geometry problems from Peruvian Team Selection Tests (TST)

for IMO, Cono Sur and IberoAmerican

with aops links in the names

(

for IMO, Cono Sur and IberoAmerican

with aops links in the names

(

**only those not in IMO Shortlist**)

[3p per day]

IMO TST 2006 - 2018

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 $w_1.$

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.

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.

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.

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.

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.

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.

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.

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.

2013 Peru IMO TST P4

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 $$ t\cdot a + h_a, \ t\cdot b + h_b , \ t\cdot c + h_c.$$

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.

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.

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 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

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 $ABC.$

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.

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.

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.

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 $\angle AIP = 90^{\circ}.$

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 $\angle AIP = 90^{\circ}.$

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 $AB$.

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 $BC$.

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 $BC$.

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 $\angle MNB = \angle AMP$ .

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 $\angle QBC = \angle PCB$.

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.

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 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 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.

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 $KA=KF$.

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 $KA=KF$.

Cono Sur TST 2002-2015

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 (also)

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 $ABC$.

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.

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 $ABC$).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.

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 $MK^2 + AL^2 = AK^2 + LN^2$

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)

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)

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 $\angle QNC = \angle PQC$.

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)

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 $MH = BQ$

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 $BC$.

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 P6In 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 $AC$.

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 $AC$.

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.

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.

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 $Y$).

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 $Y$).

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.

Ibero TST 2007 - 2018

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 $x_a + x_b + x_c = 90º$

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 $x_a + x_b + x_c = 90º$

2008 Peru Ibero TST P2 (ISL 2008 G3)

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be points inside $ABCD$ such that $PQDA$ and $QPBC$ are cyclic quadrilaterls. Suppose there is a point $E$ in the $PQ$ segment such that $\angle PAE = \angle QDE$ and $\angle PBE = \angle QCE$. Prove that the quadrilateral $ABCD$ is cyclic.

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be points inside $ABCD$ such that $PQDA$ and $QPBC$ are cyclic quadrilaterls. Suppose there is a point $E$ in the $PQ$ segment such that $\angle PAE = \angle QDE$ and $\angle PBE = \angle QCE$. Prove that the quadrilateral $ABCD$ is cyclic.

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 $M, N,P$ 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.

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 $X$.

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 $X$.

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 $\angle BAC + \angle CBD + \ DCA + \angle ADB = 180^o$

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

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.

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.

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 $\odot (QFB)$

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.

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,

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 $\angle BKA = \angle CKD = 60^o$.

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 $CH$.

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 $CH$.

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 $AI = IK$.

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.

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.

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