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