Peru TST 2006-21 (IMO - OMCS - OIM - EGMO) 92p

geometry problems from Peruvian Team Selection Tests (TST) for IMO, Cono Sur (OMCS), IberoAmerican (OIM) and European Girls (EGMO) with aops links in the names

(only those not in IMO Shortlist)

[3p per day]

collected inside aops: here

IMO TST 2006 - 21

2006 Peru IMO TST P4

In an actue-angled triangle $ABC$ draws up: its circumcircle $w$ with center $O$, the circumcircle $w_1$ of the triangle $AOC$ and the diameter $OQ$ of $w_1$. The points are chosen $M$ and $N$ on the straight lines $AQ$ and $AC$, respectively, in such a way that the quadrilateral $AMBN$ is a parallelogram. Prove that the intersection point of the straight lines $MN$ and $BQ$ belongs the circumference $w_1.$

2007 Peru IMO TST P1

Let $P$ be an interior point of the semicircle whose diameter is $AB$ ($\angle APB$ is obtuse). The incircle of $\triangle ABP$ touches $AP$ and $BP$ at $M$ and $N$ respectively. The line $MN$ intersects the semicircle in $X$ and $Y$. Prove that $\widehat{XY}= \angle APB$.

2007 Peru IMO TST P6
Let $ABC$ be a triangle such that $CA \neq CB$, the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$, respectively, and $I$ the incircle. The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$. The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$. Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.

2008 Peru IMO TST P1
Let $ABC$ be a triangle and let $I$ be the incenter. $Ia$ $Ib$ and $Ic$ are the excenters opposite to points $A$ $B$ and $C$ respectively. Let $La$ be the line joining the orthocenters of triangles $IBC$ and $IaBC$. Define $Lb$ and $Lc$ in the same way. Prove that $La$ $Lb$ and $Lc$ are concurrent.

2008 Peru IMO TST P4

Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be two non-concentric circumferences such that $\mathcal{S}_1$ is inside $\mathcal{S}_2$. Let $K$ be a variable point on $\mathcal{S}_1$. The line tangent to $\mathcal{S}_1$ at point $K$ intersects $\mathcal{S}_2$ at points $A$ and $B$. Let $M$ be the midpoint of arc $AB$ that is in the semiplane determined by $AB$ that does not contain $\mathcal{S}_1$. Determine the locus of the point symmetric to $M$ with respect to $K.$

2009 Peru IMO TST P3
Let $ABCDEF$ be a convex hexagon that has no pair of parallel sides. It is known that, for every point $P$ inside the hexagon, the sum: $\text{Area}[ABP]+\text{Area}[CDP]+\text{Area}[EFP]$ has a constant value. Prove that the triangles  $ACE$ and $BDF$ have the same barycentre.

2009 Peru IMO TST P5

Let $\mathcal{C}$ be the circumference inscribed in the triangle $ABC,$ which is tangent to sides $BC, AC, AB$ at the points $A' , B' , C' ,$ respectively. The distinct points $K$ and $L$ are taken on $\mathcal{C}$ such that $\angle AKB'+\angle BKA' =\angle ALB'+\angle BLA'=180^{\circ}.$ Prove that the points $A', B', C'$ are equidistant from the line $KL.$

2010 Peru IMO TST P1
Let $ABC$ be an acute-angled triangle and $F$ a point in its interior such that $\angle AFB = \angle BFC = \angle CFA = 120^{\circ}.$ Prove that the Euler lines of the triangles $AFB, BFC$ and $CFA$ are concurrent.

2011 Peru IMO TST P4

Let $ABC$ be an acute triangle, and $AA_1$, $BB_1$, and $CC_1$ its altitudes. Let $A_2$ be a point on segment $AA_1$ such that $\angle{BA_2C} = 90^{\circ}$. The points $B_2$ and $C_2$ are defined similarly. Let $A_3$ be the intersection point of segments $B_2C$ and $BC_2$. The points $B_3$ and $C_3$ are defined similarly. Prove that the segments $A_2A_3$, $B_2B_3$, and $C_2C_3$ are concurrent.

2012 Peru IMO TST P2

Let $a, b, c$ be the lengths of the sides of a triangle, and $h_a, h_b, h_c$ the lengths of the heights corresponding to the sides $a, b, c,$ respectively. If $t \geq \frac{1} {2}$ is a real number, show that there is a triangle with sidelengths $$t\cdot a + h_a, \  t\cdot b + h_b , \  t\cdot c + h_c.$$

2012 Peru IMO TST P4

Let $ABCD$ be a parallelogram such that $\angle{ABC} > 90^{\circ}$, and $\mathcal{L}$ the line perpendicular to $BC$ that passes through $B$. Suppose that the segment $CD$ does not intersect $\mathcal{L}$. Of all the circumferences that pass through $C$ and $D$, there is one that is tangent to $\mathcal{L}$ at $P$, and there is another one that is tangent to $\mathcal{L}$ at $Q$ (where $P \neq Q$). If $M$ is the midpoint of $AB$, prove that $\angle{PMD} = \angle{QMD}$.

2013 Peru IMO TST P3
A point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$. Let $\omega$ be the inscribed circumference of triangle $CPD$ and $I$ the centre of $\omega$. It is known that $\omega$ is tangent to the inscribed circumferences of triangles $APD$ and $BPC$ at points $K$ and $L$ respectively. Let $E$ be the point where the lines $AC$ and $BD$ intersect, and $F$ the point where the lines $AK$ and $BL$ intersect. Prove that the points $E, I, F$ are collinear.

2013 Peru IMO TST P4

Let $A$ be a point outside of a circumference $\omega$. Through $A$, two lines are drawn that intersect $\omega$, the first one cuts $\omega$ at $B$ and $C$, while the other one cuts $\omega$ at $D$ and $E$ ($D$ is between $A$ and $E$). The line that passes through $D$ and is parallel to $BC$ intersects $\omega$ at point $F \neq D$, and the line $AF$ intersects $\omega$ at $T \neq F$. Let $M$ be the intersection point of lines $BC$ and $ET$, $N$ the point symmetrical to $A$ with respect to $M$, and $K$ be the midpoint of $BC$. Prove that the quadrilateral $DEKN$ is cyclic.

2014 Peru IMO TST  P3
Let $ABC$ be an acuteangled triangle with $AB> BC$ inscribed in a circle. The perpendicular bisector of the side $AC$ cuts arc $AC,$ containing $B,$ in $Q.$ Let $M$ be a point on the segment $AB$ such that $AM = MB + BC.$ Prove that the circumcircle of the triangle $BMC$ cuts $BQ$ in its midpoint.

2014 Peru IMO TST P6
Let $ABC$ be a triangle where $AB > BC$, and $D$ and $E$ be points on sides $AB$ and $AC$ respectively, such that $DE$ and $AC$ are parallel. Consider the circumscribed circumference of triangle $ABC$. A circumference that passes through points $D$ and $E$ is tangent to the arc $AC$ that does not contain $B$ at point $P$. Let $Q$ be the reflection of point $P$ with respect to the perpendicular bisector of $AC$. The segments $BQ$ and $DE$ intersect at $X$. Prove that $AX = XC$.

2014 Peru IMO TST P10
Let $ABCDEF$ be a convex hexagon that does not have two parallel sides, such that $\angle AF B = \angle F DE,  \angle DF E = \angle BDC$ and $\angle BFC = \angle ADF.$ Prove that the lines $AB, FC$ and $DE$ are concurrent if and only if the lines $AF, BE$ and $CD$ are concurrent.

2014 Peru IMO TST P11

Let $ABC$ be a triangle, and $P$ be a variable point inside $ABC$ such that $AP$ and $CP$ intersect sides $BC$ and $AB$ at $D$ and $E$ respectively, and the area of the triangle $APC$ is equal to the area of quadrilateral $BDPE$. Prove that the circumscribed circumference of triangle $BDE$ passes through a fixed point different from $B$.

2015 Peru IMO TST P3
Let $M$ be the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC,$ $I$ the incenter of the triangle $ABC$ and $L$ a point on the side $BC$ such that $AL$ is bisector. The line $MI$ cuts the circumcircle again at $K.$ The circumcircle of the triangle $AKL$ cuts the line $BC$ again at $P.$ Prove that $\angle AIP = 90^{\circ}.$

2015 Peru IMO TST P8
Let $I$ be the incenter of triangle $ABC.$ The circle through $I$ and centered at $A$ intersects the circumcircle of triangle $ABC$ at points $M$ and $N.$ Prove that the line $MN$ is tangent to the incircle of the triangle $ABC.$

2016 Peru IMO TST P3
Let $ABCD$ a convex quadrilateral such that $AD$ and $BC$ are not parallel. Let $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The segment $BN$ intersects $AC$ and $BD$ in $K$ and $L$ respectively, Show that at least one point of the intersections of the circumcircles of $AKM$ and $BNL$ is in the line $AB$. 

2017 Peru IMO TST P2
The inscribed circle of the triangle $ABC$ is tangent to the sides $BC, AC$ and $AB$ at points $D, E$ and $F$, respectively. Let $M$ be the midpoint of $EF$. The circle circumscribed around the triangle $DMF$ intersects line $AB$ at $L$, the circle circumscribed around the triangle $DME$ intersects the line $AC$ at $K$. Prove that the circle circumscribed around the triangle  $AKL$ is tangent to the line $BC$.

2017 Peru IMO TST P9
Let $ABCD$ be a cyclie quadrilateral, $\omega$ be it's circumcircle and $M$ be the midpoint of the arc  $AB$ of $\omega$ which does not contain the vertices $C$ and $D$. The line that passes through $M$ and the point of intersection of segments $AC$ and $BD$, intersects again $\omega$ in $N$. Let $P$ and $Q$ be points in the $CD$ segment such that $\angle AQD = \angle  DAP$ and $\angle  BPC = \angle CBQ$. Prove that the circumcircle of $NPQ$ and  $\omega$ are tangent to each other.

2017 Peru IMO TST P11
Let $ABC$ be an acute and scalene of circumcircle $\Gamma$ and orthocenter $H$. Let  $A_1,B_1,C_1$ be the second points of intersection of the lines $AH, BH, CH$ with $\Gamma$, respectively. The lines that pass through $A_1,B_1,C_1$ and are parallel to $BC,CA, AB$ intersect again to $\Gamma$ at $A_2,B_2,C_2$, respectively. Let $M$ be the point of intersection of $AC_2$ and $BC_1, N$ the intersection point of $BA_2$ and $CA_1$, and $P$ the point of intersection of $CB_2$ and $AB_1$. Prove that $\angle MNB = \angle  AMP$ .

2018 Peru IMO TST P7
Let $ABC$ be, with $AC>AB$, an acute-angled triangle with circumcircle $\Gamma$ and $M$ the midpoint of side $BC$. Let $N$ be a point in the interior of $\bigtriangleup ABC$. Let $D$ and $E$ be the feet of the perpendiculars from $N$ to $AB$ and $AC$, respectively. Suppose that $DE\perp AM$. The circumcircle of $\bigtriangleup ADE$ meets $\Gamma$ at $L$ ($L\neq A$), lines $AL$ and $DE$ intersects at $K$ and line $AN$ meets $\Gamma$ at $F$ ($F\neq A$). Prove that if $N$ is the midpoint of the segment $AF$ then $KA=KF$.

2019 Peru IMO TST P3
Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$/

2021 Peru IMO TST P3

In an acute triangle $ABC$, its inscribed circle touches the sides $AB,BC$ at the points $C_1,A_1$ respectively. Let $M$ be the midpoint of the side $AC$, $N$ be the midpoint of the arc $ABC$ on the circumcircle of triangle $ABC$, and $P$ be the projection of $M$ on the segment $A_1C_1$.

Prove that the points $P,N$ and the incenter $I$ of the triangle $ABC$ lie on the same line.

Cono Sur TST  2002 - 21

1997 Peru Cono Sur TST P2 

A circle $K$ is inscribed in a square $C$ and for any point $M$ lying in $C$, which doesn't belong to $K$, is traced a tangent to $C$ that meets $K$ in the points $A$ and $B$. Let $P$ the center of the square which side is $AB$ that touch to the circle $C$ only in $M$. Find the locus of $P$ when $M$ varies along the circle $C$.

1999 Peru Cono Sur TST P2 

Let be $ABC$ a triangle with $AB < AC$. Line through $B$ parallel to $AC$ meets in $D$ to external bisector of the angle $BAC$ and the line through $C$ parallel to $AB$ meets at $E$ that bisector. Point $F$ on $AC$ satisfies $FC = AB$. Prove that $FD = FE$

2002 Peru Cono Sur TST P3 
Let $AD, BE, CF$ the angle bisectors of the triangle $ABC$, prove that if one of the angle(s) $\angle ADF$, $\angle ADE$, $\angle BED$, $\angle BEF$, $\angle CFE$, $\angle CFD$ is $30º$, therefore another angle of this angles also is $30º$.

2003 Peru Cono Sur TST P3 
Let $M$, $N$ be points in the side $BC$ of the triangle $ABC$ such that $BM = CN$ (The point $M$ is in the segment $BN$). The points $P$ and $Q$ are in the segments $AN$ and $AM$ respectively, where $\angle PMC = \angle MAB$ and $\angle QNB = \angle NAC$. Prove that $\angle QBC = \angle PCB$.

2004 Peru Cono Sur TST P4 
In the triangle $ABC$ we can put four circles $K_1, K_2, K_3, K_4$ (with the same radius), such that $K_1, K_2, K_3$ are tangent to two sides of $ABC$ and to the circle $K_4$ . Show that, the circumcenter of $K_4$ lies in the line that connects the incenter and the circumcenter of the triangle $ABC$.

2005 Peru Cono Sur TST P3
Let $D$ be the midpoint of the side $BC$ of a given triangle $ABC$. Let $M$ be a point on the side $BC$ such that $\angle BAM = \angle DAC, L$ the second intersection point of the circumcircle of the triangle $CAM$ with the side $AB$ and $K$ the second intersection point of the circumcircle of the triangle  $BAM$ with the side $AC$ . Prove that $KL$ and $BC$ are parallel.

2006 Peru Cono Sur TST P2 

Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).

2007 Peru Cono Sur TST P1

Given a square $ABCD$, let $M,K, L$ and $N$ points on the sides $AB, BC,CD$ and $DA$, respectively, such that $\angle MKA =\angle KAL = \angle  ALN = 45^o$. Prove that $MK^2 + AL^2 = AK^2 + LN^2$

2008 Peru Cono Sur TST P3

Given a triangle $ABC$, let $P$ and $Q$ be points on the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$. Let $M$ be the midpoint of $BC$ and $X$ the foot of the altirude  from $Q$ on $PM$. Prove that $\angle AXQ = \angle  QXC$

(John Cuya)

2008 Peru Cono Sur TST P8

You have the convex hexagon $ABCDEF$ such that $\angle FAB = \angle CDE = 90^o$ and the quadrilateral $BCEF$ is tangential. Prove that $AD\le BC + FE$.

(John Cuya)

2009 Peru Cono Sur TST P5 

Let $ABC$ be triangle(acute-angled), let $A_1A, B_1B, C_1C$ the altitudes of this triangle we choose two points $D$ and $E$ in the segments $BC$ and $AD$, such that $\frac{AE}{ED} = \frac{CD}{BD}$ and let $F$ be the foot of perpendicular from $D$ to the segment $BE$ and the quadrilateral $AFDC$ is cyclic. Prove that the point $E$ is on the line(s) $A_1A$ or $B_1B$ or $C_1C$

(Jorge Tipe)

2010 Peru Cono Sur TST P5
Let $ABC$ be an acute triangle . On the sides $AC$ and $AB$, liet the points $M$ and $N$ , respectively. Let $P$ be the point of intersection of segments $BM$ and $CN$, and $Q$ a point inside the quadrilateral $ANPM$ such that $\angle BQC = 90^o$ and $\angle BQP = \angle BMQ$. If the $ANPM$ quadrilateral is cyclic, prove that $\angle  QNC = \angle  PQC$.

2011 Peru Cono Sur TST P4

Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$ of a triangle $ABC$ and $G$ it's centroid. If the circles circumscribed to the triangles $AMN$ and $BGC$ are externally tangent, is it possible that the triangle $ABC$ is scalene?

(Jorge Tipe)

2011 Peru Cono Sur TST P8
Let $ABCD$ be a quadrilateral inscribed in a circle of center $O$ such that $BC$ and $AD$ are not parallel. Let $P$ be the point of intersection of the diagonals of the quadrilateral.The rays $AB$ and $DC$ intersect at  $E$. The circle of center $I$ that is inscribed in the triangle $EBC$ is tangent to side $BC$ at $T_1$. The ex-circle of the triangle $EAD$, relative to $AD$, is tangent to $AD$ at $T_2$ and has center $J$. The lines $IT_1$ and $JT_2$ intersect in $Q$. Prove that $O, P,Q$ are collinear.

2012 Peru Cono Sur TST P1 
Let $ABC$ a isosceles and $\angle ABC = 90º$, let $M$ be the midpoint of $AC$. Inside of triangle we can construct a circle where this circle is tangent to $AB$ and $BC$ in $P$ and $Q$, respectively. The line $MQ$ intersects again the circle in $T$, if $H$ is orthocenter of $AMT$ prove that $MH = BQ$

(Jorge Tipe)

2012 Peru Cono Sur TST P6
In an angle triangle  $ABC$, let  the $AP$ and $BQ$ be the altitudes, and $M$ be the midpoint of the side $AB$. If the circle circumscribed to the triangle $BMP$ is tangent to side $AC$ , prove that the circle circumscribed to the triangle $AMQ$ is tangent to the extension of the side $BC$.

2013 Peru Cono Sur TST P2
Given a triangle $ABC$, let $M, N$ and $P$ be points on the sides $AB, BC$ and $CA$, respectively, such that $MBNP$ is a parallelogram. Line $MN$ cuts the circle circumscribed to the triangle $ABC$ at the points $R$ and $S$. Prove that the circumscribed circle of the triangle $RPS$ is tangent to $AC$.

2013 Peru Cono Sur TST P5
Let $I$ be the incenter of $ABC$ and $A_1, B_1, C_1$ the point(s) in the segments $AI, BI, CI$ respectively.  The perpendicular bisectors of the segment(s) $AA_1, BB_1, CC_1$, where this segments determine the triangle $T$, if $I$ is the orthocenter of $A_1B_1C_1$ and let $O$ be the circumcenter of $T$. Prove that the $O$ is also the circumcenter of $ABC$.

2014 Peru Cono Sur TST P3 
Let $ABCD$ be a cyclic quadrilateral, suppose that the line(s) $BC$ and $AD$ intersects in $P$, and $Q$ is a point such that $P$ is midpoint of $BQ$. We can construct the parallelogram(s) $CAQR$ and $DBCS$, prove that the quadrilateral $CQRS$ is cyclic.

2014 Peru Cono Sur TST P8
Let  $\omega$ be a circle and $A$ point exterior to $\omega$  . The tangent lines to $\omega$   that pass through $A$ touch $\omega$  at points $B$ and $C$. Let $M$ be the midpoint of $AB$. Line $MC$ intersects $\omega$ again at $D$ and line $AD$ intersects $\omega$  again at $E$. Let $AB = a$ and $BC = b$, find $CE$ in terms of $a$ and $b$.

2015 Peru Cono Sur TSΤ P3 
Let $ABCD$ be a parallelogram, let $X$ and $Y$ in the segments $AB$ and $CD$, respectively. The segments $AY$ and $DX$ intersects in $P$ and the segments $BY$ and $DX$ intersects in $Q$, show that the line $PQ$ passes in a fixed point(independent of the positions of the points $X$ and $Y$).

2015 Peru Cono Sur TST P8
Let $ABCD$ be a cyclic quadrilateral such that the lines $AB$ and $CD$ intersects in $K$, let $M$ and $N$ be the midpoints of $AC$ and $CK$ respectively. Find the possible value(s) of $\angle ADC$ if the quadrilateral $MBND$ is cyclic.

2016 Peru Cono Sur TST P2
Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

2016 Peru Cono Sur TST P6
Two circles $\omega_1$ and $\omega_2$, which have centers $O_1$ and $O_2$, respectively, intersect at $A$ and $B$. A line  $\ell$ that passes through $B$ cuts to $\omega_1$ again at $C$ and cuts to $\omega_2$ again in $D$, so that points $C, B, D$ appear in that order. The tangents of $\omega_1$ and $\omega_2$ in $C$ and $D$, respectively, intersect in $E$. Line $AE$ intersects again to the circumscribed circumference of the triangle $AO_1O_2$ in $F$. Try that the length of the $EF$ segment is constant, that is, it does not depend on the choice of $\ell$.

2017 Peru Cono Sur TST P1
Every diagonal of a convex pentagon divides it on triangle and quadrilateral. Let call the diagonal good if the quadriterial is tangent. Find the maximum quantity of good diagonals in the convex pentagon

2017 Peru Cono Sur TST P5
Let $ABC$ be a triangle with circumcenter $O$. The altitude $BQ$ is drawn, with $Q$ in the $AC$ side. The parallel to the line $OC$ that passes through $Q$ intersects the line $BO$ at $X$. Prove that $X$ and the midpoints of sides $AB$ and $AC$ are collinear.

2017 Peru Cono Sur TST P9
Let $BXC$ be a triangle and $A_1, A_2, A_3$ points of the same plane such that $X$ is the orthocenter of $A_1BC, X$ is the incenter of $A_2BC$ and X is the centroid of $A_3BC$. If $A_1A_3$ is parallel to $BC$, prove that $A_2$ is the midpoint of $A_1A_3$.

2018 Peru Cono Sur TST P3
Let $I$ be the incenter of a triangle $ABC$ with $AB \ne AC$ and let $M$ be the midpoint of the arc $BAC$ of the circle of said triangle. The line perpendicular to $AI$ that passes through $I$ intersects line $BC$ at point $D$. Line $MI$ intersects the circle of the $BIC$ triangle at point $N$. Prove that the line $DN$ is tangent to the circle of the $BIC$ triangle.

2018 Peru Cono Sur TST P7

Let $ABCD$ be a fixed square and $K$ a variable point in segment $AD$. The square $KLMN$ is constructed so that $B$ is in the segment $LM$ and $C$ is in the segment $MN$. Let T be the point of intersection of lines $LA$ and $ND$. Find the locus of $T$ as $K$ varies in segment $AD$

2019 Peru Cono Sur TST P2
Let $AB$ be a diameter of a circle $\Gamma$ with center $O$. Let $CD$ be a chord where $CD$ is perpendicular to $AB$, and $E$ is the midpoint of $CO$. The line $AE$ cuts $\Gamma$ in the point $F$, the segment $BC$ cuts $AF$ and $DF$ in $M$ and $N$, respectively. The circumcircle of $DMN$ intersects $\Gamma$ in the point $K$. Prove that $KM=MB$

2020 Peru Cono Sur TST P3 (also in Ibero)

Let $ABC$ be an acute triangle with $| AB | > | AC |$. Let $D$ be the foot of the altitude from $A$ to $BC$, let $K$ be the intersection of $AD$ with the internal bisector of angle $B$, Let $M$ be the foot of the perpendicular from $B$ to $CK$ (it could be in the extension of segment $CK$) and$N$ the intersection of $BM$ and $AK$ (it could be in the extension of the segments). Let $T$ be the intersection of$AC$ with the line that passes through $N$ and parallel to $DM$. Prove that $BM$ is the internal bisector of the angle $\angle TBC$

2021 Peru Cono Sur TST P3

Let $ABC$ be a triangle and $D$ is a point in $BC$. The line $DA$ cuts the circumcircle of $ABC$ in the point $E$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. Let $F=MN\cap AD$ and $G\neq F$ is the point of intersection of the circumcircles of $\triangle DNF$ and $\triangle ECF$. Prove that $B,F$ and $G$ are collinear.

Ibero TST 2007 - 2021

1993 Peru Ibero TST P2

There are $4$ balls with radios $1$, set such a way that each one of them is tangent to the others three. Determine the radius of the smaller sphere that contains to the others four balls.

1997 Peru Ibero TST P4

Let $ABC$ be a given acute-angled triangle. Show a way to built with ruler and a compass the equilateral triangle $DEF$, with $D$ in $BC$, $E$ in $AC$ and $F$ in $AB$ such that the three lines perpendicular to $BC$ in $D$, to $AC$ in $E$ and to $AB$ in $F$ respectively, are concurrent.

1999 Peru Ibero TST P1

Let $A$, $B$ and $C$ be three points on a given circle $W$. $O$ is the incenter of triangle $ABC$ and $M$ , $N$ are the midpoints of arcs $BC$ , $CA$, respectively. The point $P$ of $W$ is such that $PC \parallel MN$ and the ray $PO$ intersects $W$ at $T$. Show that $NC \cdot MT = MC \cdot NT$

2004 Peru Ibero TST P2

Let $ABCD$ be a convex quadrilateral where the area of triangle $ABC$ is greater than or equal to the area of triangle $ACD$. Construct a point $P$ on the diagonal $AC$ such that the sum of the areas of the triangles $APB$ and $DPC$ is equal to the area of the triangle $BPC$.

2005 Peru Ibero TST (UK FST2 2006 p2)

Let $ABCD$ be a cyclic quadrilateral, and $P$ a point in its interior such that $\angle BPC= \angle PAB+\angle PDC$. Let $E$, $F$ and $G$ be the orthogonal projections of $P$ on the sides $AB$, $DA$ and $CD$, respectively. Show that triangles $BPC$ and $EFG$ are similar.

Alternative formulation.

A point $P$ is in the interior of the cyclic quadrilateral $ABCD$ and has the property $\angle BPC = \angle BAP+\angle PDC$. The feet of the perpendiculars from $P$ to $AB$, $AD$ and $DC$ are respectively denoted $E$, $F$ and $G$. Show that $\triangle FEG$ and $\triangle PBC$ are similar.

2007 Peru Ibero TST P3
Let $ABC$ be a acute-angled triangle, and let $A_1A_2A_3A_4$ be a square where this square has one vertex in $AB$ , one vertex in $AC$ and two vertices$(A_1$ or $A_2)$ in $BC$ and let $x_a = \angle A_1AA_2$. Analogously we define $x_b$ and $x_c$, show that $x_a + x_b + x_c = 90º$

2009 Peru Ibero TST P1
Let $M, N,P$ the midpoints of the sides $AB, BC, CA$ of a triangle $ABC$. Be $X$ a point inside the $MNP$ triangle. The straight lines $L_1,L_2,L_3$ passing through point $X$ are such that $L_1$ intersects segment $AB$ at point $C_1$ and segment $AC$ at point $B_2, L_2$ intersects segment $BC$ at point $A_1$ and segment $BA$ at point $C_2, L_3$ intersects the $CA$ segment at point $B_1$ and the $CB$ segment at the point $A_2$. Indicate how to constuct straight lines $L_1, L_2, L_3$ so that the sum of the triangle areas $A_1A_2X, B_1B_2X$ and $C_1C_2X$ be minimum.

2009 Peru Ibero TST P4

Let $ABC$ be a triangle such that $AB <BC$. The height $BH$ is drawn with $H$ in $AC$. Let $I$ be the incenter of the triangle $ABC$ and $M$ the midpoint of $AC$. If the line $MI$ intersects to $BH$ at point $N$, prove that $BN <IM$.

2010 Peru Ibero TST P1
Let $C_1$ and $C_2$ be two concentric circumferences of center $O$, such that the radius of $C_1$ is less than the radius of $C_2$. Let $P$ be a point other than $O$ that is inside of $C_1$, and $L$ a line that passes through $P$ and cuts to $C_1$ at $A$ and $B$. The ray $OB$ cuts to $C_2$ in $C$. Determine the geometric place that determines the circumcenter of the triangle $ABC$ as $L$ varies.

2010 Peru Ibero TST P5
The $ABCD$ trapezoid of $AB$ and $CD$ bases is inscribed in a circle $\Gamma$ . Let $X$ be a variable point of the arc $AB$ that does not contain $C$ or $D$. Let $Y$ be the point of intersection on of $AB$ and $DX$, and let $Z$ be the point of the $CX$ segment such that $\frac{XZ}{XC}=\frac{AY}{AB}$. Show that the measure of the angle $\angle AZX$ does not depend on the choice of $X$.

2011 Peru Ibero TST P2

Let $ABC$ be a scalene acute-angled triangle and $H$ is your orthocenter. The lines $BH$ and $CH$ intersects $AC$ and $AB$ in $D$ and $E$ respectively, the circumcircle of triangle $ADE$ intersects the circumcircle of the $ABC$ again in the point $F$. Show that the angle bisectors of $\angle BFC$ and $\angle BHC$ and the segment $BC$ are concurrent.

2012 Peru Ibero TST P1
Let $ABCD$ be a convex quadrilateral such that $AB \cdot CD = AD\cdot BC$.
Prove that $\angle  BAC + \angle  CBD + \ DCA + \angle ADB = 180^o$

2013 Peru Ibero TST P5

Let $C$ be a circle, $A$ and $B$ are points of $C$ (with $A \ne B$) and $\ell$ a line that does not cut $C$. Let $P$ be a variable point of $C$ such that the rays $AP$ and $BP$ cut $\ell$  at points $D$ and $E$, respectively. Prove that the circumference of diameter $DE$ is always tangent to two circumferences as $P$ varies in $C$.

2014 Peru Ibero TST P1
Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.

2014 Peru Ibero TST  P5
The incircle $\odot (I)$ of $\triangle ABC$ touch $AC$ and $AB$ at $E$ and $F$ respectively. Let $H$ be the foot of the altitude from $A$, if  $R \equiv IC \cap AH, \ \ Q \equiv BI \cap AH$ prove that the midpoint of $AH$ lies on the radical axis between $\odot (REC)$ and $\odot (QFB)$

2015 Peru Ibero TST P1
In the angle $ABC$, such that $AB \ne AC$, is $D$ the foot of the perpendicular drawn from $A$ to straight line $BC$. Let $E$ and $F$ be the midpoints of the segments $AD$ and $BC$, respectively. If $G$ is the foot of the perpendicular drawn from $B$ to the straight $AF,$ prove that $EF$ is tangent to the circle circumscribed of the trangle $GFC$ .

2015 Peru Ibero TST P6
Let $ABCD$ be a quadrilateral inscribed in a circle of center$O$. On the sides $AB$ and $CD$ are considered points $F$ and $E$, respectively, such that $EO = FO$. The lines $AD$ and $BC$ cut to the line $EF$ at points $M$ and $N$, respectively.  Finally, the point $P$ is the symmetric of $M$ with respect to the midpoint of the segment $AE$. Prove that the triangles $FBN$ and $CEP$ are similar.

2015 Peru Ibero TST P8
Let $ABC$ be a triangle($AB$ > $AC$) with circumcircle $w$, let $r$ and $s$ be the tangent line to $w$ and passes to $B$ and $C$ respectively and the line $r$ intersects the line $s$ in $P$. The perpendicular to $AP$, in $A$, intersects $BC$ in $R$ and let $S$ be a point in $PR$ such that $PS = PC$.
a) Prove that the lines $CS$, $AR$ and the circle $w$ are concurrent.
b) Let $M$ be the midpoint of $BC$ and $Q$ be the intersection of $CS$ and $AR$. If the circle $w$ and the circumcircle of $AMP$ intersects in the points $A$ and $J$, prove that $P, J$ and $Q$ are collinear.

2016 Peru Ibero TST P1
Let $ABC$ be an isosceles triangle, right in $C$. The points $M$ and N lie on segments $AC$ and $BC$, respectively, such that $MN = BC$. Be $\omega$ be a circle that is tangent to segment $AB$ and that passes through points $M$ and $N$. Find the locus of the center of $\omega$ while points $M$ and$N$ vary,

2016 Peru Ibero TST P6
In a convex quadrilateral $ABCD$,  you have $\angle  ABC = \angle  BCD = 120^o$, $M$ is the midpoint of segment $BC$, and $O$ is the point of intersection of diagonals $AC$ and $BD$. Let $K$ be the intersection point of $MO$ and $AD$. If $\angle BKC = 60$, prove that $\angle BKA = \angle  CKD = 60^o$.

2016 Peru Ibero TST P7
Let $ABC$ be a triangle with $\angle A> 90^o$, with altitude  $CH$ and medians $AM$ and $BN$. If the circle of diameter $AM$ is tangent to the line $CH$, prove that the circle of diameter $BN$ is also tangent to the line $CH$.

2017 Peru Ibero TST P1
Let $C_1$ and $C_2$ be tangent circles internally at point $A$, with $C_2$ inside of $C_1$. Let $BC$ be a chord of  $C_1$ that is tangent to $C_2$. Prove that the ratio between the length $BC$ and the perimeter of the triangle $ABC$ is constant, that is, it does not depend of the selection of the chord $BC$ that is chosen to construct the trangle.

2017 Peru Ibero TST P5
Let $ABCD$ be a trapezoid of bases $AD$ and $BC$ , with $AD> BC$, whose diagonals are cut at point $E$. Let $P$ and $Q$ be the feet of the perpendicular drawn from $E$ on the sides $AD$ and $BC$, respectively, with $P$ and $Q$ in segments $AD$ and $BC,$ respectively. Let $I$ be the center of the triangle  $AED$ and let $K$ be the point of intersection of the lines $AI$ and $CD$. If $AP + AE = BQ + BE$, show that $AI = IK$.

2018 Peru Ibero pre TST P1
Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle  APB> 90^o$ and $\angle  PAD + \angle  PBD = \angle  PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle  TRB = \angle  DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle  PAR$. Prove that $RS = RT$.

2018 Peru Ibero TST P9
Let $\Gamma$ be the circumcircle of a triangle  $ABC$ with $AB <BC$, and let $M$ be the midpoint from the side $AC$ . The median of side $AC$ cuts $\Gamma$ at points $X$ and $Y$ ($X$ in the arc $ABC$). The circumcircle of the triangle $BMY$ cuts the line $AB$ at $P$ ($P \ne B$) and the line $BC$ in $Q$ ($Q \ne B$). The circumcircles of the triangles $PBC$ and $QBA$ are cut in $R$ ($R \ne B$). Prove that points $X, B$ and $R$ are collinear.

2019 Peru Ibero TST P5

Let $\Gamma$ be the circumcircle of an acute triangle $ABC$ and let $G$ be its centroid. Let $M$ and $N$ are the midpoints of sides $AC$ and $AB$ respectively. Let $D$ be the foot of the perpendicular drawn from $A$ to side $BC$. A circle $\omega$ that passes through points $M$ and $N$ is tangent to $\Gamma$ at a point $X$ ($X\ne A$). Prove that points $X, D$ and $G$ are collinear.

2020 Peru Ibero TST P3 (also in Cono Sur)

Let $ABC$ be an acute triangle with $| AB | > | AC |$. Let $D$ be the foot of the altitude from $A$ to $BC$, let $K$ be the intersection of $AD$ with the internal bisector of angle $B$, Let $M$ be the foot of the perpendicular from $B$ to $CK$ (it could be in the extension of segment $CK$) and$N$ the intersection of $BM$ and $AK$ (it could be in the extension of the segments). Let $T$ be the intersection of$AC$ with the line that passes through $N$ and parallel to $DM$. Prove that $BM$ is the internal bisector of the angle $\angle TBC$.

EGMO TST 2018 - 2021

2018 Peru EGMO TST P3
Let $ABC$ be an acute-angled triangle with circumradius $R$ less than the sides of $ABC$, let $H$ and $O$ be the orthocenter and circuncenter of $ABC$, respectively. The angle bisectors of $\angle ABH$ and $\angle ACH$ intersects in the point $A_1$, analogously define  $B_1$ and $C_1$. If $E$ is the midpoint of $HO$, prove that $EA_1+EB_1+EC_1=p-\frac{3R}{2}$ where $p$ is the semiperimeter of $ABC$

2018 Peru EGMO TST P5

Let $I$ be the incenter of $ABC$ and $I_A$ the excenter of the side $BC$, let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BC$(with the point $A$). If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $I_AMIT$ is cyclic.

2019 Peru EGMO TST P2
Let  $\Gamma$ be the circle of an acute triangle  $ABC$ and let $H$ be its orthocenter. The circle $\omega$ with diameter $AH$ cuts  $\Gamma$   at point $D$ ($D\ne A$). Let $M$ be the midpoint of the smaller arc $BC$ of $\Gamma$ . Let $N$ be the midpoint of the largest  arc $BC$ of the circumcircle of the triangle $BHC$. Prove that there is a circle that passes through the points $D, H, M$ and $N$.

2019 Peru EGMO TST P6

Let $ABC$ be a triangle with $AB = AC$ and let $M$ be the midpoint of $BC$. Let $P$ be a point in the plane such that $PA$ is parallel to $BC$ and $PB <PC$. Let $X$ and $Y$ be points on the plane such that $B$ is in the $PX$ segment and $C$ is in the $PY$ segment. If $\angle PXM = \angle PYM$, prove that there is a circle that passes through points $A, P, X$ and $Y$.

2020 Peru EGMO TST P3

Let $ABC$ be a triangle with $AB<AC$ and $I$ be your incenter. Let $M$ and $N$ be the midpoints of the sides $BC$ and $AC$, respectively. If the lines $AI$ and $IN$ are perpendicular, prove that the line $AI$ is tangent to the circumcircle of $\triangle IMC$.

2020 Peru EGMO TST P5

Let $AD$ be the diameter of a circle $\omega$ and $BC$ is a chord of $\omega$ which is perpendicular to $AD$. Let $M,N,P$ be points on the segments $AB,AC,BC$ respectively, such that $MP\parallel AC$ and $PN\parallel AB$. The line $MN$ cuts the line $PD$ in the point $Q$ and the angle bisector of $\angle MPN$ in the point $R$. Prove that the points $B,R,Q,C$ are concyclic.

2021 Peru EGMO TST P3

The tangent lines to the circumcircle of triangle ABC passing through vertices $B$ and $C$ intersect at point $F$. Points $M$, $L$ and $N$ are the feet of the perpendiculars from vertex $A$ to the lines $FB$, $FC$ and $BC$ respectively. Show that $AM+AL \geq 2AN$

sources:
onemperu.files.wordpress.com/
selectivos-peru.blogspot.com/

http://www.geocities.com/CollegePark/7174/