Sur 1994 P6
Consider a $\triangle {ABC}$, with $AC \perp BC$. Consider a point $D$ on $AB$ such that $CD=k$, and the radius of the inscribe circles on $\triangle {ADC}$ and $\triangle {CDB}$ are equals. Prove that the area of $\triangle {ABC}$ is equal to $k^2$.

Cono Sur 1995 P3
Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$.
1. Find the distance between the centers of the circles(using $a$ and $b$).
2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.
Cono Sur 1995 P5
The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.

Cono Sur 1996 P1
In the following figure, the largest square is divided into two squares and three rectangles, as shown:
The area of ​​each smaller square is equal to $a$ and the area of each minor rectangle is equal to $b$. If $a+b=24$ and the root square from $a$ is a natural number, find all possible values ​​for the area of ​​the largest square.

Find all integers $n \leq 3$ such that there is a set $S_n$ formed by $n$ points of the plane that satisfy the following two conditions:
• Any three points are not collinear.
• No point is found inside the circle whose diameter has ends at any two points of $S_n$.
NOTE: The points on the circumference are not considered to be inside the circle.

Let $O$ be a center of a circle $C$, $AB$ a diameter of it and $R$ any point in $C$ different than $A$ and $B$. Let $P$ be the intersection of the perpendicular drawn from $O$ to $AR$. On the line $OP$ is marked the point $Q$, so that $QP$ is half of $PO$ and $Q$ does not belong to the segment $OP$. By $Q$ we draw the parallel to $AB$ that cuts the line $AR$ in $T$. We call $H$ the point of intersection of the lines $AQ$ and $OT$. Prove that $H$, $R$ and $B$ are collinear.

Let $ABC$ be a acute-angle triangle and $X$ be point in the plane of this triangle. Let $M,N,P$ be the orthogonal projections of $X$ in the lines that contains the altitudes of this triangle. Determine the positions of the point $X$ such that the triangle $MNP$ is congruent to $ABC$.

Cono Sur 1998 P2
Let H be the orthocenter of the triangle ABC, M is the midpoint of the segment BC. Let X be the point of the intersection of the line HM with arc BC(without A) of the circumcircle of ABC, let Y be the point of intersection of the line BH with the circle, show that XY = BC.

Cono Sur 1998 P2
Let $ABC$ be a triangle right in $A$. Construct a point $P$ on the hypotenuse $BC$ such that if $Q$ is the foor of the perpendicular drawn from $P$ to side $AC$, then the area of the square of side $PQ$ is equal to the area of the rectangle of sides $PB$ and $PC$. Show construction steps.

Cono Sur 2000 P4
In square $ABCD$ (labeled clockwise), let $P$ be any point on $BC$ and construct square $APRS$ (labeled clockwise). Prove that line $CR$ is tangent to the circumcircle of triangle $ABC$.

Cono Sur 2001 P3
Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Show that one can choose a vertex from each triangle so that the three chosen points determine a triangle each of whose angles is at most $90^\circ$.

A polygon of area $S$ is contained inside a square of side length $a$. Show that there are two points of the polygon that are a distance of at least $S/a$ apart.

Cono Sur 2000 P2
Given a triangle $ABC$, with right $\angle A$, we know: the point $T$ of tangency of the circumference inscribed in $ABC$ with the hypotenuse $BC$, the point $D$ of intersection of the angle bisector of $\angle B$ with side AC and the point E of intersection of the angle bisector of $\angle C$ with side $AB$ . Describe a construction with ruler and compass for points $A$, $B$, and $C$. Justify.

Cono Sur 2002 P4
Let ABCD be a convex quadrilateral such that your diagonals AC and BD are perpendiculars. Let P be the intersection of AC and BD, let M a midpoint of AB. Prove that the quadrilateral ABCD is cyclic, if and only if, the lines PM and DC are perpendicular.

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 $BR\leq{RS}$.

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 $ABC$.

Given a circle $C$ and a point $P$ on its exterior, two tangents to the circle are drawn through $P$, with $A$ and $B$ being the points of tangency. We take a point $Q$ on the minor arc $AB$ of $C$. Let $M$ be the intersection of $AQ$ with the line perpendicular to $AQ$ that goes through $P$, and let $N$ be the intersection of $BQ$ with the line perpendicular to $BQ$ that goes through $P$.
Show that, by varying $Q$ on the minor arc $AB$, all of the lines $MN$ pass through the same point.

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 $MP=RS-QW$.

Cono Sur 2005 P4
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 $r$.

Let $ABCD$ be a convex quadrilateral, let $E$ and $F$ be the midpoints of the sides $AD$ and $BC$, respectively. The segment $CE$ meets $DF$ in $O$. Show that if the lines $AO$ and $BO$ divide the side $CD$ in 3 equal parts, then $ABCD$ is a parallelogram.

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ where $D$, $E$, $F$ lie on $BC$, $AC$, $AB$, respectively. Let $M$ be the midpoint of $BC$. The circumcircle of triangle $AEF$ cuts the line $AM$ at $A$ and $X$. The line $AM$ cuts the line $CF$ at $Y$. Let $Z$ be the point of intersection of $AD$ and $BX$. Show that the lines $YZ$ and $BC$ are parallel.

Cono Sur 2007 P5
Let $ABCDE$ be a convex pentagon that satisfies all of the following:
• There is a circle $\Gamma$ tangent to each of the sides.
• The lengths of the sides are all positive integers.
• At least one of the sides of the pentagon has length $1$.
• The side $AB$ has length $2$.
Let $P$ be the point of tangency of $\Gamma$ with $AB$.
(a) Determine the lengths of the segments $AP$ and $BP$.
(b) Give an example of a pentagon satisfying the given conditions.

Cono Sur 2008 P2
Let $P$ be a point in the interior of triangle $ABC$. Let $X$, $Y$, and $Z$ be points on sides $BC$, $AC$, and $AB$ respectively, such that $<PXC=<PYA=<PZB$. Let $U$, $V$, and $W$ be points on sides $BC$, $AC$, and $AB$, respectively, or on their extensions if necessary, with $X$ in between $B$ and $U$, $Y$ in between $C$ and $V$, and $Z$ in between $A$ and $W$, such that $PU=2PX$, $PV=2PY$, and $PW=2PZ$. If the area of triangle $XYZ$ is $1$, find the area of triangle $UVW$.

Cono Sur 2008 P5
Let $ABC$ be an isosceles triangle with base $AB$. A semicircle $\Gamma$ is constructed with its center on the segment AB and which is tangent to the two legs, $AC$ and $BC$. Consider a line tangent to $\Gamma$ which cuts the segments $AC$ and $BC$ at $D$ and $E$, respectively. The line perpendicular to $AC$ at $D$ and the line perpendicular to $BC$ at $E$ intersect each other at $P$. Let $Q$ be the foot of the perpendicular from $P$ to $AB$. Show that  $\frac{PQ}{CP}=\frac{1}{2}\frac{AB}{AC}$.

Cono Sur 2009 P3
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.

Cono Sur 2008 P6
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.

Let us define  cutting  a convex polygon with $n$ sides by choosing a pair of consecutive sides $AB$ and $BC$ and substituting 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, the triangle $MBN$ is removed and a convex polygon with $n+1$ sides is obtained.
Let $P_6$ be a regular hexagon with area $1$. $P_6$ is  cut  and the polygon $P_7$ is obtained. Then $P_7$ is cut in one of seven ways and polygon $P_8$ is obtained, and so on. Prove that, regardless of how the cuts are made, the area of $P_n$ is always greater than $2/3$

The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D, E$, and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$, and $DCE$, respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$, respectively. Let $r_A$ be the line $E_{a}F_a$. Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$, $r_B$, and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$.

Let $ABC$ be an equilateral triangle. Let $P$ be a point inside of it such that the square root of the distance of $P$ to one of the sides is equal to the sum of the square roots of the distances of $P$ to the other two sides. Find the geometric place of $P$.

Let $ABC$ be a triangle and $D$ a point in $AC$. If $\angle{CBD} - \angle{ABD} = 60^{\circ}, \hat{BDC} = 30^{\circ}$ and also $AB \cdot BC = BD^{2}$, determine the measure of all the angles of triangle $ABC$.

Cono Sur 2012 P2
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 $\angle ASB=90$.

Cono Sur 2012 P6
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 $A$.

Cono Sur 2013 P2
In a triangle $ABC$, let $M$ be the midpoint of $BC$ and $I$ the incenter of $ABC$. If $IM$ = $IA$, find the least possible measure of $\angle{AIM}$.

Cono Sur 2013 P6
Let $ABCD$ be a convex quadrilateral. Let $n \geq 2$ be a whole number. Prove that there are $n$ triangles with the same area that satisfy all of the following properties:
a) Their interiors are disjoint, that is, the triangles do not overlap.
b) Each triangle lies either in $ABCD$ or inside of it.
c) The sum of the areas of all of these triangles is at least $\frac{4n}{4n+1}$ the area of $ABCD$.

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$.
Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other.
Note: the pieces can be rotated and flipped over.

Let $ABCD$ be an inscribed quadrilateral in a circumference with center $O$ such that it lies inside $ABCD$ and $\angle{BAC} = \angle{ODA}$. Let $E$ be the intersection of $AC$ with $BD$. Lines $r$ and $s$ are drawn through $E$ such that $r$ is perpendicular to $BC$, and $s$ is perpendicular to $AD$. Let $P$ be the intersection of $r$ with $AD$, and $M$ the intersection of $s$ with $BC$. Let $N$ be the midpoint of $EO$.
Prove that $M$, $N$, and $P$ lie on a line.

Cono Sur 2015 P3
Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all  integers $n \ge 1$ are defined:
• $C_n$ the foot of the perpendicular from $P$ to $A_nB_n$
• $O_n$ is the center of $\odot (PA_nB_n)$
• $A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$
• $B_{n+1} \equiv PB_n \cap O_nA_{n+1}$
If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$

Cono Sur 2015 P4
Let $ABCD$ be a convex quadrilateral such that $\angle{BAD} = 90^{\circ}$ and its diagonals $AC$ and $BD$ are perpendicular. Let $M$ be the midpoint of side $CD$, and $E$ be the intersection of $BM$ and $AC$. Let $F$ be a point on side $AD$ such that $BM$ and $EF$ are perpendicular. If $CE = AF\sqrt{2}$ and $FD = CE\sqrt{2}$, show that $ABCD$ is a square.

Cono Sur 2016 P5
Let $ABC$ be a triangle inscribed on a circle with center $O$. Let $D$ and $E$ be points on the sides $AB$ and $BC$,respectively, such that $AD = DE = EC$. Let $X$ be the intersection of the angle bisectors of $\angle ADE$ and $\angle DEC$. If $X \neq O$, show that, the lines $OX$ and $DE$ are perpendicular.

Let $A(XYZ)$ be the area of the triangle $XYZ$. A non-regular convex polygon $P_1 P_2 \ldots P_n$ is called  guayaco  if exists a point $O$ in its interior such that $A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).$  Show that, for every integer $n \ge 3$, a guayaco polygon of $n$ sides exists.

Cono Sur 2017 P4
Let $ABC$ an acute triangle with circumcenter $O$. Points $X$ and $Y$ are chosen such that:
• $\angle XAB = \angle YCB = 90^\circ$
• $\angle ABC = \angle BXA = \angle BYC$
• $X$ and $C$ are in different half-planes with respect to $AB$
• $Y$ and $A$ are in different half-planes with respect to $BC$.
Prove that $O$ is the midpoint of $XY$.

Cono Sur 2018 P1
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.

Cono Sur 2018 P5
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 $IH = BI + IC$