Peru IMO TST 2006-18 24p

geometry problems from Peruvian IMO Team Selection Tests (TST)
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.

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 $\mathcal{S}_1$ 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 $\mathcal{C}$ 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 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.

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.

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 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 $KA=KF$. 


Cono Sur TST 
under construction

2002 Peru Cono Sur TST 
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 
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 
In the triangle $ABC$ we can put four circles $K_1, K_2, K_3, K_4$ (with the same diameter), such that $K_1, K_2, K_3$ are tangents with two sides of $ABC$ and with the circle $K_4$ .
Show that, the circumcenter of $K_4$ is in the line that connect the incenter and the circumcenter of the triangle $ABC$.

2005 Peru Cono Sur TST

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

2008 Peru Cono Sur TST
2008 Peru Cono Sur TST

2009 Peru Cono Sur TST 
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 $C_1C$

2010 Peru Cono Sur TST 
2010 Peru Cono Sur TST 
2011 Peru Cono Sur TST 
2011 Peru Cono Sur TST

2012 Peru Cono Sur TST 
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 $MH = BQ$

2012 Peru Cono Sur TST
2013 Peru Cono Sur TST

2013 Peru Cono Sur TST
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 
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

2015 Peru Cono Sur TST 
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
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 
2016 Peru Cono Sur TST
2017 Peru Cono Sur TST 
2017 Peru Cono Sur TST
2018 Peru Cono Sur TST 
2018 Peru Cono Sur TST
Ibero TST 
under construction

2007 Peru Ibero TST 
2008 Peru Ibero TST 
2009 Peru Ibero TST 
2010 Peru Ibero TST 
2010 Peru Ibero TST 
2011 Peru Ibero TST 
2011 Peru Ibero TST
2012 Peru Ibero TST 
2012 Peru Ibero TST
2013 Peru Ibero TST 
2013 Peru Ibero TST 
2014 Peru Ibero TST 
2014 Peru Ibero TST 
2015 Peru Ibero TST 
2015 Peru Ibero TST 
2016 Peru Ibero TST 
2016 Peru Ibero TST 
2017 Peru Ibero TST 
2017 Peru Ibero TST 
2018 Peru Ibero TST

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