geometry problems

from Chilean Mathematical Olympiads (and a few TSTs)

1989 - 2018

under construction

Given a $\triangle ABC$, determine which is the circle with the smallest area that contains it.

2007 Chile Level 2 P6

Given an $\triangle ABC$ isoceles with base $BC$ we note with $M$ the midpoint of said base. Let $X$ be any point on the shortest arc $AM$ of the circumcircle of $\triangle ABM$ and let $T$ be a point on the inside $\angle BMA$ such that $\angle TMX = 90^o$ and $TX = BX$. Show that $\angle MTB - \angle CTM$ does not depend on $X$.

2008 Chile Level 2 P2

Let $ABC$ be right isosceles triangle with right angle in $A$. Given a point $P$ inside the triangle, denote by $a, b$ and $c$ the lengths of $PA, PB$ and $PC$, respectively. Prove that there is a triangle whose sides have a length of $a\sqrt2 , b$ and $c$

2009 Chile Level 2 P2

Consider $P$ a regular $9$-sided convex polygon with each side of length $1$. A diagonal at $P$ is any line joining two non-adjacent vertices of $P$. Calculate the difference between the lengths of the largest and smallest diagonal of $P$.

2006 Chile Level 1 P2

The vertex $ E $ of a square $ EFGH $ of side $2006$ mm is found in the center of the square $ ABCD $ of side $10$ mm. The line $ EF $ intersects $ CD $ at point $ I $. The line$ EH $ intercepts $ AD $ in $ J $. Also $ \angle EID = 60^o$. Calculate the area of the quadrilateral $ EIDJ $.

2007 Chile Level 1 P2

From a triangle $T = \triangle ABC$, we build the triangle $T_1 = \triangle A_1B_1C_1$ whose vertices they are the midpoints of the sides of $T$. The triangle $T_2 = \triangle A_2B_2C_2$ is constructed from $T_1$ in a way analogue. We build the triangles $T_3, T_4,..., T_{2007}$. Prove that the center of gravity $G$ of the triangle $T$ is inside the triangle $T_{2007}$.

2008 Chile Level 1 P2

2009 Chile Level 1 P4

On the base $AC$ of a triangle angle isosceles $ABC$, a point is taken $M$, so that $| AM | = p$ and $| MC | = q$. The inscribed circles are drawn to the $AMB$ and $CMB$ triangles, which are tangent to the $BM$ side at points $R$ and $S$ respectively. Find the distance between $R$ and $S$.

www.olimpiadadematematica.cl

www.fmat.cl/index.php?showforum=540

http://inst-mat.utalca.cl/tem/tem/inicio/inicio1.htm

inst-mat.utalca.cl/tem/tem/inicio/pruebasolim/index.html

from Chilean Mathematical Olympiads (and a few TSTs)

Olimpiada de Matemática de Chile

Olimpiada de Matemática de Chile

under construction

started in 1989

level 1 started in 1999

Level 2 (nivel mayor)

The lengths of the three sides of a $ \triangle ABC $ are rational. The altitude $ CD $ determines on the side $AB$ two segments $ AD $ and $ DB $. Prove that $ AD, DB $ are rational.

1989 Chile Level 2 P3 missing (might be geometry)

1996 Chile Level 2 P7 missing (might be geometry)

2006 Chile Level 2 P2

In a triangle $ \vartriangle ABC $ with sides integer numbers, it is known that the radius of the circumcircle circumscribed to $ \vartriangle ABC $ measures $ \dfrac {65} {8} $ centimeters and the area is $84$ cm². Determine the lengths of the sides of the triangle.

Let $ \vartriangle ABC $ be an acute triangle and scalene, with $ BC $ its smallest side. Let $ P, Q $ points on $ AB, AC $ respectively, such that $ BQ = CP = BC $. Let $ O_1, O_2 $ be the centers of the circles circumscribed to $ \vartriangle AQB, \vartriangle APC $, respectively. Sean $ H, O $ the orthocenter and circumcenter of $ \vartriangle ABC $

a) Show that $ O_1O_2 = BC $.

b) Show that $ BO_2, CO_1 $ and $ HO $ are concurrent

2007 Chile Level 2 P2The lengths of the three sides of a $ \triangle ABC $ are rational. The altitude $ CD $ determines on the side $AB$ two segments $ AD $ and $ DB $. Prove that $ AD, DB $ are rational.

1989 Chile Level 2 P3 missing (might be geometry)

1990 Chile P6

Given a regular polygon with apothem $ A $ and circumradius $ R $. Find a regular polygon of equal perimeter and with double sides, the apothem $ a $ and the circumcircle $ r $.

Given a regular polygon with apothem $ A $ and circumradius $ R $. Find a regular polygon of equal perimeter and with double sides, the apothem $ a $ and the circumcircle $ r $.

1991 Chile P2

If a polygon inscribed in a circle is equiangular and has an odd number of sides, prove that it is regular.

If a polygon inscribed in a circle is equiangular and has an odd number of sides, prove that it is regular.

1991 Chile P6

Given a triangle with $ \triangle ABC $, with: $ \angle C = 36^o$ and $ \angle A = \angle B $. Consider the points $ D $ on $ BC $, $ E $ on $ AD $, $ F $ on $ BE $, $ G $ on $ DF $ and $ H $ on $ EG $, so that the rays $ AD, BE, DF, EG, FH $ bisect the angles $ A, B, D, E, F $ respectively. It is known that $ FH = 1 $. Calculate $ AC$.

1991 Chile Level 2 P7 missing (might be geometry)

Given a triangle with $ \triangle ABC $, with: $ \angle C = 36^o$ and $ \angle A = \angle B $. Consider the points $ D $ on $ BC $, $ E $ on $ AD $, $ F $ on $ BE $, $ G $ on $ DF $ and $ H $ on $ EG $, so that the rays $ AD, BE, DF, EG, FH $ bisect the angles $ A, B, D, E, F $ respectively. It is known that $ FH = 1 $. Calculate $ AC$.

1991 Chile Level 2 P7 missing (might be geometry)

1992 Chile Level 2 P4

Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.

Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.

1992 Chile Level 2 P5

In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $.

a) Prove that $ I$ lies between $ M $ and $ H $.

b) Calculate $ AB ^ 2-AC ^ 2 $.

c) Determine $ \dfrac {AB} {AC} $.

d) Find the measure of all the sides and angles of the triangle.

In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $.

a) Prove that $ I$ lies between $ M $ and $ H $.

b) Calculate $ AB ^ 2-AC ^ 2 $.

c) Determine $ \dfrac {AB} {AC} $.

d) Find the measure of all the sides and angles of the triangle.

1993 Chile Level 2 P1

There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.

1993 Chile Level 2 P2

Given a rectangle, circumscribe a rectangle of maximum area.

1993 Chile Level 2 P6

There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.

Given a rectangle, circumscribe a rectangle of maximum area.

1993 Chile Level 2 P6

Let $ ABCD $ be a rectangle of area $ S $, and $ P $ be a point inside it. We denote by $ a, b, c, d $ the distances from $ P $ to the vertices $ A, B, C, D $ respectively. Prove that $ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2\ge 2S $. When there is equality?

1994 Chile Level 2 P2

Show that it is possible to cut any triangle into several pieces, so that a rectangle is formed when they are joined together.

Show that it is possible to cut any triangle into several pieces, so that a rectangle is formed when they are joined together.

1994 Chile Level 2 P6

On a sheet of transparent paper, draw a quadrilateral with Chinese ink, which is illuminated with a lamp. Show that it is always possible to locate the sheet in such a way that the shadow projected on the desk is a parallelogram.

1995 Chile Level 2 P2

In a circle of radius $1$, six arcs of radius $1$ are drawn, which cut the circle as in the figure. Determine the black area.

On a sheet of transparent paper, draw a quadrilateral with Chinese ink, which is illuminated with a lamp. Show that it is always possible to locate the sheet in such a way that the shadow projected on the desk is a parallelogram.

1995 Chile Level 2 P2

In a circle of radius $1$, six arcs of radius $1$ are drawn, which cut the circle as in the figure. Determine the black area.

1995 Chile Level 2 P7

In a semicircle of radius 4 three circles are inscribed, as indicated in the figure. Larger circles have radii $ R_1 $ and $ R_2 $, and the larger circle has radius $ r $.

a) Prove that $ \dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}} $

b) Prove that $ R_1 + R_2 \le 8 (\sqrt{2} -1) $

c) Prove that $ r \le \sqrt{2} -1 $

In a semicircle of radius 4 three circles are inscribed, as indicated in the figure. Larger circles have radii $ R_1 $ and $ R_2 $, and the larger circle has radius $ r $.

a) Prove that $ \dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}} $

b) Prove that $ R_1 + R_2 \le 8 (\sqrt{2} -1) $

c) Prove that $ r \le \sqrt{2} -1 $

1996 Chile Level 2 P2

Construct the $ \triangle ABC $, with $ AC <BC $, if the circumcircle is known, and the points $ D, E, F $ in it, where they intersect, respectively, the height, the median and the bisector that they start from the vertex $ C $.

1996 Chile Level 2 P6

Two circles, $ C $ and $ K $, intersect at $ A $ and $ B $. Let $ P $ a point in the arc $ AB $ in $ C $. Lines $ PA $ and $ PB $ cut back to $ K $ in $ R $ and $ S $, respectively. Let $ P_1 $ be another point in the same arc as $ P $, so that lines $ P_1A $ and $ P_1B $ re-intersect $ K $ in $ R_1 $ and $ S_1 $, respectively. Prove that the arcs $ RS $ and $ R_1S_1 $ have equal measure.

Construct the $ \triangle ABC $, with $ AC <BC $, if the circumcircle is known, and the points $ D, E, F $ in it, where they intersect, respectively, the height, the median and the bisector that they start from the vertex $ C $.

1996 Chile Level 2 P6

Two circles, $ C $ and $ K $, intersect at $ A $ and $ B $. Let $ P $ a point in the arc $ AB $ in $ C $. Lines $ PA $ and $ PB $ cut back to $ K $ in $ R $ and $ S $, respectively. Let $ P_1 $ be another point in the same arc as $ P $, so that lines $ P_1A $ and $ P_1B $ re-intersect $ K $ in $ R_1 $ and $ S_1 $, respectively. Prove that the arcs $ RS $ and $ R_1S_1 $ have equal measure.

1996 Chile Level 2 P7 missing (might be geometry)

1997 Chile Level 2 P3

Let $ ABCD $ be a quadrilateral, whose diagonals intersect at $ O $. The triangles $ \triangle AOB $, $ \triangle BOC $, $ \triangle COD $ have areas $1, 2, 4$, respectively. Find the area of $ \triangle AOD $ and prove that $ ABCD $ is a trapezoid.

Let $ ABCD $ be a quadrilateral, whose diagonals intersect at $ O $. The triangles $ \triangle AOB $, $ \triangle BOC $, $ \triangle COD $ have areas $1, 2, 4$, respectively. Find the area of $ \triangle AOD $ and prove that $ ABCD $ is a trapezoid.

1997 Chile Level 2 P5

Let: $ C_1, C_2, C_3 $ three circles , intersecting in pairs, such that the secant line common to two of them (any) passes through the center of the third. Prove that the three lines thus defined are concurrent.

Let: $ C_1, C_2, C_3 $ three circles , intersecting in pairs, such that the secant line common to two of them (any) passes through the center of the third. Prove that the three lines thus defined are concurrent.

1998 Chile Level 2 P2 (typo in question b, not posted in aops)

Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $.

a) Prove that the lines $ EF $ and $ AB $ are perpendicular.

b) Determine the LG of the point $ E $.

c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.

Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $.

a) Prove that the lines $ EF $ and $ AB $ are perpendicular.

b) Determine the LG of the point $ E $.

c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.

1998 Chile Level 2 P6

To create an equilateral triangle, cut it into four polygonal figures so that, reassembled properly, these figures form a square.

To create an equilateral triangle, cut it into four polygonal figures so that, reassembled properly, these figures form a square.

1999 Chile Level 2 P2 missing (might be geometry)

1999 Chile Level 2 P6

Prove that there are infinite pairs of non-congruent triangles that have the same angles and two of their equal sides. Develop an algorithm or rule to obtain these pairs of triangles and indicate at least one pair that satisfies the asserted.

Prove that there are infinite pairs of non-congruent triangles that have the same angles and two of their equal sides. Develop an algorithm or rule to obtain these pairs of triangles and indicate at least one pair that satisfies the asserted.

2000 Chile Level 2 P4

Let $ AD $ be the bisector of a triangle $ ABC $ $ (D \in BC) $ such that $ AB + AD = CD $ and $ AC + AD = BC $. Determine the measure of the angles of $ \vartriangle ABC $

Let $ AD $ be the bisector of a triangle $ ABC $ $ (D \in BC) $ such that $ AB + AD = CD $ and $ AC + AD = BC $. Determine the measure of the angles of $ \vartriangle ABC $

2001 Chile Level 2 P3

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that:

$$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$

For what kind of triangles does the equality hold?

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that:

$$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$

For what kind of triangles does the equality hold?

2001 Chile Level 2 P3

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that:

$$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$

For what kind of triangles does the equality hold?

2001 Chile Level 2 P6

Let $ C_1, C_2 $ be two circles of equal radius, disjoint, of centers $ O_1, O_2 $, such that $ C_1 $ is to the left of $ C_2 $. Let $ l $ be a line parallel to the line $ O_1O_2 $, secant to both circles. Let $ P_1 $ be a point of $ l $, to the left of $ C_1 $ and $ P_2 $ a point of $ l $, to the right of $ C_2 $ such that the tangents of $ P_1 $ to $ C_1 $ and of $ P_2 $ a $ C_2 $ form a quadrilateral. Show that there is a circle tangent to the four sides of said quadrilateral.

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that:

$$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$

For what kind of triangles does the equality hold?

2001 Chile Level 2 P6

Let $ C_1, C_2 $ be two circles of equal radius, disjoint, of centers $ O_1, O_2 $, such that $ C_1 $ is to the left of $ C_2 $. Let $ l $ be a line parallel to the line $ O_1O_2 $, secant to both circles. Let $ P_1 $ be a point of $ l $, to the left of $ C_1 $ and $ P_2 $ a point of $ l $, to the right of $ C_2 $ such that the tangents of $ P_1 $ to $ C_1 $ and of $ P_2 $ a $ C_2 $ form a quadrilateral. Show that there is a circle tangent to the four sides of said quadrilateral.

2002 Chile Level 2 P2

If in the $ \vartriangle ABC $, two sides are not greater than their corresponding altitudes, how much do the angles of the triangle measure?

If in the $ \vartriangle ABC $, two sides are not greater than their corresponding altitudes, how much do the angles of the triangle measure?

2002 Chile Level 2 P7

Given the segment $ AB $, let $ M $ be one point lying on it. Towards the same side of the plane and with base $ AM $ and $ MB $, the squares $ AMCD $ and $ MBEF $ are constructed. Let $ P $ and $ Q $ be the respective centers of these squares. Determine how the midpoint of the segment $ PQ $ moves as the point $ M $ moves aling the segment.

Given the segment $ AB $, let $ M $ be one point lying on it. Towards the same side of the plane and with base $ AM $ and $ MB $, the squares $ AMCD $ and $ MBEF $ are constructed. Let $ P $ and $ Q $ be the respective centers of these squares. Determine how the midpoint of the segment $ PQ $ moves as the point $ M $ moves aling the segment.

Consider a triangle $ ABC $. On the line $ AC $ take a point $ B_1 $ such that $ AB = AB_1 $ and in addition, $ B_1 $ and $ C $ are located on the same side of the line with respect to the point $ A $. The bisector of the angle $ A $ intersects the side $ BC $ at a point that we will denote as $ A_1 $. Let $ P $ and $ R $ be the circumscribed circles of the triangles $ ABC $ and $ A_1B_1C $ respectively. They intersect at points $ C $ and $ Q $. Prove that the tangent to the circle $ R $ at the point $ Q $ is parallel to the line $ AC $.

2004 Chile Level 2 P3

The perimeter, that is, the sum of the lengths of all sides of a convex quadrilateral $ ABCD $, is equal to $2004$ meters; while the length of its diagonal $ AC $ is equal to $1001$ meters. Find out if the length of the other diagonal $ BD $ can:

a) To be equal to only one meter.

b) Be equal to the length of the diagonal $ AC $.

The perimeter, that is, the sum of the lengths of all sides of a convex quadrilateral $ ABCD $, is equal to $2004$ meters; while the length of its diagonal $ AC $ is equal to $1001$ meters. Find out if the length of the other diagonal $ BD $ can:

a) To be equal to only one meter.

b) Be equal to the length of the diagonal $ AC $.

2004 Chile Level 2 P6

The $ AB, BC $ and $ CD $ segments of the polygon $ ABCD $ have the same length and are tangent to a circle $ S $, centered on the point $ O $. Let $ P $ be the point of tangency of $ BC $ with $ S $, and let $ Q $ be the intersection point of lines $ AC $ and $ BD $. Show that the point $ Q $ is collinear with the points $ P $ and $ O $.

The $ AB, BC $ and $ CD $ segments of the polygon $ ABCD $ have the same length and are tangent to a circle $ S $, centered on the point $ O $. Let $ P $ be the point of tangency of $ BC $ with $ S $, and let $ Q $ be the intersection point of lines $ AC $ and $ BD $. Show that the point $ Q $ is collinear with the points $ P $ and $ O $.

2005 Chile Level 2 P1 [figure missing]

In the center of the square of side 1 that shows the figure is an ant. At one point the ant begins to walk until it touches the left side (a), then continues walking until it reaches the bottom side (b) and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of 2.

In a triangle $ \vartriangle ABC $ with sides integer numbers, it is known that the radius of the circumcircle circumscribed to $ \vartriangle ABC $ measures $ \dfrac {65} {8} $ centimeters and the area is $84$ cm². Determine the lengths of the sides of the triangle.

Let $ \vartriangle ABC $ be an acute triangle and scalene, with $ BC $ its smallest side. Let $ P, Q $ points on $ AB, AC $ respectively, such that $ BQ = CP = BC $. Let $ O_1, O_2 $ be the centers of the circles circumscribed to $ \vartriangle AQB, \vartriangle APC $, respectively. Sean $ H, O $ the orthocenter and circumcenter of $ \vartriangle ABC $

a) Show that $ O_1O_2 = BC $.

b) Show that $ BO_2, CO_1 $ and $ HO $ are concurrent

Given a $\triangle ABC$, determine which is the circle with the smallest area that contains it.

Given an $\triangle ABC$ isoceles with base $BC$ we note with $M$ the midpoint of said base. Let $X$ be any point on the shortest arc $AM$ of the circumcircle of $\triangle ABM$ and let $T$ be a point on the inside $\angle BMA$ such that $\angle TMX = 90^o$ and $TX = BX$. Show that $\angle MTB - \angle CTM$ does not depend on $X$.

2008 Chile Level 2 P2

Let $ABC$ be right isosceles triangle with right angle in $A$. Given a point $P$ inside the triangle, denote by $a, b$ and $c$ the lengths of $PA, PB$ and $PC$, respectively. Prove that there is a triangle whose sides have a length of $a\sqrt2 , b$ and $c$

2009 Chile Level 2 P2

Consider $P$ a regular $9$-sided convex polygon with each side of length $1$. A diagonal at $P$ is any line joining two non-adjacent vertices of $P$. Calculate the difference between the lengths of the largest and smallest diagonal of $P$.

2010 Chile Level 2 P3

The sides $BC, CA$, and $AB$ of a triangle $ABC$ are tangent to a circle at points $X, Y, Z$ respectively. Show that the center of such a circle is on the line that passes through the midpoints of $BC$ and $AX$.

The sides $BC, CA$, and $AB$ of a triangle $ABC$ are tangent to a circle at points $X, Y, Z$ respectively. Show that the center of such a circle is on the line that passes through the midpoints of $BC$ and $AX$.

2010 Chile Level 2 P5

Consider a line $ \ell $ in the plane and let $ B_1, B_2, B_3 $ be different points in $ \ell$. Let $ A $ be a point that is not in $ \ell$. Show that there is $ P, Q $ in $ {B_1, B_2, B_3} $ with $ P \ne Q $ so that the distance from $ A $ to $ \ell$ is greater than the distance from $ P $ to the line that passes through $ A $ and $ Q $.

Consider a line $ \ell $ in the plane and let $ B_1, B_2, B_3 $ be different points in $ \ell$. Let $ A $ be a point that is not in $ \ell$. Show that there is $ P, Q $ in $ {B_1, B_2, B_3} $ with $ P \ne Q $ so that the distance from $ A $ to $ \ell$ is greater than the distance from $ P $ to the line that passes through $ A $ and $ Q $.

2011 Chile Level 2 P2

Let $O$ be the center of the circle circumscribed to triangle $ABC$ and let $ S_ {A} $, $ S_ {B} $, $ S_ {C} $ be the circles centered on $O$ that are tangent to the sides $BC, CA, AB$ respectively. Show that the sum of the angle between the two tangents $ S_ {A} $ from $A$ plus the angle between the two tangents $ S_ {B} $ from $B$ plus the angle between the two tangents $ S_ {C} $ from $C$ is $180$ degrees.

Let $O$ be the center of the circle circumscribed to triangle $ABC$ and let $ S_ {A} $, $ S_ {B} $, $ S_ {C} $ be the circles centered on $O$ that are tangent to the sides $BC, CA, AB$ respectively. Show that the sum of the angle between the two tangents $ S_ {A} $ from $A$ plus the angle between the two tangents $ S_ {B} $ from $B$ plus the angle between the two tangents $ S_ {C} $ from $C$ is $180$ degrees.

2012 Chile Level 2 P4

Consider an isosceles triangle $ABC$, where $AB = AC$. $D$ is a point on the $AC$ side and $P$ a point on the segment $BD$ so that the angle $\angle APC = 90^o$ and $ \angle ABP = \angle BCP $. Determine the ratio $AD: DC$.

Consider an isosceles triangle $ABC$, where $AB = AC$. $D$ is a point on the $AC$ side and $P$ a point on the segment $BD$ so that the angle $\angle APC = 90^o$ and $ \angle ABP = \angle BCP $. Determine the ratio $AD: DC$.

A conical surface $ C $ is cut by a plane $ T $ as shown in the figure on the back of this sheet. Show that $ C \cap T $ is an ellipse. You can use as an aid the fact that if you consider the two spheres tangent to $ C $ and $ T $ as shown in the figure, they intersect $ T $ in the bulbs.

2014 Chile Level 2 P2

Consider an $ABCD$ parallelogram of area $1$. Let $E$ be the center of gravity of the triangle $ABC, F$ the center of gravity of the triangle $BCD, G$ the center of gravity of the triangle $CDA$ and $H$ the center of gravity of the triangle $DAB$. Calculate the area of quadrilateral $EFGH$.

2014 Chile Level 2 P2

Consider an $ABCD$ parallelogram of area $1$. Let $E$ be the center of gravity of the triangle $ABC, F$ the center of gravity of the triangle $BCD, G$ the center of gravity of the triangle $CDA$ and $H$ the center of gravity of the triangle $DAB$. Calculate the area of quadrilateral $EFGH$.

2015 Chile Level 1 P1, Level 2 P1

On the plane, there is drawn a parallelogram $P$ and a point $X$ outside of $P$. Using only an ungraded rule, determine the point $W$ that is symmetric to $X$ with respect to the center $O$ of $P$.

On the plane, there is drawn a parallelogram $P$ and a point $X$ outside of $P$. Using only an ungraded rule, determine the point $W$ that is symmetric to $X$ with respect to the center $O$ of $P$.

2015 Chile Level 1 P5, Level 2 P5

A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $be the point of intersection of lines$ BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.

A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $be the point of intersection of lines$ BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.

2016 Chile Level 1 P2, Level 2 P2

For an equilateral triangle $\triangle ABC$, determine whether or not there is a point $P$ inside $\triangle ABC$ so that any straight line that passes through $P$ divides the triangle $\triangle ABC$ in two polygonal lines of equal length.

2016 Chile Level 2 P6

Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.

For an equilateral triangle $\triangle ABC$, determine whether or not there is a point $P$ inside $\triangle ABC$ so that any straight line that passes through $P$ divides the triangle $\triangle ABC$ in two polygonal lines of equal length.

2016 Chile Level 2 P6

Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.

2017 Chile Level 2 all year missing (there is probably geometry)

Consider $ABCD$ a square of side $1$. Points $P, Q, R,S$ are chosen in the sides $AB, BC, CD. DA$ respectively in such a way that $ |AP| = |BQ| = |CR| =|DS| = a$, with $a <1$. Segments $AQ, BR, CS$ and $DP$ are plotted. Calculate the area of the other quadrilateral , that is formed in the center of the figure.

2018 Chile Level 2 P6

Consider an acute triangle $ABC$ and its altitudes from $A$ and $B$ that intersect the respective sides in $D$ and $E$. Call $H$ the point of intersection of the altitudes . A circle is constructed with center in $H$ and radius $HE$. Let $C$ be a line tangent to the circle at point $P$. With center at $B$ and radio $BE$ another circle is plotted, and from $C$ another line is drawn tangent to the this circle at point $Q$. Show that points $D, P$, and $Q$ are collinear.

Consider an acute triangle $ABC$ and its altitudes from $A$ and $B$ that intersect the respective sides in $D$ and $E$. Call $H$ the point of intersection of the altitudes . A circle is constructed with center in $H$ and radius $HE$. Let $C$ be a line tangent to the circle at point $P$. With center at $B$ and radio $BE$ another circle is plotted, and from $C$ another line is drawn tangent to the this circle at point $Q$. Show that points $D, P$, and $Q$ are collinear.

Level 1 (nivel menor)

1999 Chile Level 1 P2

Given a pararellelogram $ ABCD $, let $ E, F $ be the midpoints of the sides $BC$ and $CD$, respectively. Prove that $ AE $ and $AF$ trisect to $BD$.

Given a pararellelogram $ ABCD $, let $ E, F $ be the midpoints of the sides $BC$ and $CD$, respectively. Prove that $ AE $ and $AF$ trisect to $BD$.

1999 Chile Level 1 P5 [figure missing]

For the rectangles in the figure we know that $a:b=1:3$ . Finf the value of the ratio of the areas of the triangles $\triangle BEC , \triangle DCF$

For the rectangles in the figure we know that $a:b=1:3$ . Finf the value of the ratio of the areas of the triangles $\triangle BEC , \triangle DCF$

2000 Chile Level 1 P5

A point $ P $ interior to a $ \vartriangle ABC $ satisfies: $ \angle PBA = \angle PCA = \dfrac {1} {3} (\angle ABC + \angle ACB) $.

Prove that $ \dfrac {AC + PB} {AB} = \dfrac {AB + PC} {AC} $A point $ P $ interior to a $ \vartriangle ABC $ satisfies: $ \angle PBA = \angle PCA = \dfrac {1} {3} (\angle ABC + \angle ACB) $.

2001 Chile Level 1 P5

The segment $ AB $ measures $9$ cm. and the point $ C $ lies on it, such that $ \dfrac {AC} {CB} = 2 $. The point $ D $ is such that $ \angle ACD = 60^o$ and $ \angle ABD = 45^o$. Determine the measures of the angles of the triangles $ \vartriangle ACD, \vartriangle CBD $ and $ \vartriangle ABD $.

The segment $ AB $ measures $9$ cm. and the point $ C $ lies on it, such that $ \dfrac {AC} {CB} = 2 $. The point $ D $ is such that $ \angle ACD = 60^o$ and $ \angle ABD = 45^o$. Determine the measures of the angles of the triangles $ \vartriangle ACD, \vartriangle CBD $ and $ \vartriangle ABD $.

2002 Chile Level 1 P3 [figure missing]

The following figure represents a river of constant horizontal width, and two houses $ A $ and $ B $ located on opposite sides of the river. There is a bridge of width equal to the width of the river that must be placed horizontally and in such a way that the distance of $ A $ to $ B $ is minimal. Determine the location of the bridge.

The following figure represents a river of constant horizontal width, and two houses $ A $ and $ B $ located on opposite sides of the river. There is a bridge of width equal to the width of the river that must be placed horizontally and in such a way that the distance of $ A $ to $ B $ is minimal. Determine the location of the bridge.

2003 Chile Level 1 P4

Investigate if there exists a tetrahedron $ ABCD $ such that all its faces are different isosceles triangles.

Investigate if there exists a tetrahedron $ ABCD $ such that all its faces are different isosceles triangles.

2004 Chile Level 1 all year missing (there is probably geometry)

2005 Chile Level 1 P3

Within a square, $2$ different points are chosen. Then, $8$ line segments are drawn connecting each of these points with the vertices of the square. Is it possible to divide this square into $9$ parts with equal area?

Within a square, $2$ different points are chosen. Then, $8$ line segments are drawn connecting each of these points with the vertices of the square. Is it possible to divide this square into $9$ parts with equal area?

The vertex $ E $ of a square $ EFGH $ of side $2006$ mm is found in the center of the square $ ABCD $ of side $10$ mm. The line $ EF $ intersects $ CD $ at point $ I $. The line$ EH $ intercepts $ AD $ in $ J $. Also $ \angle EID = 60^o$. Calculate the area of the quadrilateral $ EIDJ $.

2006 Chile Level 1 P5

Let $\vartriangle ABC $ be any triangle and $ \vartriangle MNP $ the triangle formed by the tangent points of the inscribed circle with the sides of the triangle $ \vartriangle ABC $, show that if $ \vartriangle MNP $ is equilateral, then the $\vartriangle ABC $ is equilateral.

Let $\vartriangle ABC $ be any triangle and $ \vartriangle MNP $ the triangle formed by the tangent points of the inscribed circle with the sides of the triangle $ \vartriangle ABC $, show that if $ \vartriangle MNP $ is equilateral, then the $\vartriangle ABC $ is equilateral.

From a triangle $T = \triangle ABC$, we build the triangle $T_1 = \triangle A_1B_1C_1$ whose vertices they are the midpoints of the sides of $T$. The triangle $T_2 = \triangle A_2B_2C_2$ is constructed from $T_1$ in a way analogue. We build the triangles $T_3, T_4,..., T_{2007}$. Prove that the center of gravity $G$ of the triangle $T$ is inside the triangle $T_{2007}$.

2008 Chile Level 1 P2

In a circle of radius $1$ a diameter $PQ$ is drawn and an equilateral triangle with base $AB$ parallel to $PQ$ is inscribed. The segment $PQ$ cuts to the side $BC$ at the point $R$. Is the length $PR$ smaller, equal, or greater than the length of a quarter of the circumference?

2009 Chile Level 1 P1

Consider a triangle whose sides measure $1, r$, and $r^2$.

Determine all the values of $r$ in such a way that the triangle is right.

Consider a triangle whose sides measure $1, r$, and $r^2$.

Determine all the values of $r$ in such a way that the triangle is right.

2009 Chile Level 1 P2

Consider three points inside a square on side $1$.

Show that the area of the triangle they form is less than or equal to $\frac{1}{2}$.

Consider three points inside a square on side $1$.

Show that the area of the triangle they form is less than or equal to $\frac{1}{2}$.

On the base $AC$ of a triangle angle isosceles $ABC$, a point is taken $M$, so that $| AM | = p$ and $| MC | = q$. The inscribed circles are drawn to the $AMB$ and $CMB$ triangles, which are tangent to the $BM$ side at points $R$ and $S$ respectively. Find the distance between $R$ and $S$.

2010 Chile Level 1 P6

Consider a line $L$ in the plane and let $B_1, B_2, B_3$ be points different in $L$. Let $A$ be a point that does not lie in $L$. Show that there are $P, Q$ in $\{B_1, B_2, B_3\}$ with $P \ne Q$ such that the distance from $A$ to $L$ to be greater than the distance from $P$ to the line that passes through $A$ and $Q$.

Consider a line $L$ in the plane and let $B_1, B_2, B_3$ be points different in $L$. Let $A$ be a point that does not lie in $L$. Show that there are $P, Q$ in $\{B_1, B_2, B_3\}$ with $P \ne Q$ such that the distance from $A$ to $L$ to be greater than the distance from $P$ to the line that passes through $A$ and $Q$.

2010 Chile Level 1 P3

Let $ABCD$ be a square and $M$ be its center. Consider the point $E$ on line $AC$ such that $| MC | = | CE |$. Let $S$ be the circle circumscribed to triangle $\triangle EDB$. Show that $S$ passes through the midpoint of $AM$.

Let $ABCD$ be a square and $M$ be its center. Consider the point $E$ on line $AC$ such that $| MC | = | CE |$. Let $S$ be the circle circumscribed to triangle $\triangle EDB$. Show that $S$ passes through the midpoint of $AM$.

2011 Chile Level 1 P2

Inside a cube of side $1$, two spheres are introduced that are tangent externally to each other and such that each is tangent to three faces of the cube. Determine the greater distance to which the centers of the spheres can be found.

Inside a cube of side $1$, two spheres are introduced that are tangent externally to each other and such that each is tangent to three faces of the cube. Determine the greater distance to which the centers of the spheres can be found.

2013 Chile Level 1 P5

Four points $A, B, C, D$ move in space so that always is satisfied $(AB) = (AC) =(DB) =(DC) = 1$,

- What is the greatest value that the sum $(AD)+(BC)$ can become?

- Under what conditions is this sum maximized?

Four points $A, B, C, D$ move in space so that always is satisfied $(AB) = (AC) =(DB) =(DC) = 1$,

- What is the greatest value that the sum $(AD)+(BC)$ can become?

- Under what conditions is this sum maximized?

2014 Chile Level 1 P2

The points $P,Q,R$ are the midpoints of the sides $BC,CD$ and $DA$ of a rectangle $ABCD$ respectively and $M$ is the midpoint of the segment $QR$. The area of the rectangle is $320$. Calculate the area of the triangle $APM$ .

2015 Chile Level 1 P1

On the plane, there is drawn a parallelogram $P$ and a point $X$ outside of $P$. Using only an ungraded rule, determine the point $W$ that is symmetric to $X$ with respect to the center $O$ of $P$.

The points $P,Q,R$ are the midpoints of the sides $BC,CD$ and $DA$ of a rectangle $ABCD$ respectively and $M$ is the midpoint of the segment $QR$. The area of the rectangle is $320$. Calculate the area of the triangle $APM$ .

2015 Chile Level 1 P1

On the plane, there is drawn a parallelogram $P$ and a point $X$ outside of $P$. Using only an ungraded rule, determine the point $W$ that is symmetric to $X$ with respect to the center $O$ of $P$.

Consider a triangle $\triangle ABC$ and a point $D$ in segment $BC$. The triangles $\triangle ABD$ and $\triangle ADC$ are similar in ratio $\frac{1}{\sqrt3}$. Determine the angles of the triangle $\triangle ABC$.

2015 Chile Level 1 P5, Level 2 P5

A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $be the point of intersection of lines$ BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.

A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $be the point of intersection of lines$ BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.

2016 Chile Level 1 P2, Level 2 P2

For an equilateral triangle $\triangle ABC$, determine whether or not there is a point $P$ inside $\triangle ABC$ so that any straight line that passes through $P$ divides the triangle $\triangle ABC$ in two polygonal lines of equal length.

For an equilateral triangle $\triangle ABC$, determine whether or not there is a point $P$ inside $\triangle ABC$ so that any straight line that passes through $P$ divides the triangle $\triangle ABC$ in two polygonal lines of equal length.

2016 Chile Level 1 P5

Let $\triangle ABC$ be a triangle isosceles with $AC = BC$. Let $O$ be the center of circumcircle circumscribed to the triangle and $I$ the center of the inscribed circle. If $D$ is the point on the side $BC$ such that $OD$ is perpendicular to $BI$. Show that $ID$ is parallel to $AC$.

Let $\triangle ABC$ be a triangle isosceles with $AC = BC$. Let $O$ be the center of circumcircle circumscribed to the triangle and $I$ the center of the inscribed circle. If $D$ is the point on the side $BC$ such that $OD$ is perpendicular to $BI$. Show that $ID$ is parallel to $AC$.

2017 Chile Level 1 all year missing (there is probably geometry)

2018 Chile Level 1 P1

In the drawing, the five circles are tangent to each other and tangents to the lines $L_1$ and $L_2$ as shown in the following figure. The smallest of the circles has radius $8$ and the largest has radius $18$. Calculate the radius of the circle $C$.

In the drawing, the five circles are tangent to each other and tangents to the lines $L_1$ and $L_2$ as shown in the following figure. The smallest of the circles has radius $8$ and the largest has radius $18$. Calculate the radius of the circle $C$.

2018 Chile Level 1 P6

Consider two lines $L_1, L_2$ that are cut at point $O$ and $M$ is the bisector of the angle they form, as shown in the following figure. Points $A$ and $B$ are drawn in $M$ in such a way that $OA = 8$ and $OB = 15$ and the angle $\angle L_1OL_2$ measures $45^o$ . Calculate the shortest possible path length from $A$ to $B$ by touching lines $L_1$ and $L_2$.

TSTs

sources:Consider two lines $L_1, L_2$ that are cut at point $O$ and $M$ is the bisector of the angle they form, as shown in the following figure. Points $A$ and $B$ are drawn in $M$ in such a way that $OA = 8$ and $OB = 15$ and the angle $\angle L_1OL_2$ measures $45^o$ . Calculate the shortest possible path length from $A$ to $B$ by touching lines $L_1$ and $L_2$.

TSTs

They are given in the plane a circle $ \odot (O, r) $, and a line $ L $. The distance from $ O $ to $ L $ is $ d $, with $ d> r $. Points $ M, N \in L $ are chosen such that the circle of diameter $ MN $ is tangent outside $ \odot (O, r) $. Show that there is a point $ A \notin L $ in the plane, such that all segments $MN$ subtend a constant angle in $ A $.

Let $a,b,c$ the sides of a triangle, and $A,B,C$ the respective angles (measured in degrees). Show that:

$$60^o\le \frac{a \cdot A+b\cdot B+c\cdot C}{a+b+c} \le 90^o$$

$$60^o\le \frac{a \cdot A+b\cdot B+c\cdot C}{a+b+c} \le 90^o$$

The circles $ C_1, C_2 $ are tangent internally to the circle $ C $ in the points $ A, B $, respectively. The internal tangent line common to $ C_1 $ and $ C_2 $ touches these circles in $ P $ and $ Q $, respectively. Show that $AP$ and $BQ$ intersect again $ C $ in diametrically opposite points.

Let $ I $ the incenter of the $ \vartriangle ABC$. The incircle of the $ \vartriangle ABC $ is tangent to $BC, CA, AB$ in $ K, L, M $, respectively. The line parallel to $MK$, which passes through $ B $, intersects $ LM, LK $ in $ R, S $, respectively. Prove that $ \angle RIS $ is acute.

An acute triangle $ABC$ is inscribed in a circle $ \Gamma $. Let $D$ be the point of $ \Gamma $ diametrically opposite $C$. Determine all points $X$ of the arc $BC$ (which does not contain the vertex $A$) of $ \Gamma $ such that the quadrilateral $DBXC$ and the triangle $ABC$ have equal areas.

Let $M$ br the circumcenter of an acute triangle $ABC$ and we assume that the circumscribed circle of $BMA$ intersects segment $BC$ at $P$ and $AC$ at $Q$. Show that the line $CM$ is perpendicular to $PQ$.

Let $\Gamma_1$ and $\Gamma_2$ be two circles that intersect at points $P$ and $Q$.

Construct a segment $AB$ that passes through $P$, with $A$ in $\Gamma_1, B$ in $\Gamma_2$ and such that $AP \cdot PB$ be maximum.

Suppose that $h$ is the length of the maximum height of a triangle, not an obtuse angle. Let $R$ and $r$, respectively, the radii of circumscribed and inscribed circles. Show that $R+r \le h$.

Let ABC be a triangle such that AB = AC. Let P be a point about BC. Let M, N be the feet of the perpendiculars from P to AB and AC respectively. Show that the value of the sum $PM + PN$ it does not depend on the position of the chosen point $P$.

Let $ABC$ a triangle and $l$ is a line where intersects $BC, AC$ and $BA$ in the point(s) $D, E, F$ respectively. Suppose that $l$ don't intersect a vertex in the triangle $ABC$, consider the circle(s) $C, C_b, C_a, C_c$ where are the circumcircles of triangles $ABC, DBF, AEF, DCE$ respectively. Show that this circles $C, C_a, C_b, C_c$ are concurrents.

The incircle triangle $ \vartriangle ABC$ touches $AC$ and $BC$ in $E$ and $D$ respectively. The $A$-excircle touches the extensions of $BC$ in $A_1$, of $CA$ in $B_1$ and $AB$ in $C_1$. Let $ DE \cap A_1B_1 = L$. Prove that $L$ lies on the circumcircle of the triangle $\vartriangle A_1BC_1$.

Let $ABC$ be a triangle and points $P, Q, R$ on the sides $AB, BC$ and $CA$ respectively in such a way that $\frac{AP}{AB}= \frac{BQ}{BC}= \frac{CR}{CA}= \frac{1}{n}$ for $n \in N$. Segments $AQ$ and $CP$ are cut in $D$, segments $BR$ and $AQ$ are cut at $E$ and segments $BR$ and $CP$ are cut at $F$. Calculate the ratio of areas of the triangles $ \frac{(ABC)}{(DEF)}$.

2015 Chile TST Ibero

Prove that in a acute scalene triangle, the orthocenter, the incenter and the circumcenter are not collinear.

Prove that in a acute scalene triangle, the orthocenter, the incenter and the circumcenter are not collinear.

www.olimpiadadematematica.cl

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