geometry problems from Argentinian Mathematical Olympiads (OMA)
with aops links in the names
1994 - 2020 level 1
(level 2, level 3 below)
1996 Argentina L1 p3
In a triangle $ABC, C = 45^o, Q$ is the foot of the altitude corresponding to vertex $B$ and $M$ is the midpoint of side $AB$. Let $P$ be on side BC such that $PM$ is perpendicular to $QM$. Decide if for any value of angle $A$ it is verified that $PQ^2 / PM^2 = 3$.
2005 Argentina L1 p3
Given a segment of length $d$, construct with a ruler and compass a square in which the difference between the lengths of the diagonal and the side is $d$. Indicate the construction steps and explain why the construction carried out satisfies the conditions of the problem.
2019 Argentina L1 p3
In triangle $ABC$ let $D$ and $E$ be on sides $AB$ and $AC$ respectively, such that $BD = CE$. Let $M$ and $N$ be the midpoints of $BC$ and $DE$ respectively. Show that the bisector of the angle $\angle BAC$ is parallel to the line $MN$.
2017 Argentina L2 p5
We have a convex quadrilateral $ABCD$ with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and $Q$ on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Determine the value of the radius of the circle that passes through the vertices of the triangle $DPQ$.
sources:
with aops links in the names
[Olimpíada Matemática Argentina]
collected inside aops: here
(level 3 is the hardest)
(level 2, level 3 below)
1994 Argentina L1 p4
In triangle $ABC, \angle A = 27º, \angle B = 42^o$. On the side $AB$, point $D$ is marked so that $\angle ACD = 69^o$ and on the side $AC$, point $K$ is marked such that $DK$ is parallel to $BC$. Calculate the measures of the angles $\angle CBK$ and $\angle ABK$.
1994 Argentina L1 p6
A rectangle is drawn on the plane, and three ants are standing at three of its vertices (one at each vertex). The ants move in turn. In each turn, one ant moves around the plane and the other two stay still. The moving ant walks along the straight line that is parallel to the line determined by the other two ants that are still on that turn. Is it possible that, after a few turns, the three ants are located at the midpoints of three sides of the rectangle? Justify.
Clarification: The distance an ant travels in each turn can be variable.
1995 Argentina L1 p3
Let $ABC$ be a scalene triangle of area $7$. Let $A_1$ be a point on the $BC$ side and let $B_1$ and $C_1$ be on the lines $AC$ and $AB$ respectively, such that $AA_1, BB_1$ and $CC_1$ are parallel. Find the area of the triangle $A_1B_1C_1$.
In triangle $ABC, \angle A = 27º, \angle B = 42^o$. On the side $AB$, point $D$ is marked so that $\angle ACD = 69^o$ and on the side $AC$, point $K$ is marked such that $DK$ is parallel to $BC$. Calculate the measures of the angles $\angle CBK$ and $\angle ABK$.
1994 Argentina L1 p6
A rectangle is drawn on the plane, and three ants are standing at three of its vertices (one at each vertex). The ants move in turn. In each turn, one ant moves around the plane and the other two stay still. The moving ant walks along the straight line that is parallel to the line determined by the other two ants that are still on that turn. Is it possible that, after a few turns, the three ants are located at the midpoints of three sides of the rectangle? Justify.
Clarification: The distance an ant travels in each turn can be variable.
1995 Argentina L1 p3
Let $ABC$ be a scalene triangle of area $7$. Let $A_1$ be a point on the $BC$ side and let $B_1$ and $C_1$ be on the lines $AC$ and $AB$ respectively, such that $AA_1, BB_1$ and $CC_1$ are parallel. Find the area of the triangle $A_1B_1C_1$.
1996 Argentina L1 p3
In a triangle $ABC, C = 45^o, Q$ is the foot of the altitude corresponding to vertex $B$ and $M$ is the midpoint of side $AB$. Let $P$ be on side BC such that $PM$ is perpendicular to $QM$. Decide if for any value of angle $A$ it is verified that $PQ^2 / PM^2 = 3$.
1996 Argentina L1 p4
Let $ABC$ be an area triangle $7$. The triangle $XYZ$ is constructed as follows: side $AB$ is extended such that $AX = 2AB$, side $BC$ is extended such that $BY = 3BC$, and side $CA$ is extended such that $CZ = 4CA$. Find the area of the triangle $XYZ$.
Let $ABC$ be an area triangle $7$. The triangle $XYZ$ is constructed as follows: side $AB$ is extended such that $AX = 2AB$, side $BC$ is extended such that $BY = 3BC$, and side $CA$ is extended such that $CZ = 4CA$. Find the area of the triangle $XYZ$.
1997 Argentina L1 p2
Given are a line $\ell$, two points A and B on the same side of the plane wrt line $\ell$, and a segment of length $d$. Find two points $P$ and $Q$ on line $\ell$ such that segment $PQ$ has length $d$ and the sum $AP + PQ + QB$ is as short as possible. Indicate the construction steps and explain why the shortest possible length was obtained.
Given are a line $\ell$, two points A and B on the same side of the plane wrt line $\ell$, and a segment of length $d$. Find two points $P$ and $Q$ on line $\ell$ such that segment $PQ$ has length $d$ and the sum $AP + PQ + QB$ is as short as possible. Indicate the construction steps and explain why the shortest possible length was obtained.
1997 Argentina L1 p6
Let $ABCD$ be a
parallelogram of sides $AB, BC, CD$ and $DA$. Draw a line through $D$ that cuts
the side $BC$ at $P$ and the extension of the side $AB$ at $Q$. If the area of
the quadrilateral $ABDP$ is $29$ and the area of the triangle $DPC$ is $8$,
find the area of the triangle $CPQ$.
1998 Argentina L1 p3
Points $A, B, C, D$ are considered on the sides of the vertex angle $O$ (see figure). Let $E$ be the point of intersection of segments $AD$ and $BC$. The line parallel to $AB$ is drawn by $C$ and the line parallel to $CD$ is drawn by $A$. Let $F$ be the point of intersection of the two drawn lines. Show that $S_{OAEC}= S_{BFDE}$
Points $A, B, C, D$ are considered on the sides of the vertex angle $O$ (see figure). Let $E$ be the point of intersection of segments $AD$ and $BC$. The line parallel to $AB$ is drawn by $C$ and the line parallel to $CD$ is drawn by $A$. Let $F$ be the point of intersection of the two drawn lines. Show that $S_{OAEC}= S_{BFDE}$
1998 Argentina L1 p5
Given the segments $a,b,h$ with $a<b<h$, construct a triangle $ABC$ with a ruler and compass such that, if $D$ is the foot of the height corresponding to vertex $C$, then $CD = h, AC-AD = a, BC-BD = b$. Indicate the construction steps and explain why the construction carried out satisfies the conditions of the problem.
Given the segments $a,b,h$ with $a<b<h$, construct a triangle $ABC$ with a ruler and compass such that, if $D$ is the foot of the height corresponding to vertex $C$, then $CD = h, AC-AD = a, BC-BD = b$. Indicate the construction steps and explain why the construction carried out satisfies the conditions of the problem.
1999 Argentina L1 p2
The $ABCD$ parallelogram has $AB > BC$ and $\angle DAB < \angle ABC$. The perpendicular bisectors of sides $AB$ and $BC$ intersect at point $M$, which also belongs to the extension of side $AD$. If $\angle GCD = 15^o$, find the measure of $\angle ABC$.
The $ABCD$ parallelogram has $AB > BC$ and $\angle DAB < \angle ABC$. The perpendicular bisectors of sides $AB$ and $BC$ intersect at point $M$, which also belongs to the extension of side $AD$. If $\angle GCD = 15^o$, find the measure of $\angle ABC$.
1999 Argentina L1 p6
In a convex quadrilateral $ABCD$ consider a point $M$ on the side $AB$ and a point $N$ on the side $CD$ side such that $AM / AB = CN / CD$. The segments $MD$ and $AN$ intersect at $P$, segments $NB$ and $CM$ intersect at $Q$. Show that $S_{MQNP} = S_{APD}+ S_{BQC}$.
In a convex quadrilateral $ABCD$ consider a point $M$ on the side $AB$ and a point $N$ on the side $CD$ side such that $AM / AB = CN / CD$. The segments $MD$ and $AN$ intersect at $P$, segments $NB$ and $CM$ intersect at $Q$. Show that $S_{MQNP} = S_{APD}+ S_{BQC}$.
The isosceles triangle $ABC$ has $AB = BC$. The point $P$ lies on the side $AC$ , the point $Q$ on the side $BC$ and the point $R$ on the side $AB$ are such that $PQ$ is parallel to $AB$, $RP$ is parallel to $BC$ and $RB = AP$. If $AQB = 105^o$ , calculate the angle measures of triangle $ABC$.
2000 Argentina L1 p6
2000 Argentina L1 p6
A camper has been lost in a rectangular shaped forest that is $100$ km long and $1$ km wide. He does not know where the edges of the forest are, and only has the energy to walk $2.83$ km. Find a route that will ensure you get out of the woods before exhausting your energies. Prove that the path found never fails, no matter what the starting point is.
2001 Argentina L1 p2
In a triangle $ABC$, point $D$ has been marked on the side $BC$ and point $E$ on the side $AC$ , and the bisector of the $\angle CAD$ and the bisector of the $\angle CBE$ have been plotted. These two bisectors intersect at point $F.$ Given that $\angle AFB = 84^o$, calculate the value of the sum of angles $\angle AEB + \angle ADB$.
In a triangle $ABC$, point $D$ has been marked on the side $BC$ and point $E$ on the side $AC$ , and the bisector of the $\angle CAD$ and the bisector of the $\angle CBE$ have been plotted. These two bisectors intersect at point $F.$ Given that $\angle AFB = 84^o$, calculate the value of the sum of angles $\angle AEB + \angle ADB$.
2001 Argentina L1 p5
Let $ABCDE$ be a pentagon of sides $AB, BC, CD, DE$ and $EA$
such that $S_{ABC} = S_{ABD}=S_{ACD} = S_{ADE} = 17$. Calculate the area of
the triangle $ECB$ .
2002 Argentina L1 p2
In triangle $ABC$, let $M$ be on side $AB$ such that $AM=2MB$ and $N$ be the midpoint of side $BC$. We denote $O$ at the intersection point of $AN$ and $CM$. If the area of triangle $ABC$ equals $30$, find the area of quadrilateral $MBNO$.
In triangle $ABC$, let $M$ be on side $AB$ such that $AM=2MB$ and $N$ be the midpoint of side $BC$. We denote $O$ at the intersection point of $AN$ and $CM$. If the area of triangle $ABC$ equals $30$, find the area of quadrilateral $MBNO$.
2002 Argentina L1 p5
Let $ABC$ be a triangle such that $\angle ABC=2 \angle BCA$, furthermore, if $D$ denotes the point on the side $BC$ such that $AD$ is bisector of the angle $\angle CAB$ , we have that $CD = AB $. Calculate the measures of the angles of triangle $ABC$ .
Let $ABC$ be a triangle such that $\angle ABC=2 \angle BCA$, furthermore, if $D$ denotes the point on the side $BC$ such that $AD$ is bisector of the angle $\angle CAB$ , we have that $CD = AB $. Calculate the measures of the angles of triangle $ABC$ .
2003 Argentina L1 p2
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$ and $\frac{CE}{AC}=\frac{3}{7} ,\frac{DE}{BD}=\frac{4}{9}$. Let $P$ and $Q$ be the points that divide segment $BE$ into three equal parts, with $P$ between $B$ and $Q$ , and let R be the midpoint of segment $AE$ . Calculate $\frac{S_{APQR}}{S_{ABCD}}$.
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$ and $\frac{CE}{AC}=\frac{3}{7} ,\frac{DE}{BD}=\frac{4}{9}$. Let $P$ and $Q$ be the points that divide segment $BE$ into three equal parts, with $P$ between $B$ and $Q$ , and let R be the midpoint of segment $AE$ . Calculate $\frac{S_{APQR}}{S_{ABCD}}$.
2004 Argentina L1 p6
Let $ABCD$ be a square. If $E$ is the midpoint of side $CD$ and $M$ is the interior point of the square such that $\angle MAB=\angle MBC=\angle BME$. Calculate the measure of angle $\angle MAB$.
Let $ABCD$ be a square. If $E$ is the midpoint of side $CD$ and $M$ is the interior point of the square such that $\angle MAB=\angle MBC=\angle BME$. Calculate the measure of angle $\angle MAB$.
2005 Argentina L1 p3
Given a segment of length $d$, construct with a ruler and compass a square in which the difference between the lengths of the diagonal and the side is $d$. Indicate the construction steps and explain why the construction carried out satisfies the conditions of the problem.
2006 Argentina L1 p5 (Langley's Problem)
et $ABC$ be an isosceles triangle, $AB = AC, \angle A = 20^{\circ}$. Let $D$ be a point on $AB$, and $E$ a point on $AC$ such that $\angle ACD = 20^{\circ}$ and $\angle ABE = 30^{\circ}$. What is the measure of the angle $\angle CDE$?
et $ABC$ be an isosceles triangle, $AB = AC, \angle A = 20^{\circ}$. Let $D$ be a point on $AB$, and $E$ a point on $AC$ such that $\angle ACD = 20^{\circ}$ and $\angle ABE = 30^{\circ}$. What is the measure of the angle $\angle CDE$?
Let $ABC$ be a triangle such that $\angle B =40^o$. It is known that there is a point $P$ on the bisector of the angle $\angle B$ that satisfies that $BP = BC$ and $\angle BAP=20^o$ . Determine the measures of the angles $\angle A$ and $\angle C$.
2008 Argentina L1 p3
Let $ ABC$ be a triangle, and $ P$ a point on the internal bisector of angle $ \angle BAC$. It is known that $ \angle ABP=30^{\circ}$ and $ CP=BC$. Find the measure of the angle $ \angle APC$.
Let $ ABC$ be a triangle, and $ P$ a point on the internal bisector of angle $ \angle BAC$. It is known that $ \angle ABP=30^{\circ}$ and $ CP=BC$. Find the measure of the angle $ \angle APC$.
2009 Argentina L1 p2
Let $ABC$ be a scalene triangle, $D$ a point inside the side $BC, E$ a point inside the side $CA$ and $F$ a point inside the side $AB$.
a) If $\angle FDE=\angle A, \angle DEF= \angle B, \angle EFD=\angle C$ determine if it is necessarily true that $D, E$ and $F$ are the midpoints of the sides of ABC.
b) If $\angle CED=\angle BFD, \angle AFE=\angle CDE, \angle BDF=\angle AEF$ , determine if it is necessarily true that $D, E$ and $F$ are the midpoints of the sides of $ABC$.
Let $ABC$ be a scalene triangle, $D$ a point inside the side $BC, E$ a point inside the side $CA$ and $F$ a point inside the side $AB$.
a) If $\angle FDE=\angle A, \angle DEF= \angle B, \angle EFD=\angle C$ determine if it is necessarily true that $D, E$ and $F$ are the midpoints of the sides of ABC.
b) If $\angle CED=\angle BFD, \angle AFE=\angle CDE, \angle BDF=\angle AEF$ , determine if it is necessarily true that $D, E$ and $F$ are the midpoints of the sides of $ABC$.
2009 Argentina L1 p5
We have a scalene paper triangle of area $1$. Show that three equal convex polygons can be cut from the triangle, each with an area greater than $\frac14$.
Clarification: A polygon is convex if all its angles are less than $180^o$.
We have a scalene paper triangle of area $1$. Show that three equal convex polygons can be cut from the triangle, each with an area greater than $\frac14$.
Clarification: A polygon is convex if all its angles are less than $180^o$.
2010 Argentina L1 p3
In a triangle $ABC$ point $P$ divides $AB$ side in ratio $\frac{AP}{PB}= \frac{1}{4}$. The perpendicular bisector of segment $PB$ intersects side $BC$ at point $Q$. If it is known that $S_{PQC} = \frac{4}{25}S_{ABC}$, and $AC = 7$, find $BC$.
In a triangle $ABC$ point $P$ divides $AB$ side in ratio $\frac{AP}{PB}= \frac{1}{4}$. The perpendicular bisector of segment $PB$ intersects side $BC$ at point $Q$. If it is known that $S_{PQC} = \frac{4}{25}S_{ABC}$, and $AC = 7$, find $BC$.
2011 Argentina L1 p2
Let $ABC$ be a triangle with $\angle C =90^o$. Point $P$ lies on side $AB$ is such that $PC = BC$ and point $Q$ lies on side $BC$ such that $\angle BAQ=6 \angle CAQ$ . Segment $CP$ passes through the midpoint of segment $AQ$. Find the angles of triangle $ABC$.
Let $ABC$ be a triangle with $\angle C =90^o$. Point $P$ lies on side $AB$ is such that $PC = BC$ and point $Q$ lies on side $BC$ such that $\angle BAQ=6 \angle CAQ$ . Segment $CP$ passes through the midpoint of segment $AQ$. Find the angles of triangle $ABC$.
2012 Argentina L1 p3
Let $ABC$ be an acute triangle with circumcenter $O$. Line $AO$ intersects side $BC$ at $D$. It is known that $ OD = BD = 1 $ and $ CD = 1 + \sqrt {2} $. Calculate the angles of the triangle.
Let $ABC$ be an acute triangle with circumcenter $O$. Line $AO$ intersects side $BC$ at $D$. It is known that $ OD = BD = 1 $ and $ CD = 1 + \sqrt {2} $. Calculate the angles of the triangle.
2013 Argentina L1 p3
A non-convex quadrilateral $ABCD$ with sides $AB, BC, CD, DA$ is drawn on the board, with different sidelengths and $\angle A=\angle B=\angle C=45^o$. Beto measured only one of the six segments $AB, BC, CD, DA, AC, BD$, at his choice, and obtained the value $6$. He states that with this information, he can calculate the area of the quadrilateral $ABCD$. Calculate this area.
Clarification. A quadrilateral is non convex if one of its angles is greater than $180^o$.
A non-convex quadrilateral $ABCD$ with sides $AB, BC, CD, DA$ is drawn on the board, with different sidelengths and $\angle A=\angle B=\angle C=45^o$. Beto measured only one of the six segments $AB, BC, CD, DA, AC, BD$, at his choice, and obtained the value $6$. He states that with this information, he can calculate the area of the quadrilateral $ABCD$. Calculate this area.
Clarification. A quadrilateral is non convex if one of its angles is greater than $180^o$.
2014 Argentina L1 p5
Let $ABC$ be a triangle, $AD$ be the angle bisector of $BAC$ and $M$ the midpoint of $BC$. The parallel to $AD$ drawn by $M$ cuts the line $AB$ at $E$ and the side $AC$ at F. If $AB =18, AC = 25$, calculate the lengths of the segments $BE$ and $CF$.
2015 Argentina L1 p2
Points $D$ and $E$ divide side $AB$ of equilateral triangle $ABC$ into three equal parts, $D$ is between $A$ and $E$. The point $F$ on the $BC$ side is such that $CF=AD$ . Find the value of the sum of the angles $\angle CDF + \angle CEF$.
2016 Argentina L1 p3
Let $ABC$ be a right triangle with $\angle C = 90^o$. Points $D$ and $E$ on hypotenuse $AB$ are such that $AD = AC$ and $BE = BC$. The $P$ and $Q$ points in $AC$ and $BC$ respectively are such that $AP = AE$ and $BQ = BD$. Let $M$ be the midpoint of $PQ$. Show that $M$ is the point of intersection of the bisectors of triangle $ABC$ and calculate the measure of angle $\angle AMB$.
Let $ABC$ be a triangle, $AD$ be the angle bisector of $BAC$ and $M$ the midpoint of $BC$. The parallel to $AD$ drawn by $M$ cuts the line $AB$ at $E$ and the side $AC$ at F. If $AB =18, AC = 25$, calculate the lengths of the segments $BE$ and $CF$.
2015 Argentina L1 p2
Points $D$ and $E$ divide side $AB$ of equilateral triangle $ABC$ into three equal parts, $D$ is between $A$ and $E$. The point $F$ on the $BC$ side is such that $CF=AD$ . Find the value of the sum of the angles $\angle CDF + \angle CEF$.
2016 Argentina L1 p3
Let $ABC$ be a right triangle with $\angle C = 90^o$. Points $D$ and $E$ on hypotenuse $AB$ are such that $AD = AC$ and $BE = BC$. The $P$ and $Q$ points in $AC$ and $BC$ respectively are such that $AP = AE$ and $BQ = BD$. Let $M$ be the midpoint of $PQ$. Show that $M$ is the point of intersection of the bisectors of triangle $ABC$ and calculate the measure of angle $\angle AMB$.
2017 Argentina L1 p2
The figure is made up of small squares equal to each other enclosed in a rectangle. The horizontal side of the rectangle measures $73$ and the vertical side measures $94$. Calculate the side of each small square.
The figure is made up of small squares equal to each other enclosed in a rectangle. The horizontal side of the rectangle measures $73$ and the vertical side measures $94$. Calculate the side of each small square.
Construct, using exclusively a ruler and a compass, a
trapezoid $ABCD$ of bases $AB$ and $CD$ such that if $E$ is the midpoint of the
side $AD$, $EC = BC = 4$, $CD = 2$ and $\angle EC B =
120^o$. Indicate the construction steps and calculate the area of the trapezoid $ABCD$.
2019 Argentina L1 p3
In triangle $ABC$ let $D$ and $E$ be on sides $AB$ and $AC$ respectively, such that $BD = CE$. Let $M$ and $N$ be the midpoints of $BC$ and $DE$ respectively. Show that the bisector of the angle $\angle BAC$ is parallel to the line $MN$.
In the isosceles triangle $ABC$, let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that the lines $BE$ and $CD$ intersect at $F$. Furthermore, the triangles $AEB$ and $ADC$ are equal and have $AD=AE=10$ and $AB=AC=30$. Calculate $\frac{[ADFE]}{[ABC]}$.
Clarification: $[ABC]$ denotes the area of the figure $ABC$.
1995 - 2020 level 2
1995 Argentina L2 p2
Given a triangle $ABC$, with $BC <AC$, let $K$ be the midpoint of $AB$ and $L$ be the point on the side $AC$ such that $AL = LC + CB$. Show that if $\angle KLB = 90^o$ then $AC = 3 CB$ and conversely, if $AC = 3 CB$ then $\angle KLB = 90^o$.
Given a triangle $ABC$, with $BC <AC$, let $K$ be the midpoint of $AB$ and $L$ be the point on the side $AC$ such that $AL = LC + CB$. Show that if $\angle KLB = 90^o$ then $AC = 3 CB$ and conversely, if $AC = 3 CB$ then $\angle KLB = 90^o$.
1995 Argentina L2 p5
Let be a circle with center $O$ and a parallelogram $ABCD$ such that $A, B$ and $C$ belong to the circle and $O$ belongs to the side $AD$. The lines $AD, CD$ and $BO$ intersect the circumference again at $K, M, N$ respectively. Prove that the segments $NK, NM$ and $ND$ are equal to each other.
Let be a circle with center $O$ and a parallelogram $ABCD$ such that $A, B$ and $C$ belong to the circle and $O$ belongs to the side $AD$. The lines $AD, CD$ and $BO$ intersect the circumference again at $K, M, N$ respectively. Prove that the segments $NK, NM$ and $ND$ are equal to each other.
1996 Argentina L2 p2
A trapezoid inscribed in a circle of radius $r$ has three sides of length $s$ and a fourth of length $r + s$, with $s <r$. Find the measures of the angles of the trapezoid.
A trapezoid inscribed in a circle of radius $r$ has three sides of length $s$ and a fourth of length $r + s$, with $s <r$. Find the measures of the angles of the trapezoid.
1996 Argentina L2 p4
In the right triangle $ABC, B = 90^o, AB = 3, BC = 4, AC = 5$. Let $O$ be the center of gravity of the triangle, that is, the point of intersection of the medians and let $A', B', C'$ be points on the sides $BC, AC, AB$ respectively, such that $OA'$ is perpendicular to $BC, OB'$ is perpendicular to $AC$ and $OC'$ to $AB$. Find the area of the triangle $A'B'C'$.
In the right triangle $ABC, B = 90^o, AB = 3, BC = 4, AC = 5$. Let $O$ be the center of gravity of the triangle, that is, the point of intersection of the medians and let $A', B', C'$ be points on the sides $BC, AC, AB$ respectively, such that $OA'$ is perpendicular to $BC, OB'$ is perpendicular to $AC$ and $OC'$ to $AB$. Find the area of the triangle $A'B'C'$.
1997 Argentina L2 p3
Let $ABC$ be an acute triangle, $D$ a point on the side $AB, E$ the point on the side $AC$ such that $DE$ is perpendicular to $AC$ and $F$ the point on the side $BC$ such that $DF$ is perpendicular to $BC$. Construct a point $D$ such that $EF$ is parallel to $AB$.
Let $ABC$ be an acute triangle, $D$ a point on the side $AB, E$ the point on the side $AC$ such that $DE$ is perpendicular to $AC$ and $F$ the point on the side $BC$ such that $DF$ is perpendicular to $BC$. Construct a point $D$ such that $EF$ is parallel to $AB$.
1997 Argentina L2 p4
The figure shows an equilateral triangle divided by three
lines in seven regions. The corresponding area is indicated in six of the
regions. Find the area of the seventh region, that is, of the central
triangle.
1998 Argentina L2 p3
Triangles $A_1B_1C_1$ and $A_2B_2C_2$ have parallel sides:
$A_1B_1 \parallel A_2B_2, B_1C_1 \parallel
B_2C_2, C_1A_1 \parallel C_2A_2$,
and triangle $A_2B_2C_2$ is contained inside triangle $A_1B_1C_1$. A third
triangle $ABC$ is inscribed in $A_1B_1C_1$ and circumscribed to $A_2B_2C_2$,
that is, points $A, B, C$ belong inside segments $B_1C_1, C_1A_1, A_1B_1$ and
points $A_2, B_2, C_2$ belong inside segments $BC, CA, AB$, respectively. If
the area of $A_1B_1C_1$ is $63$ and the area of $A_2B_2C_2$ is $7$, find
the area of $ABC$.
1998 Argentina L2 p5
On the board he had drawn a quadrilateral $ABCD$, with $BC = CD = DA$, which had the midpoints $M, N, P$ of those three equal sides marked. The whole figure was erased, except points $M, N$ and $P$. Reconstruct the quadrilateral, indicating the construction steps.
On the board he had drawn a quadrilateral $ABCD$, with $BC = CD = DA$, which had the midpoints $M, N, P$ of those three equal sides marked. The whole figure was erased, except points $M, N$ and $P$. Reconstruct the quadrilateral, indicating the construction steps.
1999 Argentina L2 p3
The triangle $ABC$ is isosceles, with $AB = BC$ and $\angle ABC = 82^o$. The point $M$ inside the triangle is considered such that $AM = AB$ and $\angle MAC = 11^o$. Find the measure of the angle $\angle MCB$.
The triangle $ABC$ is isosceles, with $AB = BC$ and $\angle ABC = 82^o$. The point $M$ inside the triangle is considered such that $AM = AB$ and $\angle MAC = 11^o$. Find the measure of the angle $\angle MCB$.
1999 Argentina L2 p4
The triangle $ABC$ is isosceles, with $AB = BC$ and $\angle ABC = 82^o$. The point $M$ inside the triangle is considered such that $AM = AB$ and $\angle MAC = 11^o$. Find the measure of the angle $\angle MCB$.
The triangle $ABC$ is isosceles, with $AB = BC$ and $\angle ABC = 82^o$. The point $M$ inside the triangle is considered such that $AM = AB$ and $\angle MAC = 11^o$. Find the measure of the angle $\angle MCB$.
2000 Argentina L2 p2
Let $ABCD$ be a rectangle, $M$ the midpoint of $DA$, $N$ the midpoint of $BC$. Let $P$ be the point on the extension of the side $CD$ (closer to $D$ than to $C$) such that $\angle CPN = 20^o$. Let $Q$ be the point of intersection of the line $PM$ with the diagonal $AC$. Calculate the measure of the angle $\angle PNQ$ .
Let $ABCD$ be a rectangle, $M$ the midpoint of $DA$, $N$ the midpoint of $BC$. Let $P$ be the point on the extension of the side $CD$ (closer to $D$ than to $C$) such that $\angle CPN = 20^o$. Let $Q$ be the point of intersection of the line $PM$ with the diagonal $AC$. Calculate the measure of the angle $\angle PNQ$ .
2000 Argentina L2 p5
In a parallelogram $ABCD$, we call the midpoints of sides $BC$ and $CD, M$ and $N$, respectively. Decide whether it is possible for lines $AM$ and $AN$ to divide angle $BAD$ into three equal angles. If the answer is yes, give an example of such a parallelogram. If the answer is negative, explain why.
In a parallelogram $ABCD$, we call the midpoints of sides $BC$ and $CD, M$ and $N$, respectively. Decide whether it is possible for lines $AM$ and $AN$ to divide angle $BAD$ into three equal angles. If the answer is yes, give an example of such a parallelogram. If the answer is negative, explain why.
2001 Argentina L2 p2
In triangle $ABC$, which has $\angle BAC = 63^o$, the bisector of angle $\angle BAC$ was drawn. Let $\ell$ be the line that passes through $A$ and is perpendicular to this bisector. If line $\ell$ intersects line $BC$ at $P$ such that $BP = AC + AB$, find the measures of the angles of triangle $ABC$. Give all the possibilities.
In triangle $ABC$, which has $\angle BAC = 63^o$, the bisector of angle $\angle BAC$ was drawn. Let $\ell$ be the line that passes through $A$ and is perpendicular to this bisector. If line $\ell$ intersects line $BC$ at $P$ such that $BP = AC + AB$, find the measures of the angles of triangle $ABC$. Give all the possibilities.
2001 Argentina L2 p5
Let $ABCD$ be a trapezium of bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$, such that $\angle BAD = \angle ADC = 90^o$, $AB = 54$ and $CD = 24$. It is also known that the bisector of angle $\angle ABC$ cuts the bisector of angle $\angle BCD$ at a point $P$ on the $DA$ side. Calculate the lengths of the sides $BC$ and $DA$.
Let $ABCD$ be a trapezium of bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$, such that $\angle BAD = \angle ADC = 90^o$, $AB = 54$ and $CD = 24$. It is also known that the bisector of angle $\angle ABC$ cuts the bisector of angle $\angle BCD$ at a point $P$ on the $DA$ side. Calculate the lengths of the sides $BC$ and $DA$.
2002 Argentina L2 p2
On the board was a quadrilateral $ABCD$ on which the points $P, Q, R, S$ were marked on the sides $AB, BC, CD, DA$, respectively, such that $ \frac{AP}{PB} =\frac{ BQ}{QC} =\frac{CR}{RD} = \frac{DS}{SA }= \frac12$ .The whole figure was deleted, except for the four points $P, Q, R, S$.Describe a procedure to reconstruct quadrilateral $ABCD$.
On the board was a quadrilateral $ABCD$ on which the points $P, Q, R, S$ were marked on the sides $AB, BC, CD, DA$, respectively, such that $ \frac{AP}{PB} =\frac{ BQ}{QC} =\frac{CR}{RD} = \frac{DS}{SA }= \frac12$ .The whole figure was deleted, except for the four points $P, Q, R, S$.Describe a procedure to reconstruct quadrilateral $ABCD$.
Given a circle of center $O$, four lines are drawn tangent to the circle so that these four lines determine the trapezoid $ABCD$, with bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$. If $AO = 2\sqrt6 , BO = 4\sqrt3$ and $CO = 4$, calculate the measures of the sides and angles of the trapezoid.
2004 Argentina L2 p6
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 90^o$, $\angle ACB = \angle BDC$, $\angle DBC = \angle ACD + \angle ADC$, $AC = BD$ and $CD = 3$. Calculate the area of the quadrilateral.
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 90^o$, $\angle ACB = \angle BDC$, $\angle DBC = \angle ACD + \angle ADC$, $AC = BD$ and $CD = 3$. Calculate the area of the quadrilateral.
2005 Argentina L2 p3
Let $ABC$ be a right triangle and isosceles, with $AB = AC$. We consider points $M$ and $N$ in $AB$ such that $AM = BN$. The perpendicular from $A$ on $CM$ that cuts $BC$ at $P$. If $\angle APC = 62^o$, calculate the measure of angle $\angle BNP$.
On the board was a trapezoid $ABCD$ of bases$ AB$ and $CD$, on which the four points $E, F, O$ and $P$ were marked. $E$ and $F$ are the midpoints of the non-parallel sides $AD$ and $BC$, respectively. $O$ is the point of intersection of diagonals $AC$ and $BD$, and $P$ is an arbitrary point on line $AB$. The entire figure was deleted, except for the four points $E, F, O$ and $P$. Describe a procedure that allows the trapezium $ABCD$ to be reconstructed.
2007 Argentina L2 p3
Let $ABC$ be a triangle with $\angle A=45^o$, and the bisector of $A$, the median from $B$, and the altitude from $C$ meet at a point. Calculate the measure of the angle $\angle B$ .
From a square of paper on side $1$, cut two equal equilateral triangles. Find the maximum possible value on the side of the triangles.
Let $ABC$ be a triangle such that $\angle A= 3\angle B$. If $BC = 5$ and $CA = 3$, calculate the measurement of side $AB$.
Let $ABC$ be a right triangle and isosceles, with $AB = AC$. We consider points $M$ and $N$ in $AB$ such that $AM = BN$. The perpendicular from $A$ on $CM$ that cuts $BC$ at $P$. If $\angle APC = 62^o$, calculate the measure of angle $\angle BNP$.
2007 Argentina L2 p3
Let $ABC$ be a triangle with $\angle A=45^o$, and the bisector of $A$, the median from $B$, and the altitude from $C$ meet at a point. Calculate the measure of the angle $\angle B$ .
Let $ABC$ be a triangle such that $\angle C = 45^o$ and $2AC = 3BC$. Let $k$ be the circle passing through $A, C$ and tangent to $BC$ at $C$, and let $k'$ be the circle passing through $B, C$ and tangent to $AC$ at $C$. The other point of intersection of $k$ and $k'$ is $D$. Line $CD$ intersects side $AB$ at $E$. If $AD = 6$ , calculate $AE$ and $BE$.
2010 Argentina L2 p3
Let $ABCD$ be a trapezoid with $AB\parallel CD, AB> CD$, such that $BC = CD = DA$. Points $E$ and $F$ divide $AB$ into three equal parts, $E$ is between $A$ and $F$. Lines $CF$ and $DE$ intersect $P$. Show that $\angle APB = \angle DAB$.
2011 Argentina L2 p3
Let $ABC$ be a triangle of sides $AB = 15, AC = 14$ and $BC = 13$. Let $M$ be the midpoint of side $AB$, and let $I$ be the intersection of the bisectors of triangle $ABC$. The line $MI$ cuts the atlitude corresponding to the side $AB$ of the triangle $ABC$ at the point $P$. Calculate the length of the segment $PC$.
2012 Argentina L2 p3
Let $ABC$ be a triangle with $\angle A= 105^o, \angle B= 45^o$. Let L be in $BC$ such that $AL$ is the bisector of $\angle BAC$ and $M$ the midpoint of $AC$. If $AL$ and $BM$ intersect at $P,$ calculate the ratio $\frac{AP}{AL}$ .
2010 Argentina L2 p3
Let $ABCD$ be a trapezoid with $AB\parallel CD, AB> CD$, such that $BC = CD = DA$. Points $E$ and $F$ divide $AB$ into three equal parts, $E$ is between $A$ and $F$. Lines $CF$ and $DE$ intersect $P$. Show that $\angle APB = \angle DAB$.
2011 Argentina L2 p3
Let $ABC$ be a triangle of sides $AB = 15, AC = 14$ and $BC = 13$. Let $M$ be the midpoint of side $AB$, and let $I$ be the intersection of the bisectors of triangle $ABC$. The line $MI$ cuts the atlitude corresponding to the side $AB$ of the triangle $ABC$ at the point $P$. Calculate the length of the segment $PC$.
2012 Argentina L2 p3
Let $ABC$ be a triangle with $\angle A= 105^o, \angle B= 45^o$. Let L be in $BC$ such that $AL$ is the bisector of $\angle BAC$ and $M$ the midpoint of $AC$. If $AL$ and $BM$ intersect at $P,$ calculate the ratio $\frac{AP}{AL}$ .
2013 Argentina L2 p2
Let $ABC$ be a right triangle. It is known that there are points $D$ on the side $AC$ side and $E$ on the side $BC$ such that $AB = AD = BE$ and $BD$ is perpendicular to $DE$. Calculate $\frac{AB}{BC}$ and $\frac{BC}{CA}$.
2013 Argentina L2 p6
Decide if there is a square with a side less than $1$ that can completely cover any rectangle of diagonal $1$.
2014 Argentina L2 p3
Let $ABCD$ be a parallelogram of sides $AB = 10$ and $BC = 6$. The circle $c_1$ and $c_2$ pass through $B$ and have centers $A$ and $C$ respectively. An arbitrary circle with center $D$ intersects $c_1$ at points$ P_1, Q_1$ and $c_2$ at points $P_2, Q_2$. Find the ratio $\frac{P_1Q_1}{P_2Q_2}$
2015 Argentina L2 p2
The rectangle $ABCD$ has sides $AB = 3, BC = 2$. The point $P$ on the side $AB$ is such that the bisector of $\angle CDP$ passes through the midpoint of $BC$. Find the length of the segment $BP$.
Let $ABC$ be a right triangle. It is known that there are points $D$ on the side $AC$ side and $E$ on the side $BC$ such that $AB = AD = BE$ and $BD$ is perpendicular to $DE$. Calculate $\frac{AB}{BC}$ and $\frac{BC}{CA}$.
2013 Argentina L2 p6
Decide if there is a square with a side less than $1$ that can completely cover any rectangle of diagonal $1$.
2014 Argentina L2 p3
Let $ABCD$ be a parallelogram of sides $AB = 10$ and $BC = 6$. The circle $c_1$ and $c_2$ pass through $B$ and have centers $A$ and $C$ respectively. An arbitrary circle with center $D$ intersects $c_1$ at points$ P_1, Q_1$ and $c_2$ at points $P_2, Q_2$. Find the ratio $\frac{P_1Q_1}{P_2Q_2}$
2015 Argentina L2 p2
The rectangle $ABCD$ has sides $AB = 3, BC = 2$. The point $P$ on the side $AB$ is such that the bisector of $\angle CDP$ passes through the midpoint of $BC$. Find the length of the segment $BP$.
2016 Argentina L2 p2
Point $D$ on the side $BC$ of the acute triangle $ABC$ is chosen so that $AD = AC$. Let $P$ and $Q$ respectively be the feet of the perpendiculars from $C$ and $D$ on the side $AB$. $AP^2 + 3BP^2 = AQ^2 + 3BQ^2$ is known. Calculate the measure of angle $\angle ABC$.
Point $D$ on the side $BC$ of the acute triangle $ABC$ is chosen so that $AD = AC$. Let $P$ and $Q$ respectively be the feet of the perpendiculars from $C$ and $D$ on the side $AB$. $AP^2 + 3BP^2 = AQ^2 + 3BQ^2$ is known. Calculate the measure of angle $\angle ABC$.
We have a convex quadrilateral $ABCD$ with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and $Q$ on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Determine the value of the radius of the circle that passes through the vertices of the triangle $DPQ$.
2018 Argentina L2 p3
A geometry program on the computer allows the following operations to be performed:
$\bullet$ Mark points on segments, on lines, or outside them.
$\bullet$ Draw the line that joins two points.
$\bullet$ Find the point of intersection of two lines.
$\bullet$ Given a point $P$ and a line $\ell$, trace the symmetric of $P$ with respect to $\ell$.
Given an $ABC$ triangle, using exclusively the allowed operations, construct the intersection point of the perpendicular bisectors of the triangle.
A geometry program on the computer allows the following operations to be performed:
$\bullet$ Mark points on segments, on lines, or outside them.
$\bullet$ Draw the line that joins two points.
$\bullet$ Find the point of intersection of two lines.
$\bullet$ Given a point $P$ and a line $\ell$, trace the symmetric of $P$ with respect to $\ell$.
Given an $ABC$ triangle, using exclusively the allowed operations, construct the intersection point of the perpendicular bisectors of the triangle.
Let $\Gamma$ be a circle of center $S$ and radius $r$ and $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$, and we consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.
Let $ABCD$ be a parallelogram with $\angle ABC = 105^o$. Inside the parallelogram there is a point $E$ such that the triangle $BEC$ is equilateral and $\angle CED = 135^o$. Let K be the midpoint of side $AB$. Calculate the measure of the angle $\angle BKC$.
1995 - 2020 level 3
1995 Argentina L3 p3
Let ABCD be a parallelogram, and P a point such that $2 \angle PDA=\angle ABP$ and $2 \angle PAD=\angle PCD$. Show that $AB=BP=CP$
Let ABCD be a parallelogram, and P a point such that $2 \angle PDA=\angle ABP$ and $2 \angle PAD=\angle PCD$. Show that $AB=BP=CP$
1996 Argentina L3 p3
The non-regular hexagon $ABCDEF$ is inscribed on a circle of center $O$ and $AB = CD = EF$. If diagonals $AC$ and $BD$ intersect at $M$, diagonals $CE$ and $DF$ intersect at $N$, and diagonals $AE$ and $BF$ intersect at $K$, show that the heights of triangle $MNK$ intersect at $O$.
1996 Argentina L3 p4
Let $ABCD$ be a parallelogram with center $O$ such that $\angle BAD <90^o$ and $\angle AOB> 90^o$. Consider points $A_1$ and $B_1$ on the rays $OA$ and $OB$ respectively, such that $A_1B_1$ is parallel to $AB$ and $\angle A_1B_1C = \frac12 \angle ABC$. Prove that $A_1D$ is perpendicular to $B_1C$.
The non-regular hexagon $ABCDEF$ is inscribed on a circle of center $O$ and $AB = CD = EF$. If diagonals $AC$ and $BD$ intersect at $M$, diagonals $CE$ and $DF$ intersect at $N$, and diagonals $AE$ and $BF$ intersect at $K$, show that the heights of triangle $MNK$ intersect at $O$.
1996 Argentina L3 p4
Let $ABCD$ be a parallelogram with center $O$ such that $\angle BAD <90^o$ and $\angle AOB> 90^o$. Consider points $A_1$ and $B_1$ on the rays $OA$ and $OB$ respectively, such that $A_1B_1$ is parallel to $AB$ and $\angle A_1B_1C = \frac12 \angle ABC$. Prove that $A_1D$ is perpendicular to $B_1C$.
1997 Argentina L3 p2
Let $ABC$ be a triangle and $M$ be the midpoint of $AB$. If it is known that $\angle CAM + \angle MCB = 90^o$, show that triangle $ABC$ is isosceles or right.
Let $ABC$ be a triangle and $M$ be the midpoint of $AB$. If it is known that $\angle CAM + \angle MCB = 90^o$, show that triangle $ABC$ is isosceles or right.
1997 Argentina L3 p5
Given two non-parallel segments $AB$ and $CD$ on the plane, find the locus of points $P$ on the plane such that the area of triangle $ABP$ is equal to the area of triangle $CDP$.
Given two non-parallel segments $AB$ and $CD$ on the plane, find the locus of points $P$ on the plane such that the area of triangle $ABP$ is equal to the area of triangle $CDP$.
1998 Argentina L3 p2
Let a quadrilateral $ABCD$ have an inscribed circle and let $K, L, M, N$ be the tangency points of the sides $AB, BC, CD$ and $DA$, respectively. Consider the orthocenters of each of the triangles $\vartriangle AKN, \vartriangle BLK, \vartriangle CML$ and $\vartriangle DNM$. Prove that these four points are the vertices of a parallelogram.
1998 Argentina L3 p5
Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.
Let a quadrilateral $ABCD$ have an inscribed circle and let $K, L, M, N$ be the tangency points of the sides $AB, BC, CD$ and $DA$, respectively. Consider the orthocenters of each of the triangles $\vartriangle AKN, \vartriangle BLK, \vartriangle CML$ and $\vartriangle DNM$. Prove that these four points are the vertices of a parallelogram.
1998 Argentina L3 p5
Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.
1999 Argentina L3 p2
Let $C_1$ and $C_2$ be the outer circumferences of centers $O_1$ and $O_2$, respectively. The two tangents to the circumference $C_2$ are drawn by $O_1$, intersecting $C_1$ at $P$ and $P'$. The two tangents to the circumference $C_1$ are drawn by $O_2$, intersecting $C_2$ at $Q$ and $Q'$. Prove that the segment $PP'$ is equal to the segment $QQ'$.
Let $C_1$ and $C_2$ be the outer circumferences of centers $O_1$ and $O_2$, respectively. The two tangents to the circumference $C_2$ are drawn by $O_1$, intersecting $C_1$ at $P$ and $P'$. The two tangents to the circumference $C_1$ are drawn by $O_2$, intersecting $C_2$ at $Q$ and $Q'$. Prove that the segment $PP'$ is equal to the segment $QQ'$.
2000 Argentina L3 p2
Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.
Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.
2000 Argentina L3 p6
You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.
You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.
2001 Argentina L3 p2
Let $\vartriangle ABC$ be a triangle such that angle $\angle ABC$ is less than angle $\angle ACB$. The bisector of angle $\angle BAC$ cuts side $BC$ at $D$. Let $E$ be on side $AB$ such that $\angle EDB = 90^o$ and $F$ on side $AC$ such that $\angle BED = \angle DEF$. Prove that $\angle BAD = \angle FDC$.
Let $\vartriangle ABC$ be a triangle such that angle $\angle ABC$ is less than angle $\angle ACB$. The bisector of angle $\angle BAC$ cuts side $BC$ at $D$. Let $E$ be on side $AB$ such that $\angle EDB = 90^o$ and $F$ on side $AC$ such that $\angle BED = \angle DEF$. Prove that $\angle BAD = \angle FDC$.
2002 Argentina L3 p3
In a circumference $\Gamma$ a chord $PQ$ is considered such that the segment that joins the midpoint of the smallest arc $PQ$ and the midpoint of the segment $PQ$ measures $1$. Let $\Gamma_1, \Gamma_2$ and $\Gamma_3$ be three tangent circumferences to the chord $PQ$ that are in the same half plane than the center of $\Gamma$ with respect to the line $PQ$. Furthermore, $\Gamma_1$ and $\Gamma_3$ are internally tangent to $\Gamma$ and externally tangent to$ \Gamma_2$, and the centers of $\Gamma_1$ and $\Gamma_3$ are on different halfplanes with respect to the line determined by the centers of $\Gamma$ and $\Gamma_2$.
If the sum of the radii of $\Gamma_1, \Gamma_2$ and $\Gamma_3$ is equal to the radius of $\Gamma$, calculate the radius of $\Gamma_2$.
In a circumference $\Gamma$ a chord $PQ$ is considered such that the segment that joins the midpoint of the smallest arc $PQ$ and the midpoint of the segment $PQ$ measures $1$. Let $\Gamma_1, \Gamma_2$ and $\Gamma_3$ be three tangent circumferences to the chord $PQ$ that are in the same half plane than the center of $\Gamma$ with respect to the line $PQ$. Furthermore, $\Gamma_1$ and $\Gamma_3$ are internally tangent to $\Gamma$ and externally tangent to$ \Gamma_2$, and the centers of $\Gamma_1$ and $\Gamma_3$ are on different halfplanes with respect to the line determined by the centers of $\Gamma$ and $\Gamma_2$.
If the sum of the radii of $\Gamma_1, \Gamma_2$ and $\Gamma_3$ is equal to the radius of $\Gamma$, calculate the radius of $\Gamma_2$.
2002 Argentina L3 p5
Let $\vartriangle ABC$ be an isosceles triangle with $AC = BC$. Points $D, E, F$ are considered on $BC, CA, AB$, respectively, such that $AF> BF$ and that the quadrilateral $CEFD$ is a parallelogram. The perpendicular line to $BC$ drawn by $B$ intersects the perpendicular bisector of $AB$ at $G$. Prove that $DE \perp FG$.
Let $\vartriangle ABC$ be an isosceles triangle with $AC = BC$. Points $D, E, F$ are considered on $BC, CA, AB$, respectively, such that $AF> BF$ and that the quadrilateral $CEFD$ is a parallelogram. The perpendicular line to $BC$ drawn by $B$ intersects the perpendicular bisector of $AB$ at $G$. Prove that $DE \perp FG$.
2003 Argentina L3 p4
The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.
The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.
2004 Argentina L3 p5
The pentagon $ABCDE$ has $AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o$ and $BD = 2$. Calculate the area of the pentagon.
The pentagon $ABCDE$ has $AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o$ and $BD = 2$. Calculate the area of the pentagon.
2005 Argentina L3 p5
Let $AM$ and $AN$ be the lines tangent to a circle $\Gamma$ drawn from a point $A$ $(M$ and $N$ belong to the circle). A line through $A$ cuts $\Gamma$ at $B$ and $C$ with $B$ between $A$ and $C$, and $\frac{AB}{BC} =\frac23$. If $P$ is the intersection point of $AB$ and $MN$, calculate $\frac{AP}{CP}$.
Let $AM$ and $AN$ be the lines tangent to a circle $\Gamma$ drawn from a point $A$ $(M$ and $N$ belong to the circle). A line through $A$ cuts $\Gamma$ at $B$ and $C$ with $B$ between $A$ and $C$, and $\frac{AB}{BC} =\frac23$. If $P$ is the intersection point of $AB$ and $MN$, calculate $\frac{AP}{CP}$.
In triangle $ABC, M$ is the midpoint of $AB$ and $D$ the foot of the bisector of angle $\angle ABC$. If $MD$ and $BD$ are known to be perpendicular, calculate $\frac{AB}{BC}$.
2007 Argentina L3 p3
Let $ ABCD$ be a parellogram with $ AB>AD$. Suposse the ratio between diagonals $ AC$ and $ BD$ is $ \frac {AC} {BD}=3$. Let $ r$ be the line symmetric to $ AD$ with respect to $ AC$ and $ s$ the line symmetric to $ BC$ with respect to $ BD$. If $ r$ and $ s$ intersect at $ P$ , find the ratio $ \frac {PA} {PB}$
Let $ ABCD$ be a parellogram with $ AB>AD$. Suposse the ratio between diagonals $ AC$ and $ BD$ is $ \frac {AC} {BD}=3$. Let $ r$ be the line symmetric to $ AD$ with respect to $ AC$ and $ s$ the line symmetric to $ BC$ with respect to $ BD$. If $ r$ and $ s$ intersect at $ P$ , find the ratio $ \frac {PA} {PB}$
On a circle of center $O$, let $A$ and $B$ be points on the circle such that $\angle AOB = 120^o$. Point $C$ lies on the small arc $AB$ and point $D$ lies on the segment $AB$. Let also $AD = 2, BD = 1$ and $CD = \sqrt2$. Calculate the area of triangle $ABC$.
2009 Argentina L3 p3
Isosceles trapezoid $ ABCD$, with $ AB \parallel CD$, is such that there exists a circle $ \Gamma$ tangent to its four sides. Let $ T = \Gamma \cap BC$, and $ P= \Gamma \cap AT$ ($ P \neq T$).If $ \frac{AP}{AT} =\frac{2}{5}$, compute $ \frac{AB}{CD}$.
Isosceles trapezoid $ ABCD$, with $ AB \parallel CD$, is such that there exists a circle $ \Gamma$ tangent to its four sides. Let $ T = \Gamma \cap BC$, and $ P= \Gamma \cap AT$ ($ P \neq T$).If $ \frac{AP}{AT} =\frac{2}{5}$, compute $ \frac{AB}{CD}$.
Let $ABC$ be a triangle with $\angle C = 90^o$ and $AC = 1$. The median $AM$ intersects the incircle at the points $P$ and $Q$, with $P$ between $A$ and $Q$, such that $AP = QM$. Find the length of $PQ$.
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 75^o$ and $AB = 2$. The points $P$ and $Q$ on the sides $AC$ and $BC$ respectively are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$ . Calculate the measurement of the segment $QA $.
We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.
In the triangle $ABC$ the incircle is tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively. The line $DE$ intersects the circumcircle at $P$ and $Q$, with $P$ in the small arc $AB$ and $Q$ in the small arc $AC$. If $P$ is the midpoint of the arc $AB$, find the angle A and the ratio $\frac{PQ}{BC}$.
In a convex quadrilateral $ABCD$ the angles $\angle A$ and $\angle C$ are equal and the bisector of $\angle B$ passes through the midpoint of the side $CD$. If it is known that $CD = 3AD$, calculate $\frac{AB}{BC}$.
2014 Argentina L3 p3
Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$ and $B$, with $AO=BO$. Find the distance of point $A$ to the line $OB$.
Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$ and $B$, with $AO=BO$. Find the distance of point $A$ to the line $OB$.
Consider the points $O = (0,0), A = (- 2,0)$ and $B = (0,2)$ in the coordinate plane. Let $E$ and $F$ be the midpoints of $OA$ and $OB$ respectively. We rotate the triangle $OEF$ with a center in $O$ clockwise until we obtain the triangle $OE'F'$ and, for each rotated position, let $P = (x, y)$ be the intersection of the lines $AE'$ and $BF'$. Find the maximum possible value of the $y$-coordinate of $P$.
Find the angles of a convex quadrilateral $ABCD$ such that $\angle ABD = 29^o$, $\angle ADB = 41^o$, $\angle ACB = 82^o$ and $\angle ACD = 58^o$
Let $ABC$ be a triangle of perimeter $100$ and $I$ be the point of intersection of its bisectors. Let $M$ be the midpoint of side $BC$. The line parallel to $AB$ drawn by$ I$ cuts the median $AM$ at point $P$ so that $\frac{AP}{PM} =\frac73$. Find the length of side $AB$.
Let $ABCD$ be a parallelogram. An interior circle of the $ABCD$ is tangent to the lines $AB$ and $AD$ and intersects the diagonal $BD$ at $E$ and $F$. Prove that exists a circle that passes through $E$ and $F$ and is tangent to the lines $CB$ and $CD$.
In triangle $ABC$ it is known that $\angle ACB = 2\angle ABC$. Furthermore $P$ is an interior point of the triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $\angle BAC = 3 \angle BAP$.
Let $ABC$ be a right isosceles triangle with right angle at $A$. Let $E$ and $F$ be points on A$B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$. Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$. Calculate the measure of angle $\angle FDC$.
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