geometry problems from Team Selection Tests (TST) from Ecuador, with aops links
sources:
https://omec-mat.org/entrenamiento/pruebas-anteriores/pruebas-selectivas-cono-sur-e-imo/
for Cono Sur (OMCS) and IMO
Olimpiada Matemática Ecuatoriana Selectivos (OMEC)
collected inside aops here
Cono Sur TST 2006 - 2021
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 $PQR$?
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?
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 $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$
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 $ABC$.
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.
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 $a+b+c$.
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.
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 $ABC$.
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 $.
Be the quadrilateral $ABCD$ with $\angle CAD = 45^o, \angle ACD = 30^o, \angle BAC = \angle BCA = 15^o$. Find the numerical value of $\angle DBC$.
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 $\angle BAC$.
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.
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 $\angle BAC$.
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 $BD + DC = 2DE$
In the figure, it is known that $\angle ABC = 60^o$, $\angle BCD = 70^o$ . Find $\angle CBD$.
In the figure we have that $AD = DB = 5$ , $EC = 8$, $AE = 4$ and $\angle AED$ is a right angle. Find the length of $BC$.
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 $\frac{EF}{FC}+\frac{AF}{FD}.$
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.
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 $BC$.
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 $BM = NE$.
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 $EF, AF$.
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 $\angle BFC$?
Given a regular hexagon $ABCDEF$ of side $6$, find the area of the triangle $BCE$.
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 $KL$.
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 $\angle CFD$.
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 $MPG$.
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.
Let $ABC$ be a triangle and $D$ be the midpoint of $AB$, if $\angle ACD=105^o$ and $\angle DCB=30^o$. Find $\angle ABC$.
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 $AC = BC$.
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 $EF$.
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 $r =\frac14$.
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
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 $cosA +cosB +cosC =1+\frac{r}{R}$
Find the tangents of the angles of a triangle knowing that they are positive integers.
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?
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 $AP=BC$.
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 $\frac{KP}{AP}+\frac{KQ}{BQ}=1$
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 $PO=HQ$.
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.
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.
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.
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 $BCO$.
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 $\angle AED = 90^o$.
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 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 $M$.
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 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.
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?
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 $BD$.
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.
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 $QR$.
The convex hexagon $ABCDEF$ satisfies that the triangles $ABC$, $BCD$, $CDE$, $DEF$ , $EFA$ and $FAB$ are congruent. Show that $AD = BE = CF$.
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.
https://omec-mat.org/entrenamiento/pruebas-anteriores/pruebas-selectivas-cono-sur-e-imo/
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