geometry problems

from National Mathematical Olympiad from Ecuador, Team Selection Tests (TST)

easier than IMO TSTs

2006 Ecuador TST1 p2

Consider an $ABC$ triangle. From $B$ we extend the $AB$ side up to a point $P$, such that $AB = BP$. In the same way we do with the $BC$ side then from point $C$ we extend it to point $Q$ in such a way that $BC = CQ$. Finally, starting from point $A$, we extended the $CA$ side until obtain $R$ such that $CA = AR$. Knowing that the area of triangle $ABC$ is $x$, which is the area of the triangle $PQR$?

Let $P$ be an interior point of the equilateral triangle $ABC$ such that $PA = 5, PB = 7, PC = 8$. Find the length of one side of the triangle $ABC$.

2006 Ecuador TST2 p6

Let $D$ be the point on the arc $BC$ of the circumcircle circumscribed around the isosceles triangle $ABC$ with $AB = AC$ that does not contain the vertex $A$. Let $E$ be the intersection point of the segment $CD$ and the line perpendicular to $CD$, which passes through the vertex $A$. Prove that $BD + DC = 2DE$

2011 Ecuador TST2 p6

2015 Ecuador TST1 p3

Let $ABC$ be an equilateral triangle with center $O$ and side equal to $3$. Let $M$ be a point on the $AC$ side such that $CM = 1$ and let $P$ be a point on the side $AB$ such that $AP = 1$. Calculate the measure of the internal angles of the triangle $MOP$.

2015 Ecuador TST1 p8

Let $ABC$ be an acute triangle and $H$ its orthocenter. Let $D$ and $E$ be the feet of the atlitudes from $B$ and $C$ on $AC$ and $AB$ respectively. The circumscribed circle of $ADE$ cuts to the circle circumscribed around $ABC$ in $F\ne A$ . Prove that the internal bisectors of $\angle BFC$ and $\angle BHC$ are cut in a point on segment $BC$.

2016 Ecuador TST3 p7

Let $\Gamma_1$ and $\Gamma_2$ be two circles, of centers $O_1$ and $O_2$ respectively, intersecting at $M$ and $N$. The straight line $l$ is the common tangent to $\Gamma_1$ and $\Gamma_2$, closer to $M$. Points $A$ and $B$ are the respective points of contact of $l$ with $\Gamma_1$ and $\Gamma_2, C$ the point diametrically opposite $B$ and $D$ the point of intersection of the line $O_1O_2$ with the line perpendicular to line $AM$ drawn by $B$. Show that $M, D$ and $C$ are collinear.

2017 Ecuador TST1 p2

source:

https://omec-mat.org/entrenamiento/pruebas-anteriores/pruebas-selectivas-cono-sur-e-imo/

from National Mathematical Olympiad from Ecuador, Team Selection Tests (TST)

**mostly**Selections Tests for Cono Sur (OMCS)

easier than IMO TSTs

Olimpiada Matemática Ecuatoriana

**Selectivos**(OMEC)
2006 - 2018

Consider an $ABC$ triangle. From $B$ we extend the $AB$ side up to a point $P$, such that $AB = BP$. In the same way we do with the $BC$ side then from point $C$ we extend it to point $Q$ in such a way that $BC = CQ$. Finally, starting from point $A$, we extended the $CA$ side until obtain $R$ such that $CA = AR$. Knowing that the area of triangle $ABC$ is $x$, which is the area of the triangle $PQR$?

2006 Ecuador TST1 p5

Intersecting the four lines perpendicular on the sides of a parallelogram, not rectangle, by its midpoints, you get a region of the limited plane for those $4$ lines. Under what conditions the area of this new region has area equal to the area of the parallelogram?

2006 Ecuador TST2 p4Intersecting the four lines perpendicular on the sides of a parallelogram, not rectangle, by its midpoints, you get a region of the limited plane for those $4$ lines. Under what conditions the area of this new region has area equal to the area of the parallelogram?

2006 Ecuador TST1 p7

A right triangle has legs of lengths $a$ and $b$ . A circle of radius $r$ is tangent to the two legs and has its center on the hypotenuse of the right triangle. Show that $\frac{1}{a}+ \frac{1}{b}= \frac{1}{r}$

A right triangle has legs of lengths $a$ and $b$ . A circle of radius $r$ is tangent to the two legs and has its center on the hypotenuse of the right triangle. Show that $\frac{1}{a}+ \frac{1}{b}= \frac{1}{r}$

Let $P$ be an interior point of the equilateral triangle $ABC$ such that $PA = 5, PB = 7, PC = 8$. Find the length of one side of the triangle $ABC$.

Let $ABC$ be an acute triangle. On the median $AM$ we take a point $P$ so that the rays $BP$ and $CP$ cut the sides $AC$ and $AB$ at the points $Q$ and $R$ respectively. Show that the $QR$ and $BC$ segments are parallel.

2007 Ecuador TST1 p4

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$, so that $AB = 10$ and $CD = 20$. Let $O$ be the intersection of the trapeze diagonals. By $O$ a straight line is drawn $l$ parallel to the bases. Find the length of the segment of $l$ that is found inside the trapezoid.

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$, so that $AB = 10$ and $CD = 20$. Let $O$ be the intersection of the trapeze diagonals. By $O$ a straight line is drawn $l$ parallel to the bases. Find the length of the segment of $l$ that is found inside the trapezoid.

2007 Ecuador TST1 p7

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

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

2007 Ecuador TST2 p7

Let $ABC$ be an acute triangle with angles $A,B,C$. Let $r$ be the inradius of the triangle and $R$ its circumradius. Show that $cosA +cosB +cosC =1+\frac{r}{R}$

Let $ABC$ be an acute triangle with angles $A,B,C$. Let $r$ be the inradius of the triangle and $R$ its circumradius. Show that $cosA +cosB +cosC =1+\frac{r}{R}$

2008 Ecuador TST1 p4

The length of the side $AC$ of the right triangle $ABC$ with $\angle C = 90^o$ is $1$ meter and $\angle A = 30^o$. Let $D$ be a point within the triangle $ABC$ such that $\angle BDC = 90^o$ and $\angle ACD = \angle DBA$. Let F be the point of intersection of the side $AC$ and the extension of the segment $BD$ . Find the length of the segment $AF $.

The length of the side $AC$ of the right triangle $ABC$ with $\angle C = 90^o$ is $1$ meter and $\angle A = 30^o$. Let $D$ be a point within the triangle $ABC$ such that $\angle BDC = 90^o$ and $\angle ACD = \angle DBA$. Let F be the point of intersection of the side $AC$ and the extension of the segment $BD$ . Find the length of the segment $AF $.

2008 Ecuador TST1 p7

Be the quadrilateral $ABCD$ with $\angle CAD = 45^o, \angle ACD = 30^o, \angle BAC = \angle BCA = 15^o$. Find the numerical value of $\angle DBC$.

Be the quadrilateral $ABCD$ with $\angle CAD = 45^o, \angle ACD = 30^o, \angle BAC = \angle BCA = 15^o$. Find the numerical value of $\angle DBC$.

What are all the possible areas of a hexagon that has all its angles equal and whose sides measure $1, 2, 3, 4, 5$ and $6$, in some order?

2009 Ecuador TST1 p3

In triangle $ABC$ we have $\angle ABC = \angle ACB = 80^o$. Let $P$ be a point in segment $AB$ such that $\angle BPC =30^o$. Show that $AP=BC$.

In triangle $ABC$ we have $\angle ABC = \angle ACB = 80^o$. Let $P$ be a point in segment $AB$ such that $\angle BPC =30^o$. Show that $AP=BC$.

2009 Ecuador TST1 p7

Let $ABCD$ be a quadrilateral with $AD$ parallel to $BC$, the angles $A$ and $B$ straight and such that the angle $CMD$ is straight, where $M$ is the midpoint of $AB$. Let $K$ be the foot of the perpendicular to $CD$ that passes by $M, P$ the point of intersection of $AK$ with $BD$ and $Q$ the point of intersection of $BK$ with $AC$. Show that the angle $AKB$ is straight, and that $\frac{KP}{AP}+\frac{KQ}{BQ}=1$

2009 Ecuador TST2 p7

Let $ABC$ be an acute triangle with $AB >AC$ and $\angle BAC =60^o$. Denote the circumcenter by $O$ and orthocenter by $H$. The extension of $OH$ intersects $AB$ and $AC$ in $P$ and $Q$ respectively. Prove that $PO=HQ$.

2009 Ecuador TST3 p3

Let $ABC$ be an acute triangle such that the bisector of $ \angle BAC$, the altitude taken from $B$ and the median to the side $AB$ intersect at a point. Determine the angle $\angle BAC$.

2009 Ecuador TST3 p7

Let $ABCD$ be a quadrilateral with $AD$ parallel to $BC$, the angles $A$ and $B$ straight and such that the angle $CMD$ is straight, where $M$ is the midpoint of $AB$. Let $K$ be the foot of the perpendicular to $CD$ that passes by $M, P$ the point of intersection of $AK$ with $BD$ and $Q$ the point of intersection of $BK$ with $AC$. Show that the angle $AKB$ is straight, and that $\frac{KP}{AP}+\frac{KQ}{BQ}=1$

Let $ABC$ be an acute triangle with $AB >AC$ and $\angle BAC =60^o$. Denote the circumcenter by $O$ and orthocenter by $H$. The extension of $OH$ intersects $AB$ and $AC$ in $P$ and $Q$ respectively. Prove that $PO=HQ$.

Let $ABC$ be an acute triangle such that the bisector of $ \angle BAC$, the altitude taken from $B$ and the median to the side $AB$ intersect at a point. Determine the angle $\angle BAC$.

2009 Ecuador TST3 p5

Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively of the parallelogram $ABCD$, such that $AM = CN$. Let $Q$ be the intersection point of $AN$ and $CM$. Show that $DQ$ is bisector of angle $CDA$.

Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively of the parallelogram $ABCD$, such that $AM = CN$. Let $Q$ be the intersection point of $AN$ and $CM$. Show that $DQ$ is bisector of angle $CDA$.

In triangle $ABC$, the inner bisector of angle $A$ and the median drawn from $ A$ cut $BC$ at two different points $D$ and $E$, respectively. Let $M$ be the point intersection of $AE$ and the perpendicular to $AD$ drawn from $B$. Prove that $AB$ and $DM$ are parallel.

2010 Ecuador TST1 p4

Two chords $AB$ and $CD$ of a circle are cut at point $K$. Point $A$ divides the arc $CAD$ in two equal parts. Let $AK = 2$ and $KB = 6$. Determine the measurement of the chord $AD$.

Two chords $AB$ and $CD$ of a circle are cut at point $K$. Point $A$ divides the arc $CAD$ in two equal parts. Let $AK = 2$ and $KB = 6$. Determine the measurement of the chord $AD$.

Let $ABC$ be a triangle, and let $D$ be the midpoint of $AB$ and $E$ a point on $BC$, such that $BE = 2EC$. Knowing that $\angle ADC = \angle BAE$. Determine the measure of the angle $\angle BAC$.

2010 Ecuador TST2 p7

Consider $P$ a point inside the triangle $ABC$. Let $D, E$ and $F$ be the midpoints of $AP, BP$ and $CP$, respectively, and $L, M$ and $N$ the points of intersection of $BF$ with $CE, AF$ with $CD$ and $AE$ with $BD$, respectively. Show that the segments $DL, EM$ and $FN$ are concurrent.

2011 Ecuador TST1 p6Consider $P$ a point inside the triangle $ABC$. Let $D, E$ and $F$ be the midpoints of $AP, BP$ and $CP$, respectively, and $L, M$ and $N$ the points of intersection of $BF$ with $CE, AF$ with $CD$ and $AE$ with $BD$, respectively. Show that the segments $DL, EM$ and $FN$ are concurrent.

Let $D$ be the point on the arc $BC$ of the circumcircle circumscribed around the isosceles triangle $ABC$ with $AB = AC$ that does not contain the vertex $A$. Let $E$ be the intersection point of the segment $CD$ and the line perpendicular to $CD$, which passes through the vertex $A$. Prove that $BD + DC = 2DE$

2011 Ecuador TST2 p6

Given the acute triangle $ABC$. The circle of diameter $AB$
intersects the altitude $CF$ in $M$ and $N$ (with $CM <CF$), the circle of
diameter $AC$ intersects the altitude
$BE$ in $P$ and $Q$ (with $BP <BE$). Show that the points $M, N, P$
and $Q$ are cyclic.

2012 Ecuador TST1 p7

In the figure point $E$ is on $AB$ such that $AE : EB =1: 3$ and $D$ on $BC$, such that $CD : DB= 1: 2$ .Find the value of $\frac{EF}{FC}+\frac{AF}{FD}.$

In the figure point $E$ is on $AB$ such that $AE : EB =1: 3$ and $D$ on $BC$, such that $CD : DB= 1: 2$ .Find the value of $\frac{EF}{FC}+\frac{AF}{FD}.$

2012 Ecuador TST2 p5

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$. Let $O$ be the point of intersection of its diagonals $AC$ and $BD$. If the area of triangle ABC is $150$ and the area of triangle $ACD$ is $120$. Calculate the area of triangle $BCO$.

2012 Ecuador TST2 p7

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$. Let $O$ be the point of intersection of its diagonals $AC$ and $BD$. If the area of triangle ABC is $150$ and the area of triangle $ACD$ is $120$. Calculate the area of triangle $BCO$.

Let $ABC$ be an isosceles triangle with $AB= AC$. Let $X$ and $Y$ be points on the sides $BC$ and $CA$ respectively, such that $XY // AB$. Let $D$ be the circumcenter of triangle $CXY$ and $E$ the midpoint of $BY$. Show that $\angle AED = 90^o$.

The medians drawn on the equal sides of an isosceles triangle are perpendicular to each other. If the base of the isosceles triangle is $4$, find its area.

2014 Ecuador TST missing

2015 Ecuador TST1 p3

Let $ABC$ be an equilateral triangle with center $O$ and side equal to $3$. Let $M$ be a point on the $AC$ side such that $CM = 1$ and let $P$ be a point on the side $AB$ such that $AP = 1$. Calculate the measure of the internal angles of the triangle $MOP$.

Let $ABC$ be an acute triangle and $H$ its orthocenter. Let $D$ and $E$ be the feet of the atlitudes from $B$ and $C$ on $AC$ and $AB$ respectively. The circumscribed circle of $ADE$ cuts to the circle circumscribed around $ABC$ in $F\ne A$ . Prove that the internal bisectors of $\angle BFC$ and $\angle BHC$ are cut in a point on segment $BC$.

2015 Ecuador TST2 p3

In the equilateral triangle $ABC$, let $D$ be a point on the $AC$ side such that $3 AD = AC$, and $E$ on $BC$ such that $3 CE = BC$. $BD$ and $AE$ are cut in $F$. Find the value of $\angle CFB$.

In the equilateral triangle $ABC$, let $D$ be a point on the $AC$ side such that $3 AD = AC$, and $E$ on $BC$ such that $3 CE = BC$. $BD$ and $AE$ are cut in $F$. Find the value of $\angle CFB$.

A line $l$ does not intersect the circle $\omega$ with center $O$. $E$ is the point in $l$ such that $OE$ is perpendicular to $l$. $M$ is a point in $l$ distinct from $E$. The tangents from $M$ to $\omega$ intersect at $A$ and $B$ at said circle. $C$ is the point in $AM$ such that $CE$ is perpendicular to $AM$. $D$ is the point in $BM$ such that $DE$ is perpendicular to $BM$. The line $CD$ cut $EO$ in $F$. Prove that the position of $F$ is independent of the position of $M$.

2016 Ecuador TST1 p5

Let $ABCDE$ be a convex pentagon such that the triangles $ABC, BCD, DEC$, and $EAD$ have the same area. Suppose that $AC$ and $AD$ cut $BE$ at points $M$ and $N$ respectively. Show that $BM = NE$.

2016 Ecuador TST2 p3

Let $ABCDEF$ be a hexagon such that the lengths of its sides $AB, BC, CD$ and $DE$ are $6, 4, 8$ and $9$, respectively. If it is known that their internal angles measure all $120^o$ , determine the length of the other sides $EF, AF$.

Let $ABCDE$ be a convex pentagon such that the triangles $ABC, BCD, DEC$, and $EAD$ have the same area. Suppose that $AC$ and $AD$ cut $BE$ at points $M$ and $N$ respectively. Show that $BM = NE$.

2016 Ecuador TST2 p3

Let $ABCDEF$ be a hexagon such that the lengths of its sides $AB, BC, CD$ and $DE$ are $6, 4, 8$ and $9$, respectively. If it is known that their internal angles measure all $120^o$ , determine the length of the other sides $EF, AF$.

2016 Ecuador TST2 p5

In the triangle $ABC$ , $\angle CAB = 18^o$ and $\angle BCA = 24^o$ . $E$ is a point on $CA$ such that $\angle CEB = 60^o$ and $F$ is a point on $AB$ such that $ \angle AEF = 60^o$ . What is the measure, in degrees, of $\angle BFC$?

In the triangle $ABC$ , $\angle CAB = 18^o$ and $\angle BCA = 24^o$ . $E$ is a point on $CA$ such that $\angle CEB = 60^o$ and $F$ is a point on $AB$ such that $ \angle AEF = 60^o$ . What is the measure, in degrees, of $\angle BFC$?

2016 Ecuador TST3 p3

Let $ABC$ be an acute triangle . Let $E$ be the foot of the altitude from $B$ to $AC$. Let $l$ be the tangent line to the circumscribed circle of $ABC$ at point $B$ and let $F$ be the foot of the perpendicular from $C$ to $l$. Show that $EF$ is parallel to $AB$

Let $ABC$ be an acute triangle . Let $E$ be the foot of the altitude from $B$ to $AC$. Let $l$ be the tangent line to the circumscribed circle of $ABC$ at point $B$ and let $F$ be the foot of the perpendicular from $C$ to $l$. Show that $EF$ is parallel to $AB$

Let $\Gamma_1$ and $\Gamma_2$ be two circles, of centers $O_1$ and $O_2$ respectively, intersecting at $M$ and $N$. The straight line $l$ is the common tangent to $\Gamma_1$ and $\Gamma_2$, closer to $M$. Points $A$ and $B$ are the respective points of contact of $l$ with $\Gamma_1$ and $\Gamma_2, C$ the point diametrically opposite $B$ and $D$ the point of intersection of the line $O_1O_2$ with the line perpendicular to line $AM$ drawn by $B$. Show that $M, D$ and $C$ are collinear.

2017 Ecuador TST1 p2

Given a regular hexagon $ABCDEF$ of side $6$, find the area
of the triangle $BCE$.

2017 Ecuador TST2 p2

Let $ABCDE$ be a convex pentagon (not necessarily regular) and let $M, P, N$ and $Q$ be the points means of $AB$, $BC, CD$ and $DE$, respectively. $K$ and $L$ are the midpoints of $MN$ and $P Q$ respectively. If $AE$ has length $4$, determine the length of $KL$.

Let $ABCDE$ be a convex pentagon (not necessarily regular) and let $M, P, N$ and $Q$ be the points means of $AB$, $BC, CD$ and $DE$, respectively. $K$ and $L$ are the midpoints of $MN$ and $P Q$ respectively. If $AE$ has length $4$, determine the length of $KL$.

2017 Ecuador TST2 p5

A semicircle $\Gamma$ is drawn whose diameter belongs to the line $l$. $C$ and $D$ are points in $\Gamma$ that do not belong to $l$. The tangents a $\Gamma$ by $C$ and $D$ cut to $l$ in $B$ and $A$ respectively. Let $E$ be the point intersection of $AC$ with $BD$, and $F$ the point in $l$ such that $EF$ is perpendicular to $l$. If you have to the center of $\Gamma$ belongs to the $AB$ segment, showing that $EF$ bisects to $\angle CFD$.

2017 Ecuador TST3 p1

In the $ABC$ triangle, $\angle A + \angle B = 110^o$ , and $D$ is a point on segment $AB$ such that $CD = CB$ and $\angle DCA = 10^o$ . How much does the $\angle A$ measure?

2017 Ecuador TST3 p7

A convex quadrilateral $ABCD$ has no parallel sides. The angles formed by the diagonal $AC$ and the four sides are $55^o, 55^o , 19^o$ and $16^o$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$.

2018 Ecuador TST1 p1

Let $ABC$ be a triangle and $D$ be the midpoint of $AB$, if $\angle ACD = 105^o$ and $\angle DCB = 30^o$ . Find $\angle ABC$ .

A semicircle $\Gamma$ is drawn whose diameter belongs to the line $l$. $C$ and $D$ are points in $\Gamma$ that do not belong to $l$. The tangents a $\Gamma$ by $C$ and $D$ cut to $l$ in $B$ and $A$ respectively. Let $E$ be the point intersection of $AC$ with $BD$, and $F$ the point in $l$ such that $EF$ is perpendicular to $l$. If you have to the center of $\Gamma$ belongs to the $AB$ segment, showing that $EF$ bisects to $\angle CFD$.

2017 Ecuador TST3 p1

In the $ABC$ triangle, $\angle A + \angle B = 110^o$ , and $D$ is a point on segment $AB$ such that $CD = CB$ and $\angle DCA = 10^o$ . How much does the $\angle A$ measure?

2017 Ecuador TST3 p7

A convex quadrilateral $ABCD$ has no parallel sides. The angles formed by the diagonal $AC$ and the four sides are $55^o, 55^o , 19^o$ and $16^o$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$.

Let $ABC$ be a triangle and $D$ be the midpoint of $AB$, if $\angle ACD = 105^o$ and $\angle DCB = 30^o$ . Find $\angle ABC$ .

2018 Ecuador TST2 p3

Let $ABC$ be a non-degenerate triangle and $I$ the point of intersection of its bisectors. $D$ is the midpoint of the segment $CI$. If it is known that $\frac{AD}{DB}=\frac{BC}{AC}$ . Prove that $AC = BC$

2018 Ecuador TST3 p5

Let $ABC$ be a non-degenerate triangle and $I$ the point of intersection of its bisectors. $D$ is the midpoint of the segment $CI$. If it is known that $\frac{AD}{DB}=\frac{BC}{AC}$ . Prove that $AC = BC$

2018 Ecuador TST3 p5

Six circles $C_i, 1 \le i \le 6$ are all externally tangent to the circle $C$, and $C_i$ is tangent externally to $C_{i + 1}$ for all $i$, with $C_6$ tangent to $C_1$. Let $P_i$ be the point of tangency between $C_i$ and $C$. Demonstrate that $P_1P_4, P_2P_5$, and $P_3P_6$ are concurrent.

source:

https://omec-mat.org/entrenamiento/pruebas-anteriores/pruebas-selectivas-cono-sur-e-imo/

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