Argentina TST 1996 - 2022 (IMO - OMCS - OIM) 110p

geometry problems from Argentinian Cono Sur + IMO + IberoAmerican  Team Selection Tests (TST)
with aops links in the names

(only those not in IMO Shortlist)

[3p per day]

collected inside aops: here

IMO TST 1996 - 2022

1996 Argentina IMO TST p5

Let $ABC$ be a triangle such that $$\angle A = \frac{180^o}{7},  \angle B =  \frac{360^o}{7},  \angle C =  \frac{720^o}{7}.$$ Let $D, E, F$ be the midpoints of the sides $BC, AC, AB$ respectively. Let $P, Q, R$ be the feet of the altitudes drawn from $A, B, C$ respectively. Show that the points $D, E, F, P, Q, R$ (in some order) are $6$ of the vertices of a regular heptagon.

1997 Argentina IMO TST p2

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

1997 Argentina IMO TST p5

Let $ABCDE$ be a regular pentagon and $X$ an interior point such that $<AEX = 48^o$ and $\angle BCX = 42^o$. Find $\angle AXC$.

1998 Argentina IMO TST p3

Let$AB$ be a diameter of a circle $k_1$. Another circle, $k_2$, with center at $A$ intersects $k_1$ at $E$ and $F$. An arbitrary point $D$ is chosen on the arc $EF$ of $k_1$ that is interior to k_2, and C is the intersection point between $k_2$ and $BD$. If $DE = a$ and $DF = b$, find $DC$.

1998 Argentina IMO TST p5

Let $ABCD$ be a convex quadrilateral that has side $AB$ equal to side $CD$ but does not any pairs of opposite parallel sides. Let $M$ and $N$ be the midpoints of the sides $AD$ and $BC$ respectively. Show that the angle formed by lines MN and $AB$ is equal to the angle formed by lines $MN$ and $CD$.

1999 Argentina IMO TST p2

Four circles $C_1, C_2, C_3, C_4$ are considered such that $C_1$ and $C_2$ are externally tangent at $A$, $C_2$ and $C_3$ are externally tangent at $B$, $C_3$ and $C_4$ are externally tangent at $C$, $C_4$ and $C_1$ are externally tangent at $D$ and furthermore, the lines $AB$ and $CD$ intersect at $S$. A tangent to $C_2$ that touches said circle at $P$ is drawn through $S$ and a tangent to $C_4$ that touches said circle at $Q$ is drawn through $S$. Show that $SP = SQ$.

2000 Argentina IMO TST p2

Let $ABC$ be an obtuse triangle with C> 90º, $D$ a point on side $BC$ and $O$ the midpoint of $AD$. Let $M$ and $N$ in $OD$ and $OC$, respectively, such that $MN$ is parallel to $BC$ and $AN = 2OM$. We denote $P$ the intersection point of $CM$ and the parallel to $AC$ drawn through $O$. Show that $AP$ is a bisector of the angle $\angle MAN$.

2000 Argentina IMO TST p4

Let $ABCDEF$ be a hexagon, not necessarily regular, and $S$ a circle tangent to all six sides of the hexagon, such that $S$ is tangent to $AB, CD$, and $EF$ at their midpoints $P$, $Q$, and $R$, respectively. If $X, Y, Z$ are the touchpoints of $S$ with $BC$, $DE$, and $FA$, respectively, show that $PY$, $QZ$, and $RX$ are concurrent.

2001 Argentina IMO TST p3

Given a circle and a point $A$ outside the circle, let $M$ and $N$ be the touch points of the lines tangent to the circle drawn from $A$. A line through $A$ intersects the circle at $B$ and $C$. If $D$ is the midpoint of segment $MN$, show that $MN$ is bisector of angle $\angle BDC$.

2001 Argentina IMO TST p5

Let $C$ be the smallest of the angles of triangle $ABC$. On the circle that passes through $A, B$ and $C$, we consider a variable point $X$ on the arc $AB$ that does not contain $C$. Let $D$ be on the ray $AX$ such that $AD = BC$ and let $E$ be on the ray $BX$ such that $BE = AC$ . We denote $M$ the midpoint of the segment $DE$. Determine the locus of the points $M$ when $X$ varies in the arc $AB$.

2002 Argentina IMO TST p2

In a triangle $ABC$, let $E$ on side $AC$ and $D$ on side $BC$ such that $BE$ is the bisector of angle $\angle B$ and $AD$ is the bisector of angle $\angle A$. Show that if $\angle BED = 30^o$, then triangle $ABC$ has an angle of $60^o$ or has an angle of $120^o$.

2002 Argentina IMO TST p4

In a parallelogram $ABCD$, the perpendicular to side $BC$ drawn through $A$ intersects line $BC$ at $M$ and the perpendicular to side $CD$ drawn through $A$ intersects line $CD$ at $N$. We denote by $H$ the orthocenter of triangle $AMN$. If $AC = 21$ and $MN = 18$, find the lenght of segment $AH$.

2003 Argentina IMO TST p2

Let $O$ be the center of the circle circumscribed to the triangle $ABC$. The points $M$ and $N$ on the sides $AB$ and $BC$, respectively, are such that $2\angle MON = \angle AOC$. Show that the perimeter of the triangle $BMN$ is greater than or equal to $AC$.

2003 Argentina IMO TST p5

In a triangle $ABC$, let $M$ and $N$ be on the sides $AB$ and $AC$, respectively, such that $MN$ is parallel to $BC$. Segments $BN$ and $CM$ intersect at $K$. The circumcircle through $A, K$, and $B$ intersects line $BC$ at $P$ and the circumcircle through $A, K$, and $C$ intersects line $BC$ at $Q$. If $T$ is the intersection point of the lines $PM$ and $QN$, prove that $T$ belongs to the line $AK$.

2004 Argentina IMO TST p2

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$ .Consider a point $P$ in $AB$. A variable point $Q$ moves along $CD$. Let $X$ be the intersection point of $BQ$ and $CP$, and let$Y$ be the intersection point of $AQ$ and $DP$. Find the position of $Q$ for which the area of the quadrilateral $PXQY$ is maximum.

2004 Argentina IMO TST p6

Let $A$ be a point outside a circle $\Omega$. Let $B$ and $C$ be the points of tangency of the tangents from $A$. A line $r$, tangent to $\Omega$, intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. The line parallel to $AC$ through $P$ intersects the line BC at R. Show that as r varies, the lines $QR$ pass through a fixed point.

2005 Argentina IMO TST p3

Given the triangle $ABC$ we consider the points $X,Y,Z$ such that the triangles $ABZ,BCX,CAZ$ are equilateral, and they don't have intersection with $ABC$. Let $B'$ be the midpoint of $BC$, $N'$ the midpoint of $CY$, and $M,N$ the midpoints of $AZ,CX$, respectively. Prove that $B'N' \bot MN$.

2006 Argentina IMO TST p3

In a circumference with center $O$ we draw two equal chord $AB=CD$ and if $AB \cap CD =L$ then $AL>BL$ and $DL>CL$ .We consider $M \in AL$ and $N \in DL$ such that $\widehat {ALC} =2 \widehat {MON}$ .Prove that the chord determined by extending $MN$ has the same as length as both $AB$ and $CD$

2007 Argentina IMO TST p2

Let $ABCD$ be a trapezium of parallel sides $AD$ and $BC$ and non-parallel sides $AB$ and $CD$. Let $I$ be the incenter of $ABC$. It is known that exists a point $Q \in AD$ with $Q \neq A$ and $Q \neq D$ such that if $P$ is a point of the intersection of the bisectors of  $\widehat{ CQD}$ and $\widehat{CAD}$ then $PI \parallel AD$ Prove that $PI=BQ$

2008 Argentina IMO TST p2

Triangle $ABC$ is inscript in a circumference $\Gamma$. A chord $MN=1$ of $\Gamma$ intersects the sides $AB$ and $AC$ at $X$ and $Y$ respectively, with $M$, $X$, $Y$, $N$ in that order in $MN$. Let $UV$ be the diameter of $\Gamma$ perpendicular to $MN$ with $U$ and $A$ in the same semiplane respect to $MN$. Lines $AV$, $BU$ and $CU$ cut $MN$ in the ratios $\frac{3}{2}$, $\frac{4}{5}$ and $\frac{7}{6}$ respectively (starting counting from $M$). Find $XY$

2008 Argentina IMO TST p5

Let $ABC$ be a triangle, $D$, $E$ and $F$ the points of tangency of the incircle with sides $BC$, $CA$, $AB$ respectively. Let $P$ be the second point of intersection of $CF$ and the incircle. If $ABPE$ is a cyclic quadrilateral prove that $DP$ is parellel to $AB$

2009 Argentina IMO TST p3

Let $ABC$ be a triangle, $B_1$ the midpoint of side $AB$ and $C_1$ the midpoint of side $AC$. Let $P$ be the point of intersection ($\neq A$) of the circumcircles of triangles  $ABC_1$ and $AB_1C$. Let $Q$ be the point of intersection ($\neq A$) of the line $AP$ and the circumcircle of triangle $AB_1C_1$. Prove that $\frac{AP}{AQ} = \frac{3}{2}$.

2010 Argentina IMO TST p2

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $EF$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.

2011 Argentina IMO TST p3

Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$. Show that $SO \perp XY$.

2012 Argentina IMO TST p3

Let $ABC$ be an acute and scalene triangle with $AB <AC$ and circumscribed circle $\Gamma$. Circle $\Gamma_1$ with center $A$ and radius $AB$ cuts side $BC$ at $E$ and circle $\Gamma$ at $F$. Line $EF$ cuts circle $\Gamma$ fot second time at point $D$ and side $AC$ at point $M$. Line $AD$ cuts to side $BC$ at point $K$. Finally, the circle circumscribed to triangle $BKD$ intersects line $AB$ for the second time at $L$. Show that points $K, L, M$ lie on a line parallel to $BF$.

2012 Argentina IMO TST p5

In an acute triangle $ABC$, let $M$ be the midpoint of the side $AB$ and $P, Q$ the feet of the altitudes $AP$, $BQ$, respectively. The circle through$B, M, P$ is tangent to side $AC$.Show tha t the circle through $A, M, Q$ is tangent to the extension of side $BC$.

2013 Argentina IMO TST p3

Let $ABCD$ be a quadrilateral inscribed in a circle and such that its diagonals $AC$ and $BD$ intersect at $S$. Let $k$ be a circle that passes through $S$ and $D$ and intersects sides $AD$ and $CD$ at $M$ and $N$ respectively. Let $P$ be the intersection of lines $SM$ and $AB$, and $R$ the intersection of lines $SN$ and $BC$, so that $P$ and $R$ are in the same half plane wrt line $BD$ as point $A$. Show that the line through $D$ parallel to $AC$ and the line through $S$ parallel to $PR$ intersect at circumference $k$.

2014 Argentina IMO TST p3

Two circles $\Gamma_1$ and $\Gamma_2$ of different radii intersect at two points $A$ and $B$. Let $C$ and $D$ be two points at $\Gamma_1$ and $\Gamma_2$ respectively such that $A$ is the midpoint of segment $CD$. Line $DB$ again cuts $\Gamma_1$ at $E$, with $B$ between $D$ and $E$; line $CB$ cuts $\Gamma_2$ again at $F$, with $B$ between $C$ and $F$. Let $\ell_1$ and $\ell_2$ be the bisectors of $CD$ and $EF$, respectively. It is known that $\ell_1$ and $\ell_2$ intersect at a single point $P$. Show that the lengths of $CA$, $AP$, and $PE$ are the sides of a right triangle.

2014 Argentina IMO TST p4

Let ABC be a triangle and $\Gamma$ its circumscribed circcle. Let $X, Y, Z$ be the midpoints of arcs $BC$, $CA$ and $AB$ of $\Gamma$, respectively, that do not contain the third point of the triangle. The intersection of the triangles $ABC$ and $XYZ$ forms a hexagon $DEFGHK$. Show that $DG$, $EH$, and $FK$ are concurrent.

2015 Argentina IMO TST p2

Let $A, B, C, D$ be four points on a circle, in that order, with $AD> BC$. The lines $AB$ and $CD$ intersect at $K$. It is known that the points $B, D$ and the midpoints of the segments $AC$ and $KC$ lie on the same circle. Determine all possible value of the angle $\angle ADC$.

2016 Argentina IMO TST p2

Let $ABC$ be an acute triangle with $AC \ne BC$. The altitudes $AD$ and $BE$ of this triangle intersect at $H$. Let $\omega_1$ be the circle of center $H$ and radius $HE$ and let $\omega_2$ be the circle of center $B$ and radius $BE$. The tangent to $\omega_1$ drawn from $C$ that does not pass through $E$, touches $\omega_1$ at $P$; the tangent to $\omega_2$ drawn from $C$ that does not pass through $E$, touches $\omega_2$ at $Q$. Prove that points $D, P, Q$ are collinear.

2017 Argentina IMO TST p3

The inscribed circle of triangle $ABC$ is tangent to $BC$, $AC$, $AB$ at points $D$, $E$, $F$ respectively. Let $I$ be the incenter of triangle $ABC$. Suppose that the line $EF$ intersects the lines $BI$, $CI$, $BC$, $DI$ at the points $K$, $L$, $M$, $Q$ respectively. If the line through the midpoint of $CL$ and through $M$ intersects $CK$ at $P$, show that $$PQ = \frac{AB \cdot KQ}{BI}.$$

2021 Argentina IMO TST p2

Let $\Gamma_1$ and $\Gamma_2$ be two circles of different radii, with $\Gamma_1$ the smallest radius. The two circles intersect at two different points $A$, and $B$. Let $C$ at $\Gamma_1$ and $D$ at $\Gamma_2$ such that $A$ is the midpoint of segment $CD$. It is known that line $CB$ cuts $\Gamma_2$ at $F$ so that $B$ is between $C$ and $F$, and line $DB$ cuts $\Gamma_1$ at $E$ so that $B$ is between $D$ and $E$. The bisectors of $CD$ and $EF$ intersect at $P$ .

a) Show that $\angle EPF = 2\angle CAE$.

b) Show that $AP^2 = CA^2 + PE^2$.

Cono Sur TST 1997 - 2019, 2022

1997 Argentina Cono Sur TST p3

Given triangle $ABC$ such that the smallest of its angles is $\angle A = 30^o$, let $O$ be the point of intersection of the bisectors and $I$ be the point of intersection of the bisectors. If $D$ and $E$ are points on sides $AB$ and $CA$, respectively, such that $BD = CE = BC$. Prove that $OI$ and $DE$ are perpendicular and have equal length.

1997 Argentina Cono Sur TST p6

Let $ABC$ be an acute triangle and $CD$ the altitude corresponding to vertex $C$. If $M$ is the midpoint of $BC$ and $N$ is the midpoint of $AD$, calculate $MN$ knowing that $AB = 8$ and $CD = 6$.

1998 Argentina Cono Sur TST p6

Let $ABC$ be a triangle. The angle bisector of $CAB$ intersects $BC$ at $D$ and the angle bisector of $ABC$ intersects $CA$ at $E$. If $AE + BD = AB$, show that $\angle BCA = 60^o$.

1999 Argentina Cono Sur TST p2

Triangle ABC has $\angle C = 120^o$ and side $AC$ is greater than side $BC$. Knowing that the area of the equilateral triangle with side $AB$ is $31$ and the area of the equilateral triangle with side $AC - BC$ is $19$, find the area of triangle $ABC$.

1999 Argentina Cono Sur TST p5

In the square $ABCD$, let $P$ be on side $AB$ such that $AP^2 = BP \cdot BC$ and M the midpoint of $BP$. If $N$ is the interior point of the square such that $AP = PN$ and $MN$ is parallel to $BC$, calculate the measure of angle $\angle BAN$.

2000 Argentina Cono Sur TST p1

Lucas draws a segment $AC$ and Nicolás marks a point $B$ inside the segment. Let $P$ and $Q$ be points in the same half plane wrt $AC$ such that the triangles $APB$ and $BQC$ are isosceles in $P$ and $Q$, respectively, with $\angle APB = BQC = 120^o$. Let $R$ be the point of the other half plane such that the triangle $ARC$ is isosceles in $R$, with $\angle  ARC = 120^o$. The triangle $PQR$ is drawn. Show that this triangle is equilateral.

2000 Argentina Cono Sur TST p6

Let $P$ be a point in the interior of an angle, which does not belong to its bisector. Two segments are considered by $P$: $AB$ and $CD$, with $A$ and $C$ on one side of the angle, $B$ and $D$ on the other side of the angle, such that $P$ is the midpoint of $AD$ and $CD$ is perpendicular to the bisector of the angle. Show that $AB> CD$.

2001 Argentina Cono Sur TST p2

Let $ABC$ be a right triangle at $C$. We consider $D$ in the hypotenuse $AB$ such that $CD$ is the altitude of the triangle, and $E$ in the leg $BC$ such that $AE$ is the bisector of angle $A$. If $F$ is the point of intersection of $AE$ and $CD$, and $G$ is the point of intersection of $ED$ and $BF$, prove that: area $(CEGF) =$ area $(BDG)$.

2001 Argentina Cono Sur TST p5

On the line $r$ Pablo marks, from left to right, the points $A, B, C$ and $D$. Lucas must construct, with a ruler and compass, a square $PQRS$, with sides $PQ, QR, RS$ and $SP$, contained in one of the semiplanes determined by line $r$, so that $A$ lies on line $PQ$, $B$ lies on line $RS$, $C$ lies on line $QR$ and $D$ lies on line $SP$.

Show a procedure that always allows Lucas to do the construction and justify why with this procedure the requested square is achieved.

2002 Argentina Cono Sur TST p2

Let $ABCD$ be a trapezoid with bases $AB = 5$ and $CD = 2$, and non-parallel sides $BC = 4$ and $DA = 1$. The external bisector of angle $B$ intersects the external bisector of angle $C$ at point $P$, and the external bisector of angle $A$ intersects the external bisector of angle $D$ at point $Q$. Calculate the measure of segment $PQ$.

2002 Argentina Cono Sur TST p5

Let $ABC$ be a triangle, with $AB <BC$, and we denote by $O$ the center of the circle circumscribed to the triangle. The bisector of angle $B$ intersects side $AC$ at $D$. The line perpendicular to $AC$ is drawn through $O$, which intersects side $AC$ at $M$ and the arc $AC$ containing $B$ The line perpendicular from $P$ on $BC$ that cuts side $BC$ into $N$. Show that each of the diagonals of quadrilateral $BDMN$ divides triangle $ABC$ into two figures of equal areas.

2004 Argentina Cono Sur TST p2

An ancient civilization had only one instrument of geometry. This instrument has two functions, and no more: to draw lines through two points and to draw perpendicular to a line through a given point.

Give a procedure for dividing a given $60^o$ angle into two equal angles using exclusively the instrument of the ancients.

2004 Argentina Cono Sur TST p4

Nicolás must draw a triangle $ABC$ and a point $P$ inside it so that among the $6$ triangles that $ABC$ is divided into by the lines $AP$, $BP$ and $CP$ there are $4$ that have equal areas. Decide if it is possible to do this without the $6$ triangles having equal areas.

2005 Argentina Cono Sur TST p2

Given an angle of $13^o$, construct an angle of $1^o$ using only a ruler and compass.

2005 Argentina Cono Sur TST p5

Let $ABCD$ be a square. A line $t$ intersects side $BC$ at $K$ ($K\ne  B$ and $K\ne C$), diagonal $AC$ at $L$ and the extension of side $BA$ at $M$, so that $KL = DL$. Calculate the measure of the angle $\angle KMD$.

2006 Argentina Cono Sur TST p3

Let $ABC$ be a triangle with $\angle BAC=120^o$. The angle bisectors of $A,B,C$ intersect the respective opposite sides at points $D, E, F$. Show that the angle $EDF$ is right.

2007 Argentina Cono Sur TST p5

Given an equilateral triangle $ABC$. Let $M$ be a point on the side $BC$,with $M\ne B$ and $M \ne C$. The point $N$ is considered such that the triangle $BMN$ is equilateral and points $A ,N$ lie on different half-planes wrt $BC$. Let $P, Q$ and $R$ be the midpoints of $AB$, $BN$ and $CM$ respectively. Show that triangle $PQR$ is equilateral.

2008 Argentina Cono Sur TST p4

Let $O$ be the intersection point of the angle bisectors of a triangle ABC. We denote $D$ the intersection point of the line $AO$ with the segment $BC$. If $OD=DB=\frac13 BC$ , calculate the angle measure of the triangle.

2009 Argentina Cono Sur TST p5

Let $ABCD$ be a square and $E$ a point on side $BC$. Segment $AE$ cuts diagonal $BD$ at $G$. Let $F$ lie on side $CD$ such that $FG$ is perpendicular on $AE$, and let $K$ on $FG$ such that $AK = FE$. Calculate the measure of the angle $\angle FKE$.

2010 Argentina Cono Sur TST p3

Let $ABC$ be a triangle. We consider points $E$ and $D$ inside sides $AC$ and $BC$, respectively, such that $AE = BD$. Let $M$ be the midpoint of side $AB$ and $P$ the intersection point of lines $AD$ and $BE$. Show that the symmetric of $P$ wrt $M$ lies on the bisector of the angle $ACB$.

2011 Argentina Cono Sur TST p3

Let $ABC$ be a triangle and consider its circumscribed circle. The chord $AD$ is the bisector of the angle of triangle $ABC$ and intersects side $BC$ at $L$; chord $DK$ is perpendicular to side $AC$ and cuts it at $M$. If $\frac{BL}{LC}=\frac{1}{2}$, calculate$\frac{AM}{MC}$.

2012 Argentina Cono Sur TST p5
Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$. Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$. If $ML$ is the angle bisector of $\angle KMB$, find $\angle MLC$.

2013  Argentina Cono Sur TST p5
Let $ABC$ be an equilateral triangle and $D$ a point on side $AC$. Let $E$ be a point on $BC$ such that $DE \perp BC$, $F$ on $AB$ such that $EF \perp AB$, and $G$ on $AC$ such that $FG \perp AC$. Lines $FG$ and $DE$ intersect in $P$. If $M$ is the midpoint of $BC$, show that $BP$ bisects $AM$.

2014 Argentina Cono Sur TST p5
In an acute triangle $ABC$, let $D$ be a point in $BC$ such that $AD$ is the angle bisector of $\angle{BAC}$. Let $E \neq B$ be the point of intersection of the circumcircle of triangle $ABD$ with the line perpendicular to $AD$ drawn through $B$. Let $O$ be the circumcenter of triangle $ABC$. Prove that $E$, $O$, and $A$ are collinear.

2015 Argentina Cono Sur TST p4

For an acute triangle $ABC$ let $H$ be the point of intersection of the altitudes $AA_1$, $BB_1$, $CC_1$. Let $M$ and $N$ be the midpoints of the $BC$ and $AH$ segments, respectively. Show that $MN$ is the perpendicular bisector of segment $B_1C_1$.

2016 Argentina Cono Sur TST p3

An acute triangle $ABC$ has its three vertices on a circle $\Omega$. Let $G$ be the point of intersection of the medians of triangle $ABC$ and let $AH$ be an altitude of this triangle. Ray $GH$ intersects $\Omega$ at $A'$. Show that the circle that passes through the vertices of triangle $A'HB$ is tangent to the line $AB$.

2016 Argentina Cono Sur TST p5

Let ABC be a triangle with $\angle BAC <90^o$. The perpendiculars drawn from $C$ on $AB$ and from $B$ on $AC$ intersect again the circle passing through the vertices of triangle $ABC$ at $D$ and $E$ respectively. If $DE = BC$, calculate the measure of angle $\angle BA C$.

2017 Argentina Cono Sur TST p3

On a circle with diameter $AB$ a point $M$ is chosen. On the same circle, let $X$ be such that the intersection of $MX$ and $AB$ is point $Y$, with $\angle MY B <90^o$. The chord perpendicular on $AB$ that passes through $Y$ cuts $BX$ at $P.$ Show that when $X$ varies, the point $P$ always belongs to the same line.

2017 Argentina Cono Sur TST p4

Let $ABCD$ be a quadrilateral with $AC = 20$ and $AD = 16$. Let $P$ in segment $CD$ such that the triangles $ABP$ and $ACD$ are congruent. If the area of the triangle $APD$ is $28$, calculate the value of the area of the triangle $BCP$.

(Two triangles are congruent if their sides are respectively equal.)

2018 Argentina Cono Sur TST p3

Let $ABC$ be a triangle and let $K$ and $M$ be the midpoints of the sides $AB$ and $AC$ respectively. In segments $AM$ and $BK$, equilateral triangles $AMN$ and $BKL$ are constructed, both exterior to triangle $ABC$. Let $F$ be the midpoint of segment $LN$. Determine the measure of the angle $\angle KFM$.

2018 Argentina Cono Sur TST p4

Let $ABCDE$ be a regular pentagon with center $M$. A point $P$ is chosen inside the segment MD. The circle through points $A, B$, and $P$ intersects segment $AE$ at $A,Q$ and intersects the line perpendicular on $CD$ through $P$ at $P ,R$. Show that $AR$ and $QR$ have the same length.

2019 Argentina Cono Sur TST p2

Let $ABC$ be a triangle and $I$ the intersection point of its angle bisectors. Let $\Gamma$ be the circle with center $I$ that is tangent to the three sides of the triangle . Let $D$ at $BC$ and $E$ at $AC$ be the points of tangency of $\Gamma$ with $BC$ and $AC$. Let $P$ be the intersection point of lines $AI$ and $DE$, and let $M$ and $N$ be the midpoints of $BC$ and $AB$ respectively. Show that $M, N$ and $P$ belong to a line.

2019 Argentina Cono Sur TST p5

Let $ABCD$ be a parallelogram with the angle at A acute. We consider the point $E$ inside the parallelogram such that $AE = DE$ and $\angle ABE = 90^o$. Let M be the midpoint of segment $BC$. Determine the measure of the angle $\angle DME$.

Ibero TST 1993 - 2019, 2021-22

1993 Argentina Ibero TST p1

Let $ABC$ be a triangle and let $O$ be the center of the circle that is tangent to side $AC$ and to the extensions of sides $BA$ and $BC$. If $D$ is the center of the circle through $A, B$, and $O$, show that $A, B, C$, and $D$ belong to a circle.

1994 Argentina Ibero TST p2

In a triangle $ABC$, let $H$ be the foot of the altitude drawn from $A$. The bisector of angle $B$ intersects side $AC$ at $E$ and angle $BEA$ is $45$ degrees. Calculate the measure of the angle $EHC$.

1995 Argentina Ibero TST p2

Given an angle $PAQ$, an exterior point $L$, and a length $d$, draw a line through $L$ that intersects the sides of the angle at $B$ and $C$, so that $AB + BC + CA = d$. Indicate the construction steps.

1995 Argentina Ibero TST p4

Let $ABC$ be a right triangle and $D$ be the point on hypotenuse $AC$ such that $AB = CD$. Show that in triangle $ABD$ the bisector of angle $A$, the altitude from vertex $D$ and the corresponding median to side $AD$ are concurrent.

1996 Argentina Ibero TST p2

Given in the plane four lines, $a, b, c$ and $d$ such that each three of them determine a triangle. Prove that if $a$ is parallel to a median of the triangle that determine $b, c$ and $d$, then $b$ is parallel to a median of the triangle determining $a, c$ and $d$, $c$ is parallel to a median of the triangle determining $a, b$ and $d$, and $d$ is parallel to a median of the triangle determining $a, b$ and $c$.

1996 Argentina Ibero TST p6

There are two circles $C_1$ and $C_2$ of different radii, which intersect at two points. Let $P$ be an interior point of the region common to the two circles corresponding to $C_1$ and $C_2$. Each line $\ell$ that passes through $P$ intersects the edge of the region common to both circles at $U$ and $V$. Determine the position of $\ell$ for which $PU \times PV$ is minimum.

1997 Argentina Ibero TST p1

Let $ABCD$ be a parallelogram and $O$ an interior point such that $\angle AOB +\angle COD = 180^o$. Prove that $\angle OBC = \angle ODC$.

1998 Argentina Ibero TST p1

Let $ABC$ be an acute triangle and $M$ be a point inside side $AC$. Let $\ell$ be the perpendicular on $BC$ through the midpoint of $AM$, $t$ the perpendicular to AB through the midpoint of $CM$, and $P$ the intersection point of $\ell$ and $t$. Find the locus of $P$ when $M$ varies on side $AC$.

1998 Argentina Ibero TST p5

Let $ABC$ be a triangle and $P$ an interior point such that $\angle PBC =  \angle PCA <  \angle PAB$ . Line $PB$ cuts the circle circumscribed to triangle $ABC$ at $E$, and line $CE$ cuts the circle circumscribed to triangle $APE$ at $F$. Show that area $(APEF)$ / area $(APB)$ does not depend on the choice of point $P$.

1999 Argentina Ibero TST p2

Let $ABP$ be an isosceles triangle with $AB = AP$ and the acute angle $PAB$. The line perpendicular on $BP$ is drawn through $P$, and in this perpendicular we consider a point $C$ located on the same side as $A$ wrto the line $BP$ and on the same side as $P$ wrt the line $AB$. Let $D$ be such that $DA$ is parallel to $BC$ and $DC$ is parallel to $AB$. Let $M$ be the intersection point of $PC$ and $DA$. Find $DM / DA$.

1999 Argentina Ibero TST p5

Let $ABCD$ be a rectangle with side $AB$ greater than side $AD$. The circle with center $B$ and radius $AB$ intersects line $DC$ at points $E$ and $F$.

(a) Show that the circle circumscribed to triangle $EBF$ is tangent to the circle of diameter $AD$.

(b) Let $G$ be the touchpoint between the two circles mentioned in (a). Show that points $D, G$, and $B$ are collinear.

2000 Argentina Ibero TST p2

Let $ABC$ be a triangle, $O$ its circumcenter, $P$ the midpoint of $AO$, and $Q$ the midpoint of $BC$. If $\angle OPQ = a$, $\angle CBA = 4a$, $\angle ACB = 6a$, determine the measure of $a$.

2000 Argentina Ibero TST p5

Let $A, P, X$ be three points on the plane such that $45^o< \angle APX <90^o$. Construct a square $ABCD$ with a ruler and compass so that $P$ lies on side $BC$ and the line $PX$ intersects side $CD$ at point $Q$ such that $\angle QAP = \angle PAB$.

2001 Argentina Ibero TST p2

Circles $\Gamma_1$ and $\Gamma_2$ intersect at two points, $A$ and $B$. A line is drawn through $B$ that intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangent line to $\Gamma_1$ that passes through $C$ intersects the tangent line to $\Gamma_2$ that passes by $D$ at $E$. The line symmetric to line $AE$, wrt line $AC$, intersects $\Gamma_1$ at $F$ (besides $A$). Show that $BF$ is tangent to $\Gamma_2$.

Note: Consider only the case where $C$ does not belong to the interior of $\Gamma_2$ and $D$ does not belong to the interior of $\Gamma_1$

2001 Argentina Ibero TST p5

Given a triangle with sides $a, b, c$ such that the centroid of the triangle lies on the circle inscribed in the triangle. Prove that $5 (a^2 + b^2 + c^2) = 6 (ab + bc + ca)$.

2002 Argentina Ibero TST p2

In a convex quadrilateral $ABCD$ (which is not a parallelogram) the diagonals $AC$ and $BD$ intersect at $P$. The angle bisector of $\angle ABP$ intersects side $AB$ at $Q$ and the angle bisector of $\angle APD$ intersects side $AD$ at $R$. Let $M$ and $N$ denote the midpoints of $AC$ and $BD$ respectively . Let $O$ be the point of $MN$ such that $\frac{MO}{NO}=\frac{AC}{BD}$. Show that the line $AO$ passes through the midpoint of $QR$.

2004 Argentina Ibero TST p3

Let $ABC$ be an acute triangle such that $\angle B > \angle A$ and $\angle B > \angle C$. We denote by $O$ the circumcenter of triangle $ABC$, $T$ the circumcenter of triangle $AOC$ and $M$ the midpoint of side $AC$. $D$ and $E$ are considered on sides $AB$ and $BC$ respectively such that $\angle BDM=\angle BEM=\angle ABC$. Show that $BT \perp DE$.

2004 Argentina Ibero TST p4

Given a parallelogram $ABCD$ , let $M$ on side $AB$ and $N$ on side $BC$ such that $AM = NC$ ($M$ and $N$ are not vertices). We denote by$Q$ the point of intersection of $AN$ and $CM$. Show that $DQ$ is the bisector of the angle $\angle ADC$.

2005 Argentina Ibero TST p2

Let $A, B, C$ be three points on a circle such that triangle $ABC$ has $AB <BC$. We denote by $M$ the midpoint of side $AC$ and $N$ the midpoint of arc $AC$ that contains $B$. Let $I$ be the incenter of triangle $ABC$. Show that $\angle IMA =\angle INB$.

2006 Argentina Ibero TST p2

Let $ABC$ be a triangle such that $BC > CA >AB$. Let $D$ be on side $BC$ such that $BD=AC$ and let E be on the extension of side $BA$ ($A$ is between $B$ and $E$) such that $BE=AC$. The circle through $B, E$, and $D$ intersects side $AC$ at $P$, and the line $BP$ intersects the circle through $A, B$, and $C$ at $Q$ ($Q \ne B$). Prove that $AQ+CQ=BP$.

2007 Argentina Ibero TST p3

In a triangle $ABC$, with $AB <BC$, let $D$ be on the side $AC$ such that $AB = BD$. Let $K$ and $L$ be the touchpoints of the circle inscribed in triangle $ABC$ with sides $AB$ and $AC$, respectively,. Let $J$ be the incenter of triangle $BCD$. Show that the point of intersection of $KL$ and $AJ$ is the midpoint of the segment $AJ$.

2007 Argentina Ibero TST p4

Let $A, B, C, D, E$ be five points on a circle $\Gamma$ such that the pentagon $ABCDE$ is convex. It is known that $AD$ is a diameter of $\Gamma$ and that the diagonals $BE$ and $AC$ are perpendicular. Let $P$ be the point where $CE$ and $AD$ intersect. Prove that: area $(APE)$ = area $(ABC) +$ area $(CDP)$.

2008 Argentina Ibero TST p2

Two circles, $\omega_1$ and $\omega_2$ intersect at $A$ and $B$. Let $r_1$ be the tangent to $\omega_1$ that passes through $A$ and $r_2$ be the tangent to $\omega_2$ that passes through $B$. The lines $r_1$ and $r_2$ intersect at $C$. Let $T$ be the point of intersection of $r_1$ and $\omega_2$ ($T  \ne  A$). We consider a point $X$ of $\omega_1$ (which is neither $A$ nor $B$). Line $XA$ cuts $\omega_2$ at $Y$ ($Y \ne A$). Lines $YB$ and $XC$ intersect at $Z$. Show that $TZ$ is parallel to $XY$.

2009 Argentina Ibero TST p3 (MEMO 2008)

Let $ABC$ be an isosceles triangle with $AC = BC.$ Its incircle touches $AB$ in $D$ and $BC$ in $E.$ A line distinct of $AE$ goes through $A$ and intersects the incircle in $F$ and $G.$ Line $AB$ intersects line $EF$ and $EG$ in $K$ and $L,$ respectively. Prove that $DK = DL.$

2010 Argentina Ibero TST p3

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$, and sides $BC$ and $DA$, such that $AB = 2CD$. Let $E$ be the midpoint of side $BC$. Show that $AB = BC$ if and only if the quadrilateral $AECD$ has an inscribed circle.

2010 Argentina Ibero TST p4

Let $O$ be the circumcenter of the acute triangle $ABC$ and let $\Gamma$ be its circumscribed circle. The internal angle bisector of $\angle A$ intersects Γ at $D$. The internal angle bisector of $\angle B$ intersects $\Gamma$ at $E$. Let $I$ be the incenter of triangle $ABC$. If the points $D, E, O$ and $I$ belong to the same circle, calculate the measure of the angle $\angle ACB$.

2011 Argentina Ibero TST p3

Two circles, $S_1$ and $S_2$, intersect at $L$ and $M$. Let P be a point on $S_2$. Lines $PL$ and $PM$ intersect $S_1$ again at $Q$ and $R$, respectively. The lines $QM$ and $RL$ intersect at $K$. Show that when $P$ varies in $S_2$, $K$ lies on a fixed circle.

2012 Argentina Ibero TST p2

Let $ABC$ be a triangle and $O$ be a point inside it. Let $D$ in $BC$ such that $OD$ is perpendicular to $BC$ and $E$ in $AC$ such that $OE$ is perpendicular to $AC$. If $F$ is the midpoint of side $AB$ and $DF = EF$, show that $\angle OBD = \angle OAE$.

2013 Argentina Ibero TST p3

Let $ABC$ be an isosceles triangle with base $AB$. The points $P$ on side $AC$ and $Q$ on side $BC$ are chosen such that $AP + BQ = PQ$. The line parallel to $BC$ that passes through the midpoint of segment $PQ$ intersects segment $AB$ at $N$. The circle that passes through the vertices of triangle $PNQ$ intersects line $AC$ at points $P$ and $K$, and line $BC$ at points $Q$ and $L$. If $R$ is the intersection point of lines $PL$ and $QK$, show that line $PQ$ is perpendicular to line $CR$.

2013 Argentina Ibero TST p4

Given a triangle $ABC$ with $AC =\frac{AB + BC}{2}$. Let $BL$ be the bisector of the angle $\angle ABC$.Let $K$ and $M$ be the midpoints of $AB$ and $BC$ respectively. Calculate the measure of the angle $\angle KLM$ if it is known that $\angle ABC = \beta$.

2014 Argentina Ibero TST p3

There are three collinear points $B, C, D$, with $C$ between $B$ and $D$. Another point $A$ that does not belong to the line $BD$ is such that $AB = AC = CD$. If $\frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD}$, calculate the measure of the angle $\angle BAC$.

2014 Argentina Ibero TST p5

Let $ABC$ be a right triangle at $C$ and $N$ the foot of the altitude corresponding to the hypotenuse $AB$. The angle bisectors of $\angle NCA$ and $\angle BCN$ intersect segment $AB$ at $K$ and $L$ respectively. If $S$ and $T$ are the centers of the inscribed circles of the triangles $BCN$ and $NCA$ respectively, prove that the quadrilateral $KLST$ is cyclic.

2015 Argentina Ibero TST p3

Let $O$ be the center of the circle that passes through the vertices of a triangle $ABC$. We consider the line $\ell$ that passes through the midpoint of side $BC$ and is perpendicular to the bisector of angle $\angle BAC$. Determine the value of the angle $\angle BAC$ if the line $\ell$ passes through the midpoint of the segment $AO$.

2015 Argentina Ibero TST p4

A square $ABCD$ has its vertices on a circle with center $O$. Let $E$ be the midpoint of side $AD$. Line $CE$ intersects the circle again at $F$. Lines $FB$ and $AD$ intersect at $H$. Show that $HD = 2AH$.

2016 Argentina Ibero TST p2

Let $ABCD$ be a trapezoid of bases $BC,AD$. On the diagonals $AC$ and $BD$, let $P$ and $Q$ respectively be the points such that $AC$ bisects $\angle BPD$ and$BD$ bisects $\angle AQC$. Show that $\angle  BPD =   \angle  AQC$.

2016 Argentina Ibero TST p5

Let $ABCDE$ be a convex pentagon with $\angle ABC =\angle AED = 90^o$ and $\angle ACB = \angle ADE$. Points $P$ and $Q$ are the midpoints of segments $BC$ and $DE$ respectively, and segments $CQ$ and $DP$ intersect at $O$. Show that $AO$ is perpendicular to $BE$.

2017 Argentina Ibero TST p3

Let $\omega$ be the circumscribed circcle of an acute triangle $ABC$. Point $D$ belongs to arc $BC$ of $\omega$ that does not contain point $A$. Point$E$ is inside triangle $ABC$, does not belong to line $AD$, and satisfies $\angle DBE =\angle  ACB$ and $\angle DCE =\angle ABC$. Let $F$ be a point on line $AD$ such that lines $EF$ and $BC$ are parallel, and let $G$ be a point on $\omega$ other than $A$ such that $AF = FG$. Show that the points $D, E, F, G$ belong to a circle.

2017 Argentina Ibero TST p4

A point $A$ moves along the circle with center $O$ and radius $r$. Let $BC$ be a fixed segment of the plane, outside the circle. Show that the locus of the centroid of triangle $ABC$ is a circle of radius $r/3$ and whose center is the centroid of triangle $OBC$.

2018 Argentina Ibero TST p3

Two circles $\omega_1$ and $\omega_2$, with centers O_1 and O_2 respectively, intersect at points $A$ and $B$. A line through $B$ again intersects $\omega_1$ and $\omega_2$ at $C$ and $D$ respectively. The tangent lines to $\omega_1$ and $\omega_2$ at $C$ and $D$ respectively intersect at $E$. Let $F$ be the second intersection point of the line $AE$ with the circle $\omega$ that passes through $A$, $O_1$ and $O_2$. Show that the length of the segment $EF$ is equal to the diameter of $\omega$.

2018 Argentina Ibero TST p4

Let $ABC$ be an acute triangle. In the arc $BC$ of the circle that passes through $A, B$ and $C$ and does not contain the point $A$, the points $X, Y$ are chosen such that $BX = CY$. Let $M$ be the midpoint of segment $AX$. Show that $BM + CM \ge AY$.

2019 Argentina Ibero TST p3

Let $ABCDE$ be a convex pentagon such that $DC = DE$ and $\angle DCB = \angle DEA = 90^o$. Let $F$ in segment $AB$ such that $\frac{AF}{BF} = \frac{AE}{BC}$. Show that $\angle FEC = \angle BDC$.

2019 Argentina Ibero TST p5

Let $ABC$ be a paper triangle with $AB =\frac 32$, $AC =\frac{\sqrt{5}}{2}$, and $BC = \sqrt2$. A fold is made along a line perpendicular to $AB$. Find the maximum value that the area of the overlap can have and indicate the point through which the fold line passes when this maximum is achieved.

2021 Argentina Ibero TST p2

Let $ABCD$ be a convex quadrilateral and $O$ the intersection of its diagonals. Let $O$ and $M$ be the two points of intersection of the circumcircle of triangle $OAD$ with the circumcircle of triangle $OBC$. Let $T$ and $S$ be the points of intersection of $OM$ with the circumcircles of the triangles $OAB$ and $OCD$ respectively. Prove that $M$ is the midpoint of the segment $TS$.

2022 Argentina Ibero TST p3

Construct with ruler and compass a triangle $ABC$ right at $A$ with a perimeter equal to $10$ and such that the altitude drawn from the vertex $A$ measures $2$. Indicate the construction steps and justify why the conditions are satisfied.