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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


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.

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 $AC$.

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

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 $DC$.

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 $CD$.

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 $SP = SQ$.

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 $\angle MAN$.

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.

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 $\angle BDC$.

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 $AB$.

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 $120^o$.

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 $AH$.

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 $AC$.

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 $AK$.

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.

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.

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 $B'N' \bot MN$.

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$

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$
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$

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$

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}$.

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 $\frac{EM}{EN} = \frac{BF}{BP}$.
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 $SO \perp XY$.
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 $BF$.

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 $BC$.

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 $k$.

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.

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.

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 $\angle ADC$.

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.

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$$PQ = \frac{AB \cdot KQ}{BI}.$$

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


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.

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 $CD = 6$.

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 $\angle BCA = 60^o$.

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 $ABC$.

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 $\angle BAN$.

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.

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 $AB> CD$.

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 $(BDG)$.

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.

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 $PQ$.

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.
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.

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.

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

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 $\angle KMD$.

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.

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.

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.

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 $\angle FKE$.

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 $ACB$.

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.

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 $.

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 $AB$.

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 $\angle BA C$.

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.

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.)

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 $\angle KFM$.

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.

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.

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


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.

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 $EHC$.

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.

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.

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 $c$.

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.

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

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 $AC$.

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 $P$.

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 $DM / DA$.

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.

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 $a$.

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 $\angle QAP = \angle PAB$.

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 $\Gamma_1$

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 $5 (a^2 + b^2 + c^2) = 6 (ab + bc + ca)$.

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 $QR$.

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 $BT \perp DE$.

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 $\angle ADC$.

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 $\angle IMA =\angle INB$.

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 $AQ+CQ=BP$.

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 $AJ$.

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 $(CDP)$.

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 $XY$.

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.$

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.

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 $\angle ACB$.

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.

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 $\angle OBD = \angle OAE$.

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 $CR$.

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 $\angle ABC = \beta$.

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 $\angle BAC$.

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.

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 $AO$.

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 $HD = 2AH$.

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 $\angle  BPD =   \angle  AQC$.

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 $BE$.

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.

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 $OBC$.

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 $\omega$.

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 $BM + CM \ge AY$.

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 $\angle FEC = \angle BDC$.

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.

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 $TS$.

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.


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