Ecuador TST 2006-21 (OMCS, IMO) 57p

geometry problems from Team Selection Tests (TST) from Ecuador,  with aops links

 for Cono Sur (OMCS) and IMO 

Olimpiada Matemática Ecuatoriana Selectivos (OMEC)

collected inside aops here

Cono Sur TST 2006 - 2021

2006 Ecuador Cono Sur 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 $PQR$?

2006 Ecuador Cono Sur 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 Cono Sur 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 $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$

2006 Ecuador Cono Sur TST2 p1

In the right triangle $ABC$, a square is inscribed where the length of its sides is $x$, as shown in the figure. Determine the length of the legs of the triangle $ABC$.

2006 Ecuador Cono Sur TST2 p4

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 Cono Sur TST2 p6

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 Cono Sur TST p2

The length of each of the sides of an equilateral triangle is $5$. From any interior point $P$, draw segments perpendicular to each of the three sides. If the lengths of the segments are $a, b, c$, determine the value of $a+b+c$.

2007 Ecuador Cono Sur TST 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.

2007 Ecuador Cono Sur TST 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 $ABC$.

2008 Ecuador Cono Sur TST 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 intersection point of the side $AC$ and the extension of the segment $BD$ . Find the length of the segment $AF$.

2008 Ecuador Cono Sur TST 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 $\angle DBC$.

2009 Ecuador Cono Sur TST 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 Cono Sur TST 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 $CDA$.

2009 Ecuador Cono Sur TST p7

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 Cono Sur TST 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 $AD$.

2010 Ecuador Cono Sur TST p7

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 $\angle BAC$.

2011 Ecuador Cono Sur TST 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 $BD + DC = 2DE$

2012 Ecuador Cono Sur TST p2

In the figure, it is known that $\angle ABC = 60^o$, $\angle BCD = 70^o$ . Find $\angle CBD$. 

2012 Ecuador Cono Sur TST p5

In the figure we have that $AD = DB = 5$ , $EC = 8$, $AE = 4$ and $\angle AED$ is a right angle. Find the length of $BC$.

2012 Ecuador Cono Sur TST 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 $\frac{EF}{FC}+\frac{AF}{FD}.$

2013 Ecuador Cono Sur TST p1

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.

2015 Ecuador Cono Sur TST 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 Cono Sur TST 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 Cono Sur 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 Cono Sur 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 Cono Sur 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 $\angle BFC$?

2017 Ecuador Cono Sur TST1 p2

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

2017 Ecuador Cono Sur 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 $KL$.

2017 Ecuador Cono Sur 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 $\angle CFD$.

2018 Ecuador Cono Sur TST p2

Triangle $ABC$ has area $48$. Let $P$ be the midpoint of the median $AM$ and let $N$ be the midpoint of side $AB$, if $G$ is the intersection of $MN$ and $BP$. Find the area of $MPG$.

2018 Ecuador Cono Sur TST p5

Let $ABC$ be a right triangle with right angle at $A$ and altitude $AD$. The squares $BCX_1X_2$, $CAY_1Y_2$ and $ABZ_1Z_2$ are constructed towards the outside of the triangle. Let U be the intersection point of $AX_1$ with $BY_2$ and V be the intersection point of $AX_2$ with $CZ_1$. Prove that the quadrilaterals $ABDU$, $ACDV$ and $BX_1UV$ are cyclic.

2019 Ecuador Cono Sur 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 $\angle ABC$.

2019 Ecuador Cono Sur TST2 p3

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

2020 Ecuador Cono Sur TST1 p5

Inside the convex quadrilateral $ABCD$, there are different points $P$ and $Q$ such that the triangles $APQ$ and $DPQ$ have equal areas, and the triangles $BPQ$ and $CPQ$ have equal areas. Let $E$ and $F$ be the intersection point of the line $PQ$ with the segments $AD$ and $BC$, respectively. Show that there is a line that bisects the segments $AB$, $CD$ and $EF$.

2020 Ecuador Cono Sur TST2 p4

Let $ABC$ be a triangle not equilateral, with $BC\le CA \le AB$. For each pair of sides of the triangle, a point is chosen on the long side of the pair so that the distance from that point to the common vertex is equal to the length of the short side of the pair. The $3$ points chosen form triangle $T$. Let $r$ be the ratio between the area of $T$ and the area of triangle $ABC$.

a) If $BC <CA <AB$, show that $0 <r < \frac14$

b) Find all triangles $ABC$ for which $r =\frac14$.

2022 Ecuador Cono Sur TST p3

In a triangle $ABC$ let $K$ and $L$ be points on the segment $AB$ such that $\angle ACK =\angle KCL = \angle LCB$. The point $M$ belonging to the segment $BC$ satisfies that $\angle MKC = \angle BKM$ . If $ML$ is the bisector of $\angle KMB$, find the value of $\angle MLC$.

 IMO 2006 - 2021

2007 Ecuador IMO TST 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 $cosA +cosB +cosC =1+\frac{r}{R}$

2008 Ecuador IMO TST p5

Find the tangents of the angles of a triangle knowing that they are positive integers.

2008 Ecuador IMO TST p7

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 IMO 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 $AP=BC$.

2009 Ecuador IMO TST1 p7

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

2009 Ecuador IMO 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 $PO=HQ$.

2010 Ecuador IMO TST 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 IMO TST p2

Consider an equilateral triangle of side $1$ and a circumference that passes through one of its vertices, and is tangent to the opposite side at its midpoint. Calculate the area of the shaded part.

2011 Ecuador IMO TST 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 concyclic.

2012 Ecuador IMO TST p5

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$. Let $O$ be the intersection point 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 IMO TST p7

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 $\angle AED = 90^o$.

2015 Ecuador IMO TST 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 $\angle CFB$.

2015 Ecuador IMO TST p8

A line $l$ does not intersect the circumference $\omega$ 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 IMO TST 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 $AB$

2016 Ecuador IMO TST 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 intersection point 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 IMO TST 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 IMO TST 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 IMO TST 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.

2021 Ecuador IMO TST p2

Let $A. B, C, D$, and $E$ be points on a circle (in that order) such that $AE = DE$. Let $P$ be the intersection of $AC$ and $BD$. Let $Q$ be the point on line $AB$ such that $A$ is between $B$ and $Q$ and $AQ = DP$. Similarly, let $R$ be the point on line $CD$ such that $D$ is between $C$ and $R$ and $DR = AP$. Show that $PE$ is perpendicular to $QR$.

2022 Ecuador IMO TST1 p2

The convex hexagon $ABCDEF$ satisfies that the triangles $ABC$, $BCD$, $CDE$, $DEF$ , $EFA$ and $FAB$ are congruent. Show that $AD = BE = CF$.

2022 Ecuador IMO TST2 p4

Let $ABC$ be an acute triangle and $A'$ the point diametrically opposite from$A$ to the circumscribed circle of the triangle. Through point $A$ draw the tangent to the circumscribed circle of the triangle $ABC$ that intersects the line $BC$ at point $D$. A point $E$ is taken on the segment $BC$ such that $AD = ED$. Let $A''$ be the point on the circumscribed circle of the triangle $ABC$ (distinct from $A$) that belongs to the reflection of the line $AA'$ wrt the line $AE$. Prove that the lines $A'A''$ and $BC$ are parallel.

sources:
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