geometry problems from Rioplatense Mathematical Olympiads (levels 1-3)
with aops links in the names
Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.
2004 Rioplatense level 3 P3
In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that $ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).$
2005 Rioplatense level 3 P2
In trapezoid $ABCD$, the sum of the lengths of the bases $AB$ and $CD$ is equal to the length of the diagonal $BD$. Let $M$ denote the midpoint of $BC$, and let $E$ denote the reflection of $C$ about the line $DM$. Prove that $\angle AEB=\angle ACD$.
2005 Rioplatense level 3 P4
Let $P$ be a point inside triangle $ABC$ and let $R$ denote the circumradius of triangle $ABC$. Prove that $\frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.$
2005 Rioplatense level 3 P6
collected inside aops here
1990 - 2019
level 3
level 3
(I) 1990 Rioplatense level 3 P3
Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the point of intersection of the diagonals of the trapezium, show that points $P, Q$ and $R$ are collinear.
In 1991 it did not take place.
(II) 1992 Rioplatense level 3 P2
Let $D$ be the center of the circumcircle of the acute triangle $ABC$. If the circumcircle of triangle $ADB$ intersects $AC$ (or its extension) at $M$ and also $BC$ (or its extension) at $N$, show that the radii of the circumcircles of $\triangle ADB$ and $\triangle MNC$ are equal.
(II) 1992 Rioplatense level 3 P5
Let $ABC$ be an acute triangle.
Find the locus of the centers of the rectangles which have their vertices on the sides of $ABC$.
Let $S$ be the circle of center $O$ and radius $R$, and let $A, A'$ be two diametrically opposite points in $S$. Let $P$ be the midpoint of $OA'$ and $\ell$ a line passing through $P$, different from $AA '$ and from the perpendicular on $AA '$. Let $B$ and $C$ be the points of intersection of $\ell$ with $S$ and let $M$ be the midpoint of $BC$.
a) Let $H$ be the foot of the altitude from $A$ in the triangle $ABC$. Let $D$ be the point of intersection of the line $A'M$ with $AH$. Determine the locus of point $D$ while $\ell$ varies .
b) Line $AM$ intersects $OD$ at $I$. Prove that $2 OI = ID$ and determine the locus of point $I$ while $\ell$ varies .
(VI) 1997 Rioplatense level 3 P2
(VI) 1997 Rioplatense level 3 P4
Circles $c_1$ and $c_2$ are tangent internally to circle $c$ at points $A$ and $B$ , respectively, as seen in the figure. The inner tangent common to $c_1$ and $c_2$ touches these circles in $P$ and $Q$ , respectively. Show that the $AP$ and $BQ$ lines intersect the circle $c$ at diametrically opposite points.
(VII) 1998 Rioplatense level 3 P1
Consider an arc $AB$ of a circle $C$ and a point $P$ variable in that arc $AB$. Let $D$ be the midpoint of the arc $AP$ that doeas not contain $B$ and let $E$ be the midpoint of the arc $BP$ that does not contain $A$. Let $C_1$ be the circle with center $D$ passing through $A$ and $C_2$ be the circle with center $E$ passing through $B.$ Prove that the line that contains the intersection points of $C_1$ and $C_2$ passes through a fixed point.
(VII) 1998 Rioplatense level 3 P5
(VIII) 1999 Rioplatense level 3 P1
(VIII) 1999 Rioplatense level 3 P5
The quadrilateral $ABCD$ is inscribed in a circle of radius $1$, so that $AB$ is a diameter of the circumference and $CD = 1$. A variable point $X$ moves along the semicircle determined by $AB$ that does not contain $C$ or $D$. Determine the position of $X$ for which the sum of the distances from $X$ to lines $BC, CD$ and $DA$ is maximum.
(IX) 2000 Rioplatense level 3 P2
In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$
Note: $(XYZ)$ is the area of triangle $XYZ$.
(IX) 2000 Rioplatense level 3 P5
Let $ABC$ be a triangle with $AB < AC$, let $L$ be midpoint of arc $BC$(the point $A$ is not in this arc) of the circumcircle $w$($ABC$). Let $E$ be a point in $AC$ where $AE = \frac{AB + AC}{2}$, the line $EL$ intersects $w$ in $P$. If $M$ and $N$ are the midpoints of $AB$ and $BC$, respectively, prove that $AL, BP$ and $MN$ are concurrent.
(X) 2001 Rioplatense level 3 P2
Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$, points on the sides $BC, CA$ and $AB$, respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$. Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$, respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$, prove that $AB = AC$.
(X) 2001 Rioplatense level 3 P5
Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that: $\angle ANC + \angle ABC = 180$
2002 Rioplatense level 3 P2
Let $ABC$ be a triangle with $\angle C=60^o$. The point $P$ is the symmetric of $A$ with respect to the point of tangency of the circle inscribed with the side $BC$ . Show that if the perpendicular bisector of the $CP$ segment intersects the line containing the angle - bisector of $\angle B$ at the point $Q$, then the triangle $CPQ$ is equilateral.
Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Prove that $AA'$, $BB'$ and $CC'$ are concurrent.
2003 Rioplatense level 3 P4Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the point of intersection of the diagonals of the trapezium, show that points $P, Q$ and $R$ are collinear.
In 1991 it did not take place.
Let $D$ be the center of the circumcircle of the acute triangle $ABC$. If the circumcircle of triangle $ADB$ intersects $AC$ (or its extension) at $M$ and also $BC$ (or its extension) at $N$, show that the radii of the circumcircles of $\triangle ADB$ and $\triangle MNC$ are equal.
(II) 1992 Rioplatense level 3 P5
Let $ABC$ be an acute triangle.
Find the locus of the centers of the rectangles which have their vertices on the sides of $ABC$.
Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.
Let $ABCDE$ be pentagon such that $AE = ED$ and $BC = CD$. It is known that $\angle BAE + \angle EDC + \angle CB A = 360^o$ and that $P$ is the midpoint of $AB$.Show that the triangle $ECP$ is right.
In 1994 it did not take place.
(IV) 1995 Rioplatense level 3 P2
In a circle
of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed.
It is considered a triangle $A'B'C'$ whose sides have by length the
measurements of the segments $AB, CH$ and $2r$. Determine
the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.
Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: two at the ends and $n - 1$ inside. The following set of planes is considered:
$\bullet$ those that contain the faces of the tetrahedron, and
$\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.
Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.
Consider $2n$ points in the plane. Two players $A$ and $B$ alternately choose a point on each move. After $2n$ moves, there are no points left to choose from and the game ends.
Add up all the distances between the points chosen by $A$ and add up all the distances between the points chosen by $B$. The one with the highest sum wins.
If $A$ starts the game, describe the winner's strategy.
Clarification: Consider that all the partial sums of distances between points give different numbers.
A convex polygon with $2n$ sides is called rhombic if its sides are equal and all pairs of opposite sides are parallel.
A rhombic polygon can be partitioned into rhombic quadrilaterals.
For what value of$ n$, a $2n$-sided rhombic polygon splits into $666$ rhombic quadrilaterals?
Given a family $C$ of circles of the same radius $R$, which completely covers the plane (that is, every point in the plane belongs to at least one circle of the family), prove that there exist two circles of the family such that the distance between their centers is less than or equal to $R\sqrt3$ .
a) Let $H$ be the foot of the altitude from $A$ in the triangle $ABC$. Let $D$ be the point of intersection of the line $A'M$ with $AH$. Determine the locus of point $D$ while $\ell$ varies .
b) Line $AM$ intersects $OD$ at $I$. Prove that $2 OI = ID$ and determine the locus of point $I$ while $\ell$ varies .
Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA 1 , BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .
(VI) 1997 Rioplatense level 3 P4
Circles $c_1$ and $c_2$ are tangent internally to circle $c$ at points $A$ and $B$ , respectively, as seen in the figure. The inner tangent common to $c_1$ and $c_2$ touches these circles in $P$ and $Q$ , respectively. Show that the $AP$ and $BQ$ lines intersect the circle $c$ at diametrically opposite points.
Consider an arc $AB$ of a circle $C$ and a point $P$ variable in that arc $AB$. Let $D$ be the midpoint of the arc $AP$ that doeas not contain $B$ and let $E$ be the midpoint of the arc $BP$ that does not contain $A$. Let $C_1$ be the circle with center $D$ passing through $A$ and $C_2$ be the circle with center $E$ passing through $B.$ Prove that the line that contains the intersection points of $C_1$ and $C_2$ passes through a fixed point.
(VII) 1998 Rioplatense level 3 P5
We say that $M$ is the midpoint of the open polygonal $XYZ$, formed by the segments $XY, YZ$, if $M$ belongs to the polygonal and divides its length by half. Let $ABC$ be an acute triangle with orthocenter $H$. Let $A_1, B_1,C_1,A_2, B_2,C_2$ be the midpoints of the open polygonal $CAB, ABC, BCA, BHC, CHA, AHB$, respectively. Show that the lines $A_1 A_2, B_1B_2$ and $C_1C_2$ are concurrent.
Let $k$ be a fixed positive integer. For each $n = 1, 2,...,$ we will call configuration of order $n$ any set of $kn$ points of the plane, which does not contain $3$ collinear, colored with $k$ given colors, so that there are $n$ points of each color. Determine all positive integers $n$ with the following property: in each configuration of order $n$, it is possible to select three points of each color, such that the $k$ triangles with vertices of the same color that are determined are disjoint in pairs.
Let $ABC$ be a scalene acute triangle whose orthocenter is $H$. $M$ is the midpoint of segment $BC$. $N$ is the point where the segment $AM$ intersects the circle determined by $B, C$, and $H$. Show that lines $HN$ and $AM$ are perpendicular.
(VIII) 1999 Rioplatense level 3 P5
(IX) 2000 Rioplatense level 3 P2
In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$
Note: $(XYZ)$ is the area of triangle $XYZ$.
Let $ABC$ be a triangle with $AB < AC$, let $L$ be midpoint of arc $BC$(the point $A$ is not in this arc) of the circumcircle $w$($ABC$). Let $E$ be a point in $AC$ where $AE = \frac{AB + AC}{2}$, the line $EL$ intersects $w$ in $P$. If $M$ and $N$ are the midpoints of $AB$ and $BC$, respectively, prove that $AL, BP$ and $MN$ are concurrent.
(X) 2001 Rioplatense level 3 P2
Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$, points on the sides $BC, CA$ and $AB$, respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$. Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$, respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$, prove that $AB = AC$.
(X) 2001 Rioplatense level 3 P5
Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that: $\angle ANC + \angle ABC = 180$
2002 Rioplatense level 3 P2
Let $ABC$ be a triangle with $\angle C=60^o$. The point $P$ is the symmetric of $A$ with respect to the point of tangency of the circle inscribed with the side $BC$ . Show that if the perpendicular bisector of the $CP$ segment intersects the line containing the angle - bisector of $\angle B$ at the point $Q$, then the triangle $CPQ$ is equilateral.
Let $G$ be the circumcircle and $O$ the circumcenter of a triangle $ABC$ with $AC\ne BC$. The line tangent to $G$ passing through $C$ intersects the line $AB$ in $M$. The line perpendicular to $OM$ passing through $ M$ intersects lines $BC$ and $AC$ in $P$ and $Q$, respectively. Show that the segments $PM$ and $MQ$ are equal.
Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.
Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.
2004 Rioplatense level 3 P3
In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that $ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).$
In trapezoid $ABCD$, the sum of the lengths of the bases $AB$ and $CD$ is equal to the length of the diagonal $BD$. Let $M$ denote the midpoint of $BC$, and let $E$ denote the reflection of $C$ about the line $DM$. Prove that $\angle AEB=\angle ACD$.
2005 Rioplatense level 3 P4
Let $P$ be a point inside triangle $ABC$ and let $R$ denote the circumradius of triangle $ABC$. Prove that $\frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.$
2005 Rioplatense level 3 P6
Let $k$ be a positive integer. Show that for all $n>k$ there exist convex figures $F_{1},\ldots, F_{n}$ and $F$ such that there doesn't exist a subset of $k$ elements from $F_{1},..., F_{n}$ and $F$ is covered for this elements, but $F$ is covered for every subset of $k+1$ elements from $F_{1}, F_{2},....., F_{n}$.
2006 Rioplatense level 3 P2
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.
2006 Rioplatense level 3 P4
The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.
2006 Rioplatense level 3 P4
The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.
2007 Rioplatense level 3 P2
Let $ABC$ be a triangle with incenter $I$ . The circle of center $I$ which passes through $B$ intersects $AC$ at points $E$ and $F$, with $E$ and $F$ between $A $ and $C$ and different from each other. The circle circumscribed to triangle $IEF$ intersects segments $EB$ and $FB$ at $Q$ and $R$, respectively. Line $QR$ intersects the sides $A B$ and $B C$ at $P$ and $S$, respectively.
If $a , b$ and $c$ are the measures of the sides $B C, CA$ and $A B$, respectively, calculate the measurements of $B P$ and $B S$.
Let $ABC$ be a triangle with incenter $I$ . The circle of center $I$ which passes through $B$ intersects $AC$ at points $E$ and $F$, with $E$ and $F$ between $A $ and $C$ and different from each other. The circle circumscribed to triangle $IEF$ intersects segments $EB$ and $FB$ at $Q$ and $R$, respectively. Line $QR$ intersects the sides $A B$ and $B C$ at $P$ and $S$, respectively.
If $a , b$ and $c$ are the measures of the sides $B C, CA$ and $A B$, respectively, calculate the measurements of $B P$ and $B S$.
Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments.
Clarification : the vertices of the original triangle are not considered points of intersection or division.
In triangle $ABC$, where $AB<AC$, let $X$, $Y$, $Z$ denote the points where the incircle is tangent to $BC$, $CA$, $AB$, respectively. On the circumcircle of $ABC$, let $U$ denote the midpoint of the arc $BC$ that contains the point $A$. The line $UX$ meets the circumcircle again at the point $K$. Let $T$ denote the point of intersection of $AK$ and $YZ$. Prove that $XT$ is perpendicular to $YZ$.
2009 Rioplatense level 3 P2
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$ be nine points in space such that $ABCDE$, $ABFGH$, and $GFCDI$ are each regular pentagons with side length $1$. Determine the lengths of the sides of triangle $EHI$.
Acute triangle $ABP$, where $AB > BP$, has altitudes $BH$, $PQ$, and $AS$. Let $C$ denote the intersection of lines $QS$ and $AP$, and let $L$ denote the intersection of lines $HS$ and $BC$. If $HS = SL$ and $HL$ is perpendicular to $BC$, find the value of $\frac{SL}{SC}$.
2011 Rioplatense level 3 P2
Let $ABC$ an acute triangle and $H$ its orthocenter. Let $E$ and $F$ be the intersection of lines $BH$ and $CH$ with $AC$ and $AB$ respectively, and let $D$ be the intersection of lines $EF$ and $BC$. Let $\Gamma_1$ be the circumcircle of $AEF$, and $\Gamma_2$ the circumcircle of $BHC$. The line $AD$ intersects $\Gamma_1$ at point $I \neq A$. Let $J$ be the feet of the internal bisector of $\angle{BHC}$ and $M$ the midpoint of the arc $\stackrel{\frown}{BC}$ from $\Gamma_2$ that contains the point $H$. The line $MJ$ intersects $\Gamma_2$ at point $N \neq M$. Show that the triangles $EIF$ and $CNB$ are similar.
We consider $\Gamma_1$ and $\Gamma_2$ two circles that intersect at points $P$ and $Q$ . Let $A , B$ and $C$ be points on the circle $\Gamma_1$ and $D , E$ and $F$ points on the circle $\Gamma_2$ so that the lines $A E$ and $B D$ intersect at $P$ and the lines $A F$ and $C D$ intersect at $Q$. Denote $M$ and $N$ the intersections of lines $A B$ and $D E$ and of lines $A C$ and $D F$ , respectively. Show that $A M D N$ is a parallelogram.
2012 Rioplatense level 3 P2
A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?
2012 Rioplatense level 3 P3
Let $T$ be a non-isosceles triangle and $n \ge 4$ an integer . Prove that you can divide $T$ in $n$ triangles and draw in each of them an inner bisector so that those $n$ bisectors are parallel.
Let $ABCD$ be a square, and let $E$ and $F$ be points in $AB$ and $BC$ respectively such that $BE=BF$. In the triangle $EBC$, let N be the foot of the altitude relative to $EC$. Let $G$ be the intersection between $AD$ and the extension of the previously mentioned altitude. $FG$ and $EC$ intersect at point $P$, and the lines $NF$ and $DC$ intersect at point $T$. Prove that the line $DP$ is perpendicular to the line $BT$.
Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game.
For each positive integer $n$, find a winning strategy for one of the players.
Let $ABC$ be an acute scalene triangle, $H$ its orthocenter and $G$ its geocenter. The circumference with diameter $AH$ cuts the circumcircle of $BHC$ in $A'$ ($A' \neq H$). Points $B'$ and $C'$ are defined similarly. Show that the points $A'$, $B'$, $C'$, and $G$ lie in one circumference.
2014 Rioplatense level 3 P3
Kiko and Ñoño play with a rod of length $2n$ where $n \le 3$ is an integer. Kiko cuts the rod in $ k \le 2n$ pieces of integer lengths. Then Ñoño has to arrange these pieces so that they form a hexagon of equal opposite sides and equal angles. The pieces can not be split and they all have to be used. If Ñoño achieves his goal, he wins, in any other case, Kiko wins. Determine which victory can be secured based on $k$.
In the segment $A C$ a point $B$ is taken. Construct circles $T_1, T_2$ and $T_3$ of diameters $A B, BC$ and $AC$ respectively. A line that passes through $B$ cuts $T_3$ in the points $P$ and $Q$, and the circles $T_1$ and $T_2$ respectively at points $R$ and $S$. Prove that $PR = Q S$.
2015 Rioplatense level 3 P1
Let $ABC$ be a triangle and $P$ a point on the side $BC$. Let $S_1$ be the circumference with center $B$ and radius $BP$ that cuts the side $AB$ at $D$ such that $D$ lies between $A$ and $B$. Let $S_2$ be the circumference with center $C$ and radius $CP$ that cuts the side $AC$ at $E$ such that $E$ lies between $A$ and $C$. Line $AP$ cuts $S_1$ and $S_2$ at $X$ and $Y$ different from $P$, respectively. We call $T$ the point of intersection of $DX$ and $EY$. Prove that $\angle BAC+ 2 \angle DTE=180$
2015 Rioplatense level 3 P6
Let $A B C$ be an acut-angles triangle of incenter $I$, circumcenter $O$ and inradius $r.$ Let $\omega$ be the inscribed circle of the triangle $A B C$. $A_1$ is the point of ω such that $A IA_1O$ is a convex trapezoid of bases $A O$ and $IA_1$. Let $\omega_1$ be the circle of radius $r$ which goes through $A_1$, tangent to the line $A B$ and is different from $\omega$ . Let $\omega_2$ be the circle of radius $r$ which goes through $A_1$, is tangent to the line $A C$ and is different from $\omega$ . Circumferences $\omega_1$ and $\omega_2$ they are cut at points $A_1$ and $A_2$. Similarly are defined points $B_2$ and $C_2$. Prove that the lines $A A_2, B B_2$ and $CC2$ they are concurrent.
Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.
One have $n$ distinct circles(with the same radius) such that for any $k+1$ circles there are (at least) two circles that intersects in two points. Show that for each line $l$ one can make $k$ lines, each one parallel with $l$, such that each circle has (at least) one point of intersection with some of these lines.
Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$, and let $D$ be the reflection of $B$ to circumcenter of the circumcircle of$ABC$. The line $DI$ intersects again the circumcircle of $ABC$ in the point $Q$, show that $QP$ is angle bisector of the angle ∠AQC.
2018 Rioplatense level 3 P2
Let $P$ be a point outside a circumference $\Gamma$, and let $PA$ be one of the tangents from $P$ to $\Gamma$. Line $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$, with $B$ between $P$ and $C$. Let $D$ be the symmetric of $B$ with respect to $P$. Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$. Line $EB$ cuts back to $\omega _1 $ in $F$. Prove that $CF = AB$.
2018 Rioplatense level 3 P4
Let $ABC$ be an acute triangle with $AC> AB$. be $\Gamma$ the circumcircle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smallest arc $BC$ of this circumference. Let $E$ and $F$ points of the segments $AB$ and $AC$ respectively such that $AE = AF$. Let $P \neq A$ be the second point of intersection of the circumcircle circumscribed to $AEF$ with $\Gamma$. Let $G$ and $H$ be the intersections of lines $PE$ and $PF$ with $\Gamma$ other than $P$, respectively. Let $J$ and $K$ be the points of intersection of lines $DG$ and $DH$ with lines $AB$ and $AC$ respectively. Show that the $JK$ line passes through the midpoint of $BC$.
2019 Rioplatense level 3 P1
Let $ABCDEF$ be a regular hexagon, in the sides $AB$, $CD$, $DE$ and $FA$ we choose four points $P,Q,R$ and $S$ respectively, such that $PQRS$ is a square. Prove that $PQ$ and $BC$ are parallel.
(I) 1990 Rioplatense level 2 P32019 Rioplatense level 3 P1
2019 Rioplatense level 3 P5
Let $ABC$ be a triangle with $AB<AC$ and circuncircle $\omega$. Let $M$ and $N$ be the midpoints of $AC$ and $AB$ respectively and $G$ is the centroid of $ABC$. Let $P$ be the foot of perpendicular of $A$ to the line $BC$, and the point $Q$ is the intersection of $GP$ and $\omega$($Q,P,G$ are collinears in this order). The line $QM$ cuts $\omega$ in $M_1$ and the line $QN$ cuts $\omega$ in $N_1$. If $K$ is the intersection of $BM_1$ and $CN_1$ prove that $P$, $G$ and $K$ are collinear.
1990 - 2019
level 2
Let $ABC$ be a right triangle in $A$. Let $X$ be the base of the altitude corresponding to A and let $Y$ be the midpoint of $XC$. On the extension of the side $AB, D$ is the point such that $AB = BD$. Prove that the line determined by $D$ and $X$ is perpendicular to $AY$.
In 1991 it did not take place.
(II) 1992 Rioplatense level 2 P1
Let $P$ be a point inside the equilateral triangle $ABC$ and let $P_1, P_2$ and $P_3$ be the feet of the perpendiculars traced by $P$ to the sides $AB, BC$ and $CA$ respectively. Determine the locus of the points $P$ for which there exists a triangle $MNQ$ whose sides are congruent with the segments $PP_1, PP_2$ and $PP_3$.
(II) 1992 Rioplatense level 2 P6
$P$ is a point inside the square $ABCD$ such that $PA = 1, PB = 2$ and $PC = 3$ How much is the angle $\angle APB$?
(III) 1993 Rioplatense level 2 P3
Let $ABCD$ be a quadrilateral and $P$ a point inside it such that the triangles $ABP, BCP, CDP, DAP$ have the same area. Find the conditions that the quadrilateral must fulfill so that point $P$. exists.
In the figure $BMNP$ is a square of area $\sqrt{40}$. Further, $3 S_{(MCN)} - 2 S_{(PNA) }=11$.
Find the area of the triangle $ABC$.
Let $A, B$ and $C$ be three not collinear points.
Describe a procedure to construct a convex pentagon $ABCDE$ such that each of the lines containing a vertex and the midpoint of the opposite side divides said pentagon into two quadrilaterals of equal areas.
For each choice of $A, B$ and $C$ as above, is there such a pentagon? Is it unique?
Describe a procedure to construct a convex pentagon $ABCDE$ such that each of the lines containing a vertex and the midpoint of the opposite side divides said pentagon into two quadrilaterals of equal areas.
For each choice of $A, B$ and $C$ as above, is there such a pentagon? Is it unique?
Let $P$ be the point belonging to segment $AB$, such that $AP =\frac{AB}{3}$. An triangle $PAM$ isosceles at $A$, is constructed and the line $t$ is drawn perpendicular to $MP$ from $P$. Let $M'$ lie on $t$, in the same semiplane as $M$ wrt $AB$, such that the triangle $PBM' $ is isosceles at $B$. The lines $MM'$ and $AB$ intersect at $Q$. Find the locus of the centroid of the triangle $QM'B$.
Given a regular tetrahedron with edge $a$, its edges are divided into n equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:
$\bullet$ those that contain the faces of the tetrahedron, and
$\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.
Now all those points $P$ that belong (simultaneously) to four planes of that set are considered.
Two of these points $P$ are said to be “neighbors” if the distance between them is $\frac{a}{n}$ .
It is desired to color the points $P$ so that the neighboring points have different colors.
What is the minimum number of colors necessary and sufficient to do so? Justify your answer.
Given an $ABCD$ tetrahedron, determine all the interior points $P$ such that the product of the distances from $P$ to each of the faces of $ABCD$ is maximum.
(VI) 1997 Rioplatense level 2 P3Let $ABCD$ be a regular tetrahedron, $P$ and $Q$ different points in the planes $BCD$ and $ACD$ respectively. Prove that there is a triangle whose sides of measure $AP, PQ$ and $QB$.
(VII) 1998 Rioplatense level 2 P3
Given a circle $C$, choose a diameter $AB$ and mark an arbitrary point $P,$ other than $A$ and $B$. In one of the two arcs determined by the diameter $AB$, consider two points $M$ and $N$ such that $\angle APM =\angle BPN = 60^o$. Draw the segments $MP$ and $NP$ to obtain three curvilinear triangles $APM, MPN$ and $NPB$. (The sides of the curvilinear triangle $APM$ are the segments $AP$ and $PM$ and the arc $AM$). In each triangle inscirbe a circle.
Show that the sum of the radii of the three constructed circles is less than or equal to the radius of $C$.
(VII) 1998 Rioplatense level 2 P4
Let $ABCD$ be a square. In the semiplane determined by $AC$ containing $B$ a point $P$ is chosen such that $\angle APC = 90^o$ and $\angle PAC> 45^o$. Let $Q$ be the intersection point of $PC$ with $AB$ and $H$ the foot of the altitude corresponding to $Q$ in triangle $AQC$. Show that points $P, H$, and $D$ are collinear.
(VIII) 1999 Rioplatense level 2 P1
Consider two points $A$ and $B$ in the plane and $\ell$ a line such that $A$ and $B$ are in the same semi-plane determined by $\ell$ . Let $C$ be such that $A$ and $C$ are symmetric with respect to the line $\ell$ , $D$ the foot of the perpendicular from $B$ to $\ell$ , and $s$ the perpendicular of $BC$. The circumference of diameter $BC$ intersects $s$ in $E$ and $F$. Let $G$ and $H$ be the intersections of $\ell$ with the segments $CE$ and $CF$, respectively. Show that : $\angle AGC = 2 \angle BGE$ and $\angle AHC = 2 \angle BHF$
Consider two points $A$ and $B$ in the plane and $\ell$ a line such that $A$ and $B$ are in the same semi-plane determined by $\ell$ . Let $C$ be such that $A$ and $C$ are symmetric with respect to the line $\ell$ , $D$ the foot of the perpendicular from $B$ to $\ell$ , and $s$ the perpendicular of $BC$. The circumference of diameter $BC$ intersects $s$ in $E$ and $F$. Let $G$ and $H$ be the intersections of $\ell$ with the segments $CE$ and $CF$, respectively. Show that : $\angle AGC = 2 \angle BGE$ and $\angle AHC = 2 \angle BHF$
(VIII) 1999 Rioplatense level 2 P5
Let $ABC$ be a triangle and $D, E$ and $F$ points on the sides $BC, CA$ and $AB$, respectively. Let $D ', E'$ and $F'$ be the symmetric points to $D, E$ and $F$ with respect to the midpoints of $BC, CA$ and $AB$, respectively. Show that if triangle $DEF$ is congruent to triangle $D'E'F '$ and the following angles are equal: $D = D', E = E '$ and $F = F'$, then triangle $DEF$ is similar to triangle $ABC$.
Let $ABC$ be a triangle and $D, E$ and $F$ points on the sides $BC, CA$ and $AB$, respectively. Let $D ', E'$ and $F'$ be the symmetric points to $D, E$ and $F$ with respect to the midpoints of $BC, CA$ and $AB$, respectively. Show that if triangle $DEF$ is congruent to triangle $D'E'F '$ and the following angles are equal: $D = D', E = E '$ and $F = F'$, then triangle $DEF$ is similar to triangle $ABC$.
A square of side $1$ is divided into two parts by a straight line. Then one of those two parts is divided into two parts by another straight line, leaving three pieces altogether. Thus following, at each step a piece is taken and divided into two, by means of a straight line. After $1999$ of these divisions, the square is divided into $2000$ pieces. Prove that among the $2000$ pieces there is at least one that can completely cover a square of side $1/2000$.
Given a triangle $ABC$, let$ D$ and $E$ be interior points of the sides AB and AC respectively, such that $B, D, E$, and $C$ are concyclic. Let $F$ be the intersection of lines $BE$ and $CD$. The circles circumscribed to the triangles $ADF$ and $BCD$ are cut into $G$ and $D$. Show that the $GE$ line cuts the $AF$ segment at its midpoint.
Consider a triangle $ABC$ with acute angles $A$ and $B$. The bisectors of angles $A$ and $B$ cut $BC$ and $AC$ into $M$ and $N$ respectively. Let $P$ and $Q$ be points on the side $AB$ such that $MP$ is perpendicular to $AB$ and $NQ$ is perpendicular to $AB$. Knowing that the angle $\angle C> 2\angle PCQ$, calculate the measure of angle $\angle C$.
Let $\lambda$ be a segment with the following property: For any set consisting of red and blue segments such that the sum of the lengths of the red segments is $1$ and the sum of the lengths of the blue segments is also $1$, there is a way to locate all segments on $\lambda$ such that:
$\bullet$ Two segments of the same color have no interior points in common.
$\bullet$ Two segments of different colors have no interior points in common or one is included in the other.
Find the shortest possible length of segment $\lambda$ .
Esmeralda and her brother, Diamantino, play a variant of the game hot or cold. Initially, Diamantino places a box of chocolates somewhere on the plane and only informs Esmeralda that the box is at most $2001$ meters from Esmeralda's starting position.
Esmeralda is nearsighted and can only see the box if it is within $10$ centimeters of it. Her goal is to find the box using a sequence of steps, each one in any direction and measuring no more than $10$ centimeters (it's very short!). After each step, Esmeralda may or may not ask, “hot or cold?”. Diamantino must respond “hot” if the point where Esmeralda is is closer to the box than the point where she was before the last step and “cold” otherwise.
Prove that Esmeralda can find the box by taking no more than $2016$ steps and asking no more than $13$ questions.
Points $A, B, C$ and $D$, in that order, lie on the same line $r$. We consider all the triples of circles $\Gamma_1, \Gamma_2$ and $\Gamma$ with the property of $\Gamma_1$ passing through $A$ and $B$, $\Gamma_2$ passing through $C$ and $D$, $\Gamma$ passing through $B$ and $D$, and $\Gamma_1, \Gamma_2$ intersecting at points $X$ and $Y$ that are in different semiplanes with respect to $r$ and satisfy $\angle AXB =\angle CYD$.
Show that the line $XY$ passes through a fixed point.
2002 Rioplatense level 2 P2
Let $ABC$ be a triangle wth the angle \angle BAC=45^o. Let $P$ and $Q$ be interior points of triangle ABC such that $\angle ABQ = \angle QBP = \angle PBC$ and $ \angle ACQ = \angle QCP = \angle PCB$. Let $D$ and $E$ be the feet of the perpendiculars drawn from $P$ to the sides $CA$ and $AB$, respectively. Prove that $Q$ is the orthocenter of the triangle. $ADE$
2002 Rioplatense level 2 P5
Given a quadrilateral $ABCD$, isosceles triangles $ABK, BCL, CDM$ and $DAN$ are constructed, whose bases are the sides $AB, BC, CD$ and $DA$, and such that $K, L, M$ and $N$ are distinct points and there are not three of them aligned. The perpendicular on the line $KL$ traced by $B$ cuts the perpendicular on the line $LM$ traced from $C$ at the point $P$, the perpendicular on the line $MN$ drawn by $D$ cuts the perpendicular to the line $NK$ drawn from $A$ at point $Q$. Show that, if $P$ and $Q$ are different points, then $PQ$ is perpendicular to $KM$.
2003 Rioplatense level 2 P1
Let $ABC$ be a triangle with $AB=30, BC=50, CA=40$. Lines $\ell_a,\ell_b,\ell_c$ are parallel to $BC,CA,AB$ respectively and intersect the triangle. The distances between $\ell_a$ and $BC, \ell_b$ and $CA, \ell_c$ and $AB$ are $1,2,3$ respectively. Find the angles of the triangle that determine $\ell_a,\ell_b,\ell_c$ .
The quadrilateral $ABCD$ has its diagonals perpendicular and is inscribed in a circle $\Gamma$ of center $O$ . A line parallel to $BD$ intersects the segments $AO$ and $AD$ at $P$ and $Q$, respectively. Show that the lines $BP$ and $CQ$ intersect at a point of $\Gamma$.
2004 Rioplatense level 2 P1
Let $O$ be the circumcenter of triangle $ABC, c_1$ the circle passing through $B$ and tangent to the side $AC$ at point $A$ and $c_2$ the circle passing through $C$ and tangent to side $AB$ at point $A, c_1$ and $c_2$ are cut at points $A$ and $P$. Prove that if $O \ne P \ne A$ then $\angle OPA=90^o$.
2004 Rioplatense level 2 P4
Let a right angle ABC with $\angle C=90$, let $D$ the midpoint of $AC$, and $E,F$ points of the sides $AB, BC$ respectively such that $\angle DEA=\angle FEB$ , $\angle EFB=\angle AFC$, and $BC=3,AC=4$ .Find $\frac{DE+EF}{AF}$
2005 Rioplatense level 2 P1
Let $ABC$, a triangle such that $\angle A>45$, $\angle B>45$, If $PQRS$, is a square with $A,P,Q,B$ in a line in that order, $R$ in $BC$, $S$ in $AC$, Let $Q_{1}$ the foot of the perpendicular from $Q$ to $AC$, and $P_{1}$ the foot of the perpendicular from $P$ to $BC$, let $H$ the intersection of the lines $QQ_{1}$ and $PP_{1}$, prove that $CH$ is perpendicular to $AB$.
2008 Rioplatense level 2 P3
Let $ ABC$ be a triangle with $\angle BAC$ acute, $ O$ and $ H$ its circumcenter and orthocenter, respectively. The bisector of $ \angle BAC$ meets $ BC$ at $ P$. Find the product of the sides of the triangle, given that $ BC=1$ and $ AHPO$ and $ BHOC$ are concyclic.
2009 Rioplatense level 2 P1
Let $X$ be an interior point of triangle $ABC$, and let $Y$ be an interior point of triangle $AXC$ such that $\angle YAC= \angle XAB$ and $\angle YCA=\angle XCB$. Let $P$ be the symmetric of $X$ with respect to line $AB$. Let $Q$ be the symmetric of $X$ with respect to line $BC$. Show that the segments $PY$ and $QY$ are equal .
2009 Rioplatense level 2 P5
Let $ABCD$ be a convex quadrilateral, let $I_1$ be the incenter of triangle $ABD$, and let $I_2$ be the incenter of triangle $BDC$. It is known that the quadrilaterals $ABI_2D$ and $CBI_1D$ are cyclic. Show that lines $AC, BD$ and $I_1I_2$ are concurrent if and only if $ABCD$ is a parallelogram.
2010 Rioplatense level 2 P2
In the parallelogram $ABCD, G$ is located on the $AB$ side. Consider the circle that passes through $A$ and $G$ and is tangent to the extension of $CB$ at a point $P$. The extension of $DG$ intersects the circle at $L$. If the quadrilateral $GLBC$ is cyclic, prove that $AB = PC$.
2010 Rioplatense level 2 P4
In triangle $ABC$ the points $M$ and $N$ are in segments $AB$ and $AC$, respectively, so that $MN$ is parallel to $BC$ and tangent to the circle inscribed to triangle $ABC$. Let $K$ be the point at which the inscribed circle of the triangle $AMN$ is tangent to $MN$. It is known that $MN = 4, BC = 12$ and that $MK$ and $KN$ have integer lengths. Prove that triangle $ABC$ is equilateral or right..
2015 Rioplatense level 2 P3
Let $B$ and $C$ be two fixed points and $\Gamma$ a fixed circle such that line $BC$ has no common points with $\Gamma$. A point $A$ is chosen in $\Gamma$ such that $AB \ne BC$. Let $H$ be the orthocenter of triangle $ABC$. Let $X \ne H$ be the second point of intersection of the circumscribed circle of the triangle $BHC$ and the circle of diameter $AH$. Find the locus of point $X$ when $A$ varies by $\Gamma$.
2015 Rioplatense level 2 P5
Let $ABC$ be a triangle, $I$ its incenter and $D$ the foot of the perpendicular from $I$ to the side $BC$. Let $P$ and $Q$ be the orthocenters of triangles $AIB$ and $AIC$, respectively. Show that $P, Q$ and D$ are collinear.
Let $ABC$ be an acute-angled triangle, $\omega$ your incircle and $\omega'$ your excircle to the vertex $A$. The circles $\omega$ and $\omega'$ are tangents to $BC$ in $P$ and $P'$ respectively. Let $\Gamma$ be the circle that passes by $B$ and $C$ and is tangent to $\omega$ in $Q$, let $\Gamma'$ be the circle that passes by $B$ and $C$ and is tangent to $\omega'$ in $Q'$. The lines $PQ$ and $P'Q'$ intersects in $N$, prove that $AN$ is perpendicular to $BC$.
2018 Rioplatense level 2 P5
Let $ABC$ be an acute triangle and scalene triangle. The altitudes $BE$ and $CD$ that intersect at $H$ are drawn. The bisector of the angle $\angle BAC$ cuts the altitudes $BE$ and $CD$ at $P$ and $Q$ respectively. Let $T$ be the orthocenter of the triangle $HPQ$. Prove that $TDA$ and $ASD$ have the same area.
2019 Rioplatense level 2 P2
In a circle we mark, in this order, the points $A,B,C,D,E$ and $F$ such that $AB=BD, CE=EF$ and $BC>DE$. Let $H$ be a point in $AC$ such that $BH$ is perpendicular to $AE$. Prove that the lines $CD$, $BE$ and the parallel line to $AF$ by $H$ are concurrents
2004 Rioplatense level 2 P1
Let $O$ be the circumcenter of triangle $ABC, c_1$ the circle passing through $B$ and tangent to the side $AC$ at point $A$ and $c_2$ the circle passing through $C$ and tangent to side $AB$ at point $A, c_1$ and $c_2$ are cut at points $A$ and $P$. Prove that if $O \ne P \ne A$ then $\angle OPA=90^o$.
2004 Rioplatense level 2 P4
Let a right angle ABC with $\angle C=90$, let $D$ the midpoint of $AC$, and $E,F$ points of the sides $AB, BC$ respectively such that $\angle DEA=\angle FEB$ , $\angle EFB=\angle AFC$, and $BC=3,AC=4$ .Find $\frac{DE+EF}{AF}$
A closed broken line of the plane has $2004$ vertices, among which no three are aligned. Furthermore, no three of its sides intersect at a point. Determine the maximum number of crossings that this broken line can have.
Let $ABC$, a triangle such that $\angle A>45$, $\angle B>45$, If $PQRS$, is a square with $A,P,Q,B$ in a line in that order, $R$ in $BC$, $S$ in $AC$, Let $Q_{1}$ the foot of the perpendicular from $Q$ to $AC$, and $P_{1}$ the foot of the perpendicular from $P$ to $BC$, let $H$ the intersection of the lines $QQ_{1}$ and $PP_{1}$, prove that $CH$ is perpendicular to $AB$.
2005 Rioplatense level 2 P5
Let $ABC$ be such a triangle that ,when you construct the squares $ABB_1A_2, BCC_1B_2$ and $CAA_1C_2$ outside the triangle, points $A, B$, and $C$ are interior of the triangles $A_1B_1C_1$ and $A_2B_2C_2$. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ have the same area.
Let $ABC$ be such a triangle that ,when you construct the squares $ABB_1A_2, BCC_1B_2$ and $CAA_1C_2$ outside the triangle, points $A, B$, and $C$ are interior of the triangles $A_1B_1C_1$ and $A_2B_2C_2$. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ have the same area.
2006 Rioplatense level 2 P1
Let $ABC$ be a right triangle with right angle at $A$. Consider all the isosceles triangles $XYZ$ with right angle at $X$ , where $X$ lies on the segment $BC , Y$ lies on $AB,$ and $Z$ is on the segment $AC$ . Determine the locus of the midpoints of the hypotenuses $YZ$ of such triangles $XY Z$.
Let $ABC$ be a right triangle with right angle at $A$. Consider all the isosceles triangles $XYZ$ with right angle at $X$ , where $X$ lies on the segment $BC , Y$ lies on $AB,$ and $Z$ is on the segment $AC$ . Determine the locus of the midpoints of the hypotenuses $YZ$ of such triangles $XY Z$.
2006 Rioplatense level 2 P5
A circle $\Gamma$ is tangent to the sides $AB$ and $AC$ of triangle $ABC$ at $E$ and $F$ , respectively. Let $BF$ and $EC$ intersect at $X$ , let $\Gamma$ intersect $AX$ at $H$ , and let $EH$ and $FH$ intersect $BC$ at $Z$ and $T$ , respectively. The lines $ET$ and $FZ$ intersect at $Q$. Show that $Q$ lies on the line $AX$ .
2007 Rioplatense level 2 P2
Let $\Gamma$ be the circumscribed circle of the acute triangle $ABC$ and $P$ a point on the arc $BC$ that does not contain $A$. Let $K, L$ and $S$ be the feet of the perpendiculars from $P$ to the straight lines $AB, AC$ and $BC$ respectively. Let $M \ne P$ and $N \ne P$ be the intersections of $PK$ and$PL$ with $\Gamma$ respectively and $T$ the intersection of lines $KL$ and $MN$. Show that $OS = OT$, where $O$ is the center of $\Gamma$.
2007 Rioplatense level 2 P4
Let $ABC$ be an acute triangle, such that$ AB <AC$. A circle with diameter $AC$ is drawn, and on it a point $P$ such that $AP = AB$ and $P$ is in the semiplane determined by $AC$ that does not contain $B$. $BP$ cuts the circle again in $Q$, and $AQ$ cuts in $R$ the line perpendicular to $BC$ passing through $B$. Show that $BC$ and the bisectors of the angles $\angle BRC$ and $\angle BAC$ are concurrent.
A circle $\Gamma$ is tangent to the sides $AB$ and $AC$ of triangle $ABC$ at $E$ and $F$ , respectively. Let $BF$ and $EC$ intersect at $X$ , let $\Gamma$ intersect $AX$ at $H$ , and let $EH$ and $FH$ intersect $BC$ at $Z$ and $T$ , respectively. The lines $ET$ and $FZ$ intersect at $Q$. Show that $Q$ lies on the line $AX$ .
2007 Rioplatense level 2 P2
Let $\Gamma$ be the circumscribed circle of the acute triangle $ABC$ and $P$ a point on the arc $BC$ that does not contain $A$. Let $K, L$ and $S$ be the feet of the perpendiculars from $P$ to the straight lines $AB, AC$ and $BC$ respectively. Let $M \ne P$ and $N \ne P$ be the intersections of $PK$ and$PL$ with $\Gamma$ respectively and $T$ the intersection of lines $KL$ and $MN$. Show that $OS = OT$, where $O$ is the center of $\Gamma$.
2007 Rioplatense level 2 P4
Let $ABC$ be an acute triangle, such that$ AB <AC$. A circle with diameter $AC$ is drawn, and on it a point $P$ such that $AP = AB$ and $P$ is in the semiplane determined by $AC$ that does not contain $B$. $BP$ cuts the circle again in $Q$, and $AQ$ cuts in $R$ the line perpendicular to $BC$ passing through $B$. Show that $BC$ and the bisectors of the angles $\angle BRC$ and $\angle BAC$ are concurrent.
Let $ ABC$ be a triangle with $\angle BAC$ acute, $ O$ and $ H$ its circumcenter and orthocenter, respectively. The bisector of $ \angle BAC$ meets $ BC$ at $ P$. Find the product of the sides of the triangle, given that $ BC=1$ and $ AHPO$ and $ BHOC$ are concyclic.
2009 Rioplatense level 2 P1
Let $X$ be an interior point of triangle $ABC$, and let $Y$ be an interior point of triangle $AXC$ such that $\angle YAC= \angle XAB$ and $\angle YCA=\angle XCB$. Let $P$ be the symmetric of $X$ with respect to line $AB$. Let $Q$ be the symmetric of $X$ with respect to line $BC$. Show that the segments $PY$ and $QY$ are equal .
Let $ABCD$ be a convex quadrilateral, let $I_1$ be the incenter of triangle $ABD$, and let $I_2$ be the incenter of triangle $BDC$. It is known that the quadrilaterals $ABI_2D$ and $CBI_1D$ are cyclic. Show that lines $AC, BD$ and $I_1I_2$ are concurrent if and only if $ABCD$ is a parallelogram.
2010 Rioplatense level 2 P2
In the parallelogram $ABCD, G$ is located on the $AB$ side. Consider the circle that passes through $A$ and $G$ and is tangent to the extension of $CB$ at a point $P$. The extension of $DG$ intersects the circle at $L$. If the quadrilateral $GLBC$ is cyclic, prove that $AB = PC$.
2010 Rioplatense level 2 P4
There are $1000$ different points on a circle. We have to select $k$ of them so that there are not two adjacent ones among the chosen ones. In how many ways can it be done?
2011 Rioplatense level 2 P3
Let $ABCD$ be a convex quadrilateral and $M$ any point on the segment $AB$. The $BDM$ circuncírculos $ACM$ and cut again in $N$. Show that the line $MN$ passes through a fixed point by varying $M$ on $AB$.
2011 Rioplatense level 2 P4
Given three collinear points $A, B, C$, in that order, with $BC = 2AB$, we consider the circle $G$ of diameter $AB$. Given $P$ in $G$, let Q be the symmetric of $B$ with respect to $P,$ and let $R$ be the symmetric of $Q$ with respect to $A$. Finally, let $D$ be the point of intersection of lines $CR$ and $BQ$. Prove that triangles $BDC$ and $QPR$ are congruent.
2012 Rioplatense level 2 P1Let $ABCD$ be a convex quadrilateral and $M$ any point on the segment $AB$. The $BDM$ circuncírculos $ACM$ and cut again in $N$. Show that the line $MN$ passes through a fixed point by varying $M$ on $AB$.
Given three collinear points $A, B, C$, in that order, with $BC = 2AB$, we consider the circle $G$ of diameter $AB$. Given $P$ in $G$, let Q be the symmetric of $B$ with respect to $P,$ and let $R$ be the symmetric of $Q$ with respect to $A$. Finally, let $D$ be the point of intersection of lines $CR$ and $BQ$. Prove that triangles $BDC$ and $QPR$ are congruent.
In triangle $ABC$ the points $M$ and $N$ are in segments $AB$ and $AC$, respectively, so that $MN$ is parallel to $BC$ and tangent to the circle inscribed to triangle $ABC$. Let $K$ be the point at which the inscribed circle of the triangle $AMN$ is tangent to $MN$. It is known that $MN = 4, BC = 12$ and that $MK$ and $KN$ have integer lengths. Prove that triangle $ABC$ is equilateral or right..
2013 Rioplatense level 2 P3
Let $ABCD$ be a cyclic convex quadrilateral with all its different sides. Let $I$ and $J$ be the incenters of the triangles $ABC$ and $ADC$, respectively. Prove that $ABCD$ is tangential (that is, it has a circle tangent to its four sides) if and only if the quadrilateral $BIJD$ is cyclic.
2013 Rioplatense level 2 P5
Let $AB$C be a triangle. The $A$-excircle of $ABC$ is tangent to the side $BC$ in $D$, and to the extensions of the sides $AC$ and $AB$ in $E$ and $F$, respectively. Let $M$ be a point on the circumcircle circumscribed to the triangle $ABC$, such that the arcs $BM$ and $CM$ are equal, and that $A$ and $M$ belong to the same semiplane with respect to the line $BC$. The line $MD$ intersects the circle circumscribed to $ABC$ again in $G$, and the line $AG$ cuts to $EF$ in $H$. Show that $HD$ is perpendicular to $EF$.
2014 Rioplatense level 2 P4
The diagonals $AC$ and $BD$ of a parallelogram $ABCD$ are cut in $O$. The circle passing through points $A, B$ and $O$ intersects the line $AD$ in $E$. The circle that passes through points $D, O$ and $E$ intersects the line $BE$ in $F$. Prove that $\angle BCA = \angle FCD$.
Let $ABCD$ be a cyclic convex quadrilateral with all its different sides. Let $I$ and $J$ be the incenters of the triangles $ABC$ and $ADC$, respectively. Prove that $ABCD$ is tangential (that is, it has a circle tangent to its four sides) if and only if the quadrilateral $BIJD$ is cyclic.
2013 Rioplatense level 2 P5
Let $AB$C be a triangle. The $A$-excircle of $ABC$ is tangent to the side $BC$ in $D$, and to the extensions of the sides $AC$ and $AB$ in $E$ and $F$, respectively. Let $M$ be a point on the circumcircle circumscribed to the triangle $ABC$, such that the arcs $BM$ and $CM$ are equal, and that $A$ and $M$ belong to the same semiplane with respect to the line $BC$. The line $MD$ intersects the circle circumscribed to $ABC$ again in $G$, and the line $AG$ cuts to $EF$ in $H$. Show that $HD$ is perpendicular to $EF$.
Consider a convex polygon with $2013$ sides, in which there are no three diagonals that have a common point inside the polygon. Several diagonals are drawn so that all the parts into which the polygon is divided are triangles. Find the maximum number of diagonals drawn.
In the plane $77$ lines are given in general position (there are not two parallel and there are not three concurrent). The plane is divided into disjoint regions that none of these lines crosses. Two of those bounded regions that determine are polygons of $k$ sides each. What is the largest value of $k$ for which this is possible?
The diagonals $AC$ and $BD$ of a parallelogram $ABCD$ are cut in $O$. The circle passing through points $A, B$ and $O$ intersects the line $AD$ in $E$. The circle that passes through points $D, O$ and $E$ intersects the line $BE$ in $F$. Prove that $\angle BCA = \angle FCD$.
Pedro wants to color the vertices of a $2015$-sided regular polygon with $3$ colors so that there is no triangle with vertices of three different colors that contains the center of the polygon. Decide if this can be achieved.
Clarification: Each of the three colors must be used at least once.
Let $B$ and $C$ be two fixed points and $\Gamma$ a fixed circle such that line $BC$ has no common points with $\Gamma$. A point $A$ is chosen in $\Gamma$ such that $AB \ne BC$. Let $H$ be the orthocenter of triangle $ABC$. Let $X \ne H$ be the second point of intersection of the circumscribed circle of the triangle $BHC$ and the circle of diameter $AH$. Find the locus of point $X$ when $A$ varies by $\Gamma$.
Let $ABC$ be a triangle, $I$ its incenter and $D$ the foot of the perpendicular from $I$ to the side $BC$. Let $P$ and $Q$ be the orthocenters of triangles $AIB$ and $AIC$, respectively. Show that $P, Q$ and D$ are collinear.
2016 Rioplatense level 2 P2
Let $ABC$ be an acute triangle and $X$ an interior point such that $\angle AXB = 90^o + \angle ACB$. Let $M$ and $N$ be the feet of the perpendiculars drawn from $X$ to the sides $BC$ and $CA$, respectively. Let $Y$ be the point of intersection of the straight lines passing through $M$ and $N$ perpendicular to $AC$ and $BC$ respectively. Show that $\angle AYB = 90^o$
2016 Rioplatense level 2 P5
Let $n$ be a positive integer. In the interior of a convex polygon there are exactly $4n^2+1$ points of integer coordinates. Each of these points is painted red or blue. Prove that it is possible to choose $n+1$ points of the same color that are collinear.
Let $ABC$ be an acute triangle and $X$ an interior point such that $\angle AXB = 90^o + \angle ACB$. Let $M$ and $N$ be the feet of the perpendiculars drawn from $X$ to the sides $BC$ and $CA$, respectively. Let $Y$ be the point of intersection of the straight lines passing through $M$ and $N$ perpendicular to $AC$ and $BC$ respectively. Show that $\angle AYB = 90^o$
2016 Rioplatense level 2 P5
Let $n$ be a positive integer. In the interior of a convex polygon there are exactly $4n^2+1$ points of integer coordinates. Each of these points is painted red or blue. Prove that it is possible to choose $n+1$ points of the same color that are collinear.
2017 Rioplatense level 2 P2
Let $ABCD$ be a convex quadrilateral with sides $AB, BC, CD, DA$, inscribed in a circle. Lines $BC$ and $AD$ are cut in $G$ ($C$ is between $B$ and $G$). Lines $AB$ and $CD$ are cut at $E$ ($A$ is between $B$ and $E$). The circle of diameter $EG$ cuts the line $BC$ at $H$ ($H \ne G$) and $EC$ at $I$ ($I \ne E$). Show that line $HI$ passes through the midpoint of $BD$.
2017 Rioplatense level 2 P4
Let $ABCD$ be a convex quadrilateral, with sides $AB, BC, CD, DA$, such that $\angle A = \angle C$ and $AB = AD$. On the bisector of the angle $\angle BCD$ are the points $P$ and $Q$, other than$ C$, such that $BC = BP$ and $DC = DQ$. Show that $AP = AQ$.
2018 Rioplatense level 2 P3Let $ABCD$ be a convex quadrilateral with sides $AB, BC, CD, DA$, inscribed in a circle. Lines $BC$ and $AD$ are cut in $G$ ($C$ is between $B$ and $G$). Lines $AB$ and $CD$ are cut at $E$ ($A$ is between $B$ and $E$). The circle of diameter $EG$ cuts the line $BC$ at $H$ ($H \ne G$) and $EC$ at $I$ ($I \ne E$). Show that line $HI$ passes through the midpoint of $BD$.
Let $ABCD$ be a convex quadrilateral, with sides $AB, BC, CD, DA$, such that $\angle A = \angle C$ and $AB = AD$. On the bisector of the angle $\angle BCD$ are the points $P$ and $Q$, other than$ C$, such that $BC = BP$ and $DC = DQ$. Show that $AP = AQ$.
Let $ABC$ be an acute-angled triangle, $\omega$ your incircle and $\omega'$ your excircle to the vertex $A$. The circles $\omega$ and $\omega'$ are tangents to $BC$ in $P$ and $P'$ respectively. Let $\Gamma$ be the circle that passes by $B$ and $C$ and is tangent to $\omega$ in $Q$, let $\Gamma'$ be the circle that passes by $B$ and $C$ and is tangent to $\omega'$ in $Q'$. The lines $PQ$ and $P'Q'$ intersects in $N$, prove that $AN$ is perpendicular to $BC$.
2018 Rioplatense level 2 P5
Let $ABC$ be an acute triangle and scalene triangle. The altitudes $BE$ and $CD$ that intersect at $H$ are drawn. The bisector of the angle $\angle BAC$ cuts the altitudes $BE$ and $CD$ at $P$ and $Q$ respectively. Let $T$ be the orthocenter of the triangle $HPQ$. Prove that $TDA$ and $ASD$ have the same area.
2019 Rioplatense level 2 P2
In a circle we mark, in this order, the points $A,B,C,D,E$ and $F$ such that $AB=BD, CE=EF$ and $BC>DE$. Let $H$ be a point in $AC$ such that $BH$ is perpendicular to $AE$. Prove that the lines $CD$, $BE$ and the parallel line to $AF$ by $H$ are concurrents
1990 - 2019
level 1
Calculate the striped area. Given:
$A$ center of the cicle $I$. $B$ center of the circle $II$.
$B$ belongs to the circle $I$. $CD = 4$ diameter of the circle $I$
In 1991 it did not take place.$A$ center of the cicle $I$. $B$ center of the circle $II$.
$B$ belongs to the circle $I$. $CD = 4$ diameter of the circle $I$
The pieces of a rectangular puzzle are $9$ squares with sides $1$, $4$, $7$, $8$, $9$, $10$, $14$, $15$ and $18$. How should the $9$ pieces be placed to assemble the puzzle?
The
triangles $AHF, HCF, DEG, EGB$, are isosceles with the measurement of their
bases equal to those of their heights and equal to $b$. Calculate the area of
the striped area in terms of $b$.
Mark on the map where you should place the deposit and justify why.
Consider a regular polygon with $70$ sides.
(a) Prove that there are diagonals that intersect inside the polygon not forming angles of $72^\circ$.
(b) Determine how many are the pairs of diagonals existing in the conditions of (a).
.
It is considered a rectangle $ABCD$ whose sides measure $a$ and $b$.
Four right isosceles triangles are constructed externally to the rectangle:
$BAQ$ rectangle in $A$, $CBR$ rectangle in $B$ , $DCS$ rectangle in $C$, $ADP$ rectangle in $D$. Let $T$ be the foot of the perpendicular to the line $PQ$ passing through$ R$.
Show that perimeter $(ABCD)=2 \sqrt{QP \cdot RT - ab}$
(IV) 1995 Rioplatense level 1 P5
Four right isosceles triangles are constructed externally to the rectangle:
$BAQ$ rectangle in $A$, $CBR$ rectangle in $B$ , $DCS$ rectangle in $C$, $ADP$ rectangle in $D$. Let $T$ be the foot of the perpendicular to the line $PQ$ passing through$ R$.
Show that perimeter $(ABCD)=2 \sqrt{QP \cdot RT - ab}$
(IV) 1995 Rioplatense level 1 P5
Let ABC be an isosceles triangle with $AB = AC$ and $\angle A = 36^o$. Draw the bisector of $B$ that cuts $AC$ into $D$ and draw the bisector of $BDC$ that cuts $BC$ into $P$. A point $R$ is marked on line $BC$ such that $B$ is the midpoint of segment PR. Explain why the $RD$ and $AP$ segments have the same measure.
On a table there are $n$ equal coins, all of radius $1$. A dwarf ant walks on the table and skirts all the coins. In certain places, he can't see any other coin, apart from the one he are skirting, because it obstructs his vision. These places correspond to stretches of the edge of some coins, that is, to arches of some coins. Find the sum of the lengths of all these arcs.
Two regular polygons of side $1$ are called friendly if:
$\bullet$ They have a side in common that leaves them in opposite half-planes:
$\bullet$ The two sides, one from each polygon, that meet at a vertex of the common side, are sides of an equilateral triangle
Find all pairs of friendly polygons.
Ana and Celia play as follows: Ana must color all the points of a circumference in red and blue. Celia must choose three points on the colored circumference that determine a triangle whose angles measure $30^o$, $50^o$ and $100^o$. Celia wins if that triangle has all three vertices of the same color (red or blue). Ana wins otherwise. Can Ana color the circumference so that Celia cannot win? If not, explain why. If so, it shows a coloration.
With two equal pyramids with a square base and all their edges (sides) equal, a regular octahedron is formed by gluing the two bases together. On each edge of the octahedron, two points are marked that divide them into three equal segments. The $24$ marked points are the vertices of a new polyhedron, which turns out to cut out six small equal pyramids, one for each vertex of the octahedron. How many interior diagonals does the new polyhedron have?
Note: We call the interior diagonal of a polyhedron any segment that joins two vertices and is not contained in any face.
A circumference has three points painted red $A$, $B$, $C$, in a clockwise direction. We will paint another $1996$ points red as follows:
We traverse the circumference clockwise, starting from $C$. We go through a painted point ($A$) and paint $P_1$ at the midpoint of the arc $AB$. We continue to traverse the circumference in the same direction, passing through two painted points ($B$ and $C$), and we paint $P_2$ at the midpoint of the arc $CA$. Next we go through three painted points ($A$, $P_1$ and $B$) and paint $P_3$ at the midpoint of the arc $BC$, and so on until, after having painted $P_{1995}$, we go through $1996$ painted points and paint $P_{1996}$at the midpoint of the corresponding arc.
Determine how many of the $1996$ points were painted on each of the arcs $AB$, $BC$ and $CA$.
In a rectangle $ABCD$, the midpoint of the $CD$ side is $F$ and $E$ is a point on the $BC$ side such that $AF$ is bisector of the angle $\angle EAD$ . Show that $AF$ is perpendicular to $EF$
(VI) 1997 Rioplatense level 1 P1In a square $ABCD$ of area $1, E$ is the midpoint of $DC, G$ is the midpoint of $AD, F$ is the point of side $BC$ such that $3CF = FB$ and $O$ is the point of intersection between $FG$ and $AE$. Find the area of the $EFO$ triangle.
There is a triangle $T$ drawn on a sheet of white paper and two sheets of light blue paper are available. On each sheet of blue paper it is allowed to draw a single triangle, similar to $T$ (that is, with angles equal to those of $T$), but smaller, and then cut it out. Is it always possible to get two triangles of light blue paper (perhaps different from each other) and place them on the white sheet, so that triangle $T$ is completely covered? Justify.
Clarification: The light blue triangles may or may not overlap, and may or may not protrude from the drawn triangle
Agustina and Santiago play the following game on a rectangular sheet:
Agustina says a natural number $n$.
Santiago marks $n$ points on the sheet.
Agustina chooses some of the points marked by Santiago.
Santiago wins the game if he manages to draw a rectangle with sides parallel to the edges of the sheet, which contains all the points chosen by Agustina and does not contain any of the remaining points. Otherwise, Gustine wins the game.
What is the smallest number Gustine can say to ensure that she can win the game regardless of how James drew the points? Justify.
In a regular $ABCDE$ pentagon, we draw the diagonals $AC$ and $BE$, which intersect at point $P.$ We cut the triangle APB and get the hexagon $APBCDE$. We have an infinite collection of pieces equal to the $APBCDE$ hexagon in size and shape. It shows that the plane can be tiled with these pieces.
NOTE: Tiling the plane means covering it with pieces without overlapping or leaving gaps.
We have a square of side $1999$. Is it possible to divide it completely into several squares (more than one) that have sides of integer lengths greater than $35$? Justify your answer.
Clarification: The squares can be of different sizes.
In an $ABCD$ trapezoid of bases $AB$ and $CD$ the points $M$ and $N$ are chosen on the sides $AD$ and $BC$, respectively, so that $AM / MD = CN / NB$ . If $MN$ intersects the diagonals $AC$ and $BD$ in $P$ and $Q$, respectively, it shows that $MP = NQ$.
(VIII) 1999 Rioplatense level 1 P2
Let $AB$ be a segment with midpoint $M$. On the bisector of $AB$, take a point $O$ such that $OM = AM$. Let $C$ be a circle of center $O$ and radius smaller than $OM$. For $A$ the straight line $AP$ tangent to $C$ in $P$ is drawn and it does not cut the $OM$ segment. By $B$ the straight line $BQ$ tangent to $C$ is drawn in $Q$ and that cuts the $OM$ segment. Show that $AP$ and $BQ$ are perpendicular.
Let $AB$ be a segment with midpoint $M$. On the bisector of $AB$, take a point $O$ such that $OM = AM$. Let $C$ be a circle of center $O$ and radius smaller than $OM$. For $A$ the straight line $AP$ tangent to $C$ in $P$ is drawn and it does not cut the $OM$ segment. By $B$ the straight line $BQ$ tangent to $C$ is drawn in $Q$ and that cuts the $OM$ segment. Show that $AP$ and $BQ$ are perpendicular.
On three sheets of paper there is an identical drawing to the following:
Pablo must draw a circle on the same side as the segment $AC$ with respect to straight line $\ell$, and Sofia must build parallelograms that have the segment $AC$ as a diagonal, a vertex on the circumfurence and the opposite vertex on the line $\ell$.
On each page Paul draws a circle so that in the first, Sofia can draw exactly two different parallelograms with the indicated conditions, in the second only one and in the third none (the circles are not necessarily equal or in the same position).
Where does Paul draw the circles and how does Sofia construct the parallelograms in each case? Indicate in detail the constructions of Sofia.
Determine all positive integers $n$ with the following property: every convex polygon with $n$ sides can be divided into triangles by some diagonals that do not intersect inside the polygon, in such a way that each vertex of the polygon is an endpoint of an even number of these diagonals.
Let $ABC$ be a triangle with $AB = AC$. With center in a point of the side $BC$, the circle $S$ is constructed that is tangent to the sides $AB$ and $AC$. Let $P$ and $Q$ be any points on the sides $AB$ and $AC$ respectively, such that $PQ$ is tangent to $S$. Show that $PB \cdot CQ = \left(\frac{BC}{2}\right)^2$
(X) 2001 Rioplatense level 1 P1
In a triangle $ABC, M$ is the midpoint of the side $AC$ and $N$ is the point of the side $BC$ such that $CN = 2BN$. If $P$ is the point of intersection of lines $AB$ and $MN$, show that the line $AN$ cuts the segment $PC$ at its midpoint.
(X) 2001 Rioplatense level 1 P5
Let $AD$ be the altitude related to the side $BC$ of an acute triangle $ABC$. $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. Let $E$ be the second point of intersection of the circles circumscribed around the triangles $BDM$ and $CDN$. Show that the line $DE$ passes through the midpoint of $MN$.
2002 Rioplatense level 1 P2
Let $ABCD$ be a rectangle with $AB> BC$, and $O$ the point of intersection of its diagonals $AC$ and $BD$. The bisector of the angle $\angle BAC$ intersects $BD$ to $E$. Let $M$ be the midpoint of $AB$. From point $E$ the perpendicular on $AB$, cuts $AB$ at $F$, from $E$ the perpendicular on $AE$, cuts $AC$ into $H$. If $OH=a$ and $MF= 4/3 a$ are given, calculate the area of the rectangle $ABCD$ in terms of $a$.
2002 Rioplatense level 1 P4
There are two equal convex quadrilaterals of paper: $ABCD$ and $A'B'C'D '$ ($AB = A'B', BC = B'C ', CD = C'D', DA = D'A '$). The quadrilateral $ABCD$ is cut by the diagonal $AC$ and the quadrilateral $A'B'C'D '$ is cut by the diagonal $B'D'$, thus obtaining four pieces of paper.
(a) Indicates a procedure, which does not depend on the particular shape of the convex quadrilateral $ABCD$, which allows to assemble a parallelogram with the four pieces of paper, without gaps or overlaps.
(b) If the sides of the quadrilaterals measure $3, 3, 4$ and $6$, shows that a parallelogram can be built such that its perimeter is greater than $16$ and less than $28$, whatever the order of the sides
2003 Rioplatense level 1 P5
Let $ABC$ be a triangle inscribed in a circle. Let $M$ be the midpoint of the arc $AB$ that does not contain $C$ and $N$ the midpoint of the arc $AC$ that does not contain $B$. Let $E$ and $F$ be the points where the line $MN$ intersects the sides $AB$ and $AC$ respectively. Show that if $ME = EF = FN$, then triangle $ABC$ is equilateral.
2004 Rioplatense level 1 P2
The trapezium $ABCD$ of bases $AB$ and $CD$ and non-parallel sides $BC$ and $AD$, has $\angle BCD = \angle ADB =90^o$. The line perpendicular to $BD$ that passes through $C$ and the line perpendicular to $AC$ passing through $D$ intersect at a point on the side $AB$. Show that $\frac{AD}{AB}-\frac{AB}{AD}=1$
2004 Rioplatense level 1 P4
Let $ABC$ be an acute triangle, $O$ an interior point of the triangle and $M$ a point on the segment $AB$. Circles circumscribed to triangles $AOM$ and $BOM$ cut segments $AC$ and $BC$ into $P$ and into $N$, respectively. Show that $\angle PON + \angle PCN =180^o$
2005 Rioplatense level 1 P6
Let $I$ be the incenter of a triangle $ABC$, and $D$ the point of intersection of $AI$ with the circle circumscribed around $ABC$. Let $M$ be the midpoint of $AD$, and $E$ the point of the segment $BD$ such that $IE$ is perpendicular to $BD$. If $IB + IE = \frac{AD}{2}$, $ME$ is parallel to $AB$, and the point $M$ is inside the segment $AI$, determine the measure of the angles of the triangle $ABC$.
Let $ABC$ be an obtuse triangle at $C$ so that, $2 \angle BAC=\angle ABC$. $P$ is a point at side $AB$ so that $BP=2BC$. $M$ is the midpoint of $AB$ ($M$ is between $P$ and $B$). Prove that the perpendicular to the side $AC$ from $M$ cuts $PC$ at his midpoint.
Let $ABC$ be a triangle inscribed in a circle. Let $M$ be the midpoint of the arc $AB$ that does not contain $C$ and $N$ the midpoint of the arc $AC$ that does not contain $B$. Let $E$ and $F$ be the points where the line $MN$ intersects the sides $AB$ and $AC$ respectively. Show that if $ME = EF = FN$, then triangle $ABC$ is equilateral.
2004 Rioplatense level 1 P2
The trapezium $ABCD$ of bases $AB$ and $CD$ and non-parallel sides $BC$ and $AD$, has $\angle BCD = \angle ADB =90^o$. The line perpendicular to $BD$ that passes through $C$ and the line perpendicular to $AC$ passing through $D$ intersect at a point on the side $AB$. Show that $\frac{AD}{AB}-\frac{AB}{AD}=1$
2004 Rioplatense level 1 P4
Let $ABC$ be an acute triangle, $O$ an interior point of the triangle and $M$ a point on the segment $AB$. Circles circumscribed to triangles $AOM$ and $BOM$ cut segments $AC$ and $BC$ into $P$ and into $N$, respectively. Show that $\angle PON + \angle PCN =180^o$
2005 Rioplatense level 1 P6
Let $I$ be the incenter of a triangle $ABC$, and $D$ the point of intersection of $AI$ with the circle circumscribed around $ABC$. Let $M$ be the midpoint of $AD$, and $E$ the point of the segment $BD$ such that $IE$ is perpendicular to $BD$. If $IB + IE = \frac{AD}{2}$, $ME$ is parallel to $AB$, and the point $M$ is inside the segment $AI$, determine the measure of the angles of the triangle $ABC$.
2006 Rioplatense level 1 P2
Let $ABC$ be a triangle and $H$ be the intersection point of it's altitudes. It is known that $\angle BAC=60^o$. Let $J$ a point that lies on $AC$ such that $AJ = 2 JC$. It is also true thath $JH=JC$.
Given the location of points $A$ and $H$, construct the triangle $ABC$ using a ruler and a compass.
2007 Rioplatense level 1 P2
A polygon with twelve sides, whose vertices belong to a circle $C$, has six sides of length $2$ and six sides of length $\sqrt3$. Calculate the radius of the circle $C$.
2008 Rioplatense level 1 P2Let $ABC$ be a triangle and $H$ be the intersection point of it's altitudes. It is known that $\angle BAC=60^o$. Let $J$ a point that lies on $AC$ such that $AJ = 2 JC$. It is also true thath $JH=JC$.
Given the location of points $A$ and $H$, construct the triangle $ABC$ using a ruler and a compass.
A polygon with twelve sides, whose vertices belong to a circle $C$, has six sides of length $2$ and six sides of length $\sqrt3$. Calculate the radius of the circle $C$.
Let $ABC$ be an obtuse triangle at $C$ so that, $2 \angle BAC=\angle ABC$. $P$ is a point at side $AB$ so that $BP=2BC$. $M$ is the midpoint of $AB$ ($M$ is between $P$ and $B$). Prove that the perpendicular to the side $AC$ from $M$ cuts $PC$ at his midpoint.
2008 Rioplatense level 1 P6
Is it possible to color the points in the plane that have integer coordinates with three colors (all three colors must be used) such that there is no right triangle with all three vertices of different colors?
2009 Rioplatense level 1 P1
$ABC$ is a triangle whose sides measure $BC = 5, AC = 4$ and $AB = 3$. Let $I$ be the center of the circle tangent internally to the three sides of the triangle $ABC$ and $M$ be the midpoint of $BC$. The perpendiculars on the side $AB$ that pass through $M$ and $I$ intersect on the side $AB$ on $R$ and $S$, respectively. Line $MI $ cuts the side $AB$ at point $P$. Show that the segments $RS$ and $SP$ have the same measure.
2010 Rioplatense level 1 P1
Consider a semicircle with diameter $AB$, center $O$ and radius $r$. Let $C$ be the point of segment $AB$ such that $AC = \frac{2r}{3}$. Line $\ell$ is perpendicular to $AB$ on $C$ and $D$ is the intersection point of $\ell$ and the semicircle. Let $H$ be the foot of the perpendicular drawn from $O$ on $AD$ and $E$ the point of intersection of lines $CD$ and $OH$.
a) Calculate $AD$ in terms of $r$.
b) If $M$ and $N$ are the midpoints of $AE$ and $OD$ respectively, find the measure of the angle $MHN$ .
2011 Rioplatense level 1 P5
Let $ABCD$ be a rectangle , with sides $AB < BC$. On the outside of the rectangle is the point $E$ such that $ABEC$ is an isosceles trapezoid ($BE$, $AC$ are parallel and $BA = EC$). Prove that if the area of the triangle $BEC$ is the fourth part of the area of the rectangle $ABCD$, then the triangle $AED$ is equilateral.
2012 Rioplatense level 1 P1
In triangle $ABC, \angle ABC = 45^o , BC = 1$ and $E$ is the point on the $AC$ side such that $EC = 1$. The perpendicular to $AC$ that passes through $E$ cuts the extension of $BC$ at point $D$ so that $CD = 2$ and $C$ is inside $DB$. Determine the measure of the angles of triangle $ABD$.
2012 Rioplatense level 1 P5
Let $ABCD$ be a convex quadrilateral. The points $P$ and $Q$ of the sides $AB$ and $AD$, respectively, are such that area $(ABQ) =$ area $(ADP) = 1/3$ area $(ABCD)$. The intersection of $PQ$ and the diagonal $AC$ is the point $R$. Calculate the ratio $AR/RC$
$ABC$ is a triangle whose sides measure $BC = 5, AC = 4$ and $AB = 3$. Let $I$ be the center of the circle tangent internally to the three sides of the triangle $ABC$ and $M$ be the midpoint of $BC$. The perpendiculars on the side $AB$ that pass through $M$ and $I$ intersect on the side $AB$ on $R$ and $S$, respectively. Line $MI $ cuts the side $AB$ at point $P$. Show that the segments $RS$ and $SP$ have the same measure.
Consider a semicircle with diameter $AB$, center $O$ and radius $r$. Let $C$ be the point of segment $AB$ such that $AC = \frac{2r}{3}$. Line $\ell$ is perpendicular to $AB$ on $C$ and $D$ is the intersection point of $\ell$ and the semicircle. Let $H$ be the foot of the perpendicular drawn from $O$ on $AD$ and $E$ the point of intersection of lines $CD$ and $OH$.
a) Calculate $AD$ in terms of $r$.
b) If $M$ and $N$ are the midpoints of $AE$ and $OD$ respectively, find the measure of the angle $MHN$ .
2011 Rioplatense level 1 P5
Let $ABCD$ be a rectangle , with sides $AB < BC$. On the outside of the rectangle is the point $E$ such that $ABEC$ is an isosceles trapezoid ($BE$, $AC$ are parallel and $BA = EC$). Prove that if the area of the triangle $BEC$ is the fourth part of the area of the rectangle $ABCD$, then the triangle $AED$ is equilateral.
2012 Rioplatense level 1 P1
In triangle $ABC, \angle ABC = 45^o , BC = 1$ and $E$ is the point on the $AC$ side such that $EC = 1$. The perpendicular to $AC$ that passes through $E$ cuts the extension of $BC$ at point $D$ so that $CD = 2$ and $C$ is inside $DB$. Determine the measure of the angles of triangle $ABD$.
2012 Rioplatense level 1 P5
Let $ABCD$ be a convex quadrilateral. The points $P$ and $Q$ of the sides $AB$ and $AD$, respectively, are such that area $(ABQ) =$ area $(ADP) = 1/3$ area $(ABCD)$. The intersection of $PQ$ and the diagonal $AC$ is the point $R$. Calculate the ratio $AR/RC$
2013 Rioplatense level 1 P2 (Austrian-Polish 1997)
In a trapezoid $ABCD$ with $AB // CD$, the diagonals $AC$ and $BD$ intersect at point $E$. Let $F$ and $G$ be the orthocenters of the triangles $EBC$ and $EAD$. Prove that the midpoint of $GF$ lies on the perpendicular from $E$ to $AB$.
2014 Rioplatense level 1 P4
In the trapezoid $ABCD$ the parallel sides are $AD$ and $BC$. Let $P$ be a point on the side $AB$ such that area $(APD) = $ area $(BPC) = s$ and let $E$ be the point of intersection of the diagonals $AC$ and $BD$. Show what area $(CED) = s$.
2015 Rioplatense level 1 P1
Let $ABCD$ be a square of center $O$ and sides $AB, BC, CD$, and $DA$. The triangles $BJC$ and $CKD$ are constructed, out of the square, with $BJ = CJ = CK = DK$. Let $M$ be the midpoint of $CJ$. Prove that the lines $OM$ and $BK$ are perpendicular.
2015 Rioplatense level 1 P5
Let $ABC$ be an acute triangle with $AB <BC$ and let $D$ and $E$ be points on the sides $AB$ and $BC$, respectively, such that $AD = DE = EC$. Segments $AE$ and $DC$ are cut at point $X$. If $AX = XC$, determine the measure of angle $ABC$.
2016 Rioplatense level 1 P2
Let $ABCD$ be a parallelogram. The square $BDXY$ is constructed, which has no interior points in common with the triangle $ABD$. The square $ACZW$ is constructed, which has no interior points in common with the triangle $ADC$ . Let $P$ and $Q$ be the centers of the squares $BDXY$ and $ACZW$ respectively. Show that $AP = DQ$.
2019 Rioplatense level 1 P5
Let $ABC$ be a triangle and $M$ be the midpoint of the side $BC$. Suppose that $\angle AMC= 60^o$ and that the length of $AM$ is greater than the length of $MC$. Let $D$ be the point on segment $AM$ such that $AD=MC$. Prove that $AC= BD$.
In a trapezoid $ABCD$ with $AB // CD$, the diagonals $AC$ and $BD$ intersect at point $E$. Let $F$ and $G$ be the orthocenters of the triangles $EBC$ and $EAD$. Prove that the midpoint of $GF$ lies on the perpendicular from $E$ to $AB$.
In the trapezoid $ABCD$ the parallel sides are $AD$ and $BC$. Let $P$ be a point on the side $AB$ such that area $(APD) = $ area $(BPC) = s$ and let $E$ be the point of intersection of the diagonals $AC$ and $BD$. Show what area $(CED) = s$.
2015 Rioplatense level 1 P1
Let $ABCD$ be a square of center $O$ and sides $AB, BC, CD$, and $DA$. The triangles $BJC$ and $CKD$ are constructed, out of the square, with $BJ = CJ = CK = DK$. Let $M$ be the midpoint of $CJ$. Prove that the lines $OM$ and $BK$ are perpendicular.
2015 Rioplatense level 1 P5
Let $ABC$ be an acute triangle with $AB <BC$ and let $D$ and $E$ be points on the sides $AB$ and $BC$, respectively, such that $AD = DE = EC$. Segments $AE$ and $DC$ are cut at point $X$. If $AX = XC$, determine the measure of angle $ABC$.
Let $ABC$ be a right and isosceles triangle with $\angle C = 90^o$. Let $P$ be a point on the $BC$ side (which is neither $B$ nor $C$). Points $N$ and $L$ are marked in the $AP$ segment such that $CN$ is perpendicular to $AP$ and $AL = CN$. Let $M$ be the midpoint of $AB$.
i) Determine the measure of the angle $\angle LMN$.
ii) If the area of the triangle$ ABC$ is equal to four times the area of the triangle $LMN$, determine the measure of the angle $\angle CAP$.
2017 Rioplatense level 1 P4
Let $ABCD$ be a rectangle of sides$ AB, BC, CD$ and $DA$. Let $P$ be a point on the side $AB$ and $Q$ a point on the diagonal $BD$ such that $PQ$ is parallel to $AD$ and $AD = PQ + 2BQ$. The line perpendicular to $AC$ passing through $P$ cuts $AC$ into $R$. Show that $PR = 2PB$.
2018 Rioplatense level 1 P5i) Determine the measure of the angle $\angle LMN$.
ii) If the area of the triangle$ ABC$ is equal to four times the area of the triangle $LMN$, determine the measure of the angle $\angle CAP$.
Let $ABCD$ be a rectangle of sides$ AB, BC, CD$ and $DA$. Let $P$ be a point on the side $AB$ and $Q$ a point on the diagonal $BD$ such that $PQ$ is parallel to $AD$ and $AD = PQ + 2BQ$. The line perpendicular to $AC$ passing through $P$ cuts $AC$ into $R$. Show that $PR = 2PB$.
Let $ABCD$ be a parallelogram. The square $BDXY$ is constructed, which has no interior points in common with the triangle $ABD$. The square $ACZW$ is constructed, which has no interior points in common with the triangle $ADC$ . Let $P$ and $Q$ be the centers of the squares $BDXY$ and $ACZW$ respectively. Show that $AP = DQ$.
2019 Rioplatense level 1 P5
Let $ABC$ be a triangle and $M$ be the midpoint of the side $BC$. Suppose that $\angle AMC= 60^o$ and that the length of $AM$ is greater than the length of $MC$. Let $D$ be the point on segment $AM$ such that $AD=MC$. Prove that $AC= BD$.
www.oma.org.ar/enunciados/index.htm
omaforos.com.ar/archivo.php?id=17
oc.uan.edu.co/component/k2/itemlist/category/64-olimpiada-matematica-rioplatense
www.elnumerodeoro.cl/bbs/viewforum.php?f=76
thanks to Ercolle Suppa and Jorge Tipe for their help
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