Puerto Rico TST 2007-21 32p

 geometry problems from Puerto Rico Team Selection Test  with aops links in the names

collected inside aops here

Cono Sur TST 2007-21

2007 Puerto Rico Cono Sur TST p4

In a circle, if two chords $AB$ and $CD$ intersect at a point $M$, then $AM \cdot MB = CM \cdot MD$.

2008 Puerto Rico Cono Sur TST p6

Consider a triangle $ABC$, with $\angle A = 90^{\circ}$, and $AC > AB$. Let $D$ be a point in $AC$ such that $\angle ACB = \angle ABD$. Draw an altitude $DE$ in triangle $BCD$. If $AC = BD + DE$, find $\angle ABC$ and $\angle ACB$.

2009 Puerto Rico Cono Sur TST p3

On an arbitrary triangle $ABC$ let $E$ be a point on the height from $A$. Prove that $(AC)^2 - (CE)^2 = (AB)^2 - (EB)^2$.

2009 Puerto Rico Cono Sur TST p5

Let $ABCD$ be a quadrilateral inscribed in a circle. The diagonal $BD$ bisects $AC$. If $AB = 10$, $AD = 12$ and $DC = 11$, find $BC$.

2010 Puerto Rico Cono Sur TST p1

The circles in the figure have their centers at $C$ and $D$ and intersect at $A$ and $B$. Let $\angle ACB =60$, $\angle ADB =90^o$ and $DA = 1$ . Find the length of $CA$.

2011 Puerto Rico Cono Sur TST p5

Point A, which is within an acute, is reflected with respect to both sides of angle A to obtain the points B and C. the segment BC intersects the sides of angle A at points D and E respectively. Prove that BC/2>DE.

2012 Puerto Rico Cono Sur TST p3

$ABC$ is a triangle that is inscribed in a circle. The angle bisectors of $A, B, C$ meet the circle at $D,

E, F$, respectively. Show that$AD$is perpendicular to$EF$.

2012 Puerto Rico Cono Sur TST p5

A point $P$ is outside of a circle and the distance to the center is $13$. A secant line from $P$ meets the circle at $Q$ and $R$ so that the exterior segment of the secant, $PQ$, is $9$ and $QR$ is $7$. Find the radius of the circle.

2013 Puerto Rico Cono Sur TST p5

Given an equilateral triangle we select an arbitrary point on its interior. We draw theperpendiculars from that point to the three sides of the triangle. Show that the sum of the lengths of these perpendiculars is equal to the height of the triangle.

2014 Puerto Rico Cono Sur TST p1

Let $ABCD$ be a parallelogram with $AB>BC$ and $\angle DAB$ less than $\angle ABC$. The perpendicular bisectors of sides $AB$ and $BC$ intersect at the point $M$ lying on the extension of $AD$. If $\angle MCD=15^{\circ}$, find the measure of $\angle ABC$

2014 Puerto Rico Cono Sur TST p7

Consider $N$ points in the plane such that the area of a triangle formed by any three of the points does not exceed $1$. Prove that there is a triangle of area not more than $4$ that contains all $N$ points.

2015 Puerto Rico Cono Sur TST p1

In the triangle $ABC$, let $P$, $Q$, and $R$ lie on the sides $BC$, $AC$, and $AB$ respectively, such that $AQ = AR$, $BP = BR$ and $CP = CQ$. Let $\angle PQR=75^o$ and $\angle PRQ=35^o$. Calculate the measures of the angles of the triangle $ABC$.

2015 Puerto Rico Cono Sur TST p7

Let $ABCD$ be a rectangle with sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ intersects $BD$ at the point $H$. We denote by $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the length of segment $MN$.

2016 Puerto Rico Cono Sur TST p3

Let $ABCD$ be a cyclic quadrilateral. Let $P$ be the intersection of the lines $BC$ and $AD$. Line $AC$ cuts the circumscribed circle of the triangle $BDP$ in $S$ and $T$, with $S$ between $A$ and $C$. The line $BD$ intersects the circumscribed circle of the triangle $ACP$ in $U$ and $V$, with $U$ between $B$ and $D$. Prove that $PS = PT = PU = PV$.

2016 Puerto Rico Cono Sur TST p5

$ABCD$ is a quadrilateral, $E, F, G, H$ are the midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Find the point $P$ such that area $(PHAE) =$ area $(PEBF) =$ area $(PFCG) =$ area $(PGDH).$

2017 Puerto Rico Cono Sur TST p2

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 $B_1C_1$.

2017 Puerto Rico Cono Sur TST p5

Miguel has a square piece of paper $ABCD$ that he folded along a line $EF$, $E$ on $AB$, and $F$ on $CD$. This fold sent $A$ to point $A'$ on $BC$, distinct from $B$ and $C$. Also, it brought $D$ to point $D'$. $G$ is the intersection of $A'D'$ and $DC$. Prove that the inradius of $GCA'$ is equal to the sum of the inradius of $D'GF$ and $A'BE$.

2018 Puerto Rico Cono Sur TST p3

Let $M$ be the point of intersection of diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let $K$ be the point of intersection of the extension of side $AB$ (beyond$A$) with the bisector of the angle $ACD$. Let $L$ be the intersection of $KC$ and $BD$. If $MA \cdot CD = MB \cdot LD$, prove that the angle $BKC$ is equal to the angle $CDB$.

2019 Puerto Rico Cono Sur TST p2

Let $ABCD$ be a square. Let $M$ and $K$ be points on segments $BC$ and $CD$ respectively, such that $MC = KD$. Let $P$ be the intersection of the segments $MD$ and $BK$. Prove that $AP$ is perpendicular to $MK$.

2019 Puerto Rico Cono Sur TST p4

Rectangle $ABCD$ has sides $AB = 3$, $BC = 2$. Point $P$ lies on side $AB$ is such that the bisector of the angle $CDP$ passes through the midpoint $M$ of $BC$. Find $BP$.

2019 Puerto Rico Cono Sur TST p6

Starting from a pyramid $T_0$ whose edges are all of length $2019$, we construct the Figure $T_1$

when considering the triangles formed by the midpoints of the edges of each face of $T_0$, building

in each of these new pyramid triangles with faces identical to base. Then the bases of these new pyramids

are removed. Figure $T_2$ is constructed by applying the same process from $T_1$ on each triangular

face resulting from $T_1$, and so on for $T_3, T_4, ...$. Let $D_0= \max \{d(x,y)\}$, where $x$ and

$y$ are vertices of $T_0$ and $d(x,y)$ is the distance between $x$ and $y$. Then we define

$D_{n + 1} = \max \{d (x, y) |d (x, y) \notin \{D_0, D_1,...,D_n\}$, where $x, y$ are vertices of

$T_{n+1}$. Find the value of $D_n$ for all $n$.

2020 Puerto Rico Cono Sur TST p3

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$, such that $CD=BC$. The side $CA$ is extended beyond $A$ to $E$, such that $AE=2CA$. Prove that if $AD=BE$, then the triangle $ABC$ is right.

2021 Puerto Rico Cono Sur TST p2

Let $ABC$ be a right triangle with right angle at $B$ and $\angle C=30^o$. If $M$ is midpoint of the hypotenuse and I the incenter of the triangle, show that$\angle IMB=15^o$.

2021 Puerto Rico Cono Sur TST p5

Circle $o$ contains the circles $m$ , $p$ and $r$, such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$. Find the value of $x$ .

IberoAmerican TST

2009 Puerto Rico Ibero TST p3

Show that if $h_A, h_B,$ and $h_C$ are the altitudes of $\triangle ABC$, and $r$ is the radius of the incircle, then $$h_A + h_B + h_C \ge 9r$$

2009 Puerto Rico Ibero TST p4

The point $M$ is chosen inside parallelogram $ABCD$. Show that $\angle MAB$ is congruent to $\angle MCB$, if and only if $\angle MBA$ and $\angle MDA$ are congruent.

2010 Puerto Rico Ibero TST p5

Let $ABC$ be an acute triangle such that $AB>BC>AC$. Let $D$ be a point different from $C$ on the segment $BC$, such that $AC=AD$. Let $H$ be the orthocenter of triangle $ABC$ and let $A_1$ and $B_1$ be the intersections of the heights from $A$ and $B$ to the opposite sides, respectively. Let $E$ be the intersection of the lines $A_1B_1$ and $DH$. Prove that $B$, $D$, $B_1$, $E$ are concyclic.

2011 Puerto Rico Ibero TST p4

Let $P$ be a point inside the triangle $ABC$, such that the angles $\angle CBP$ and $\angle PAC$ are equal. Denote the intersection of the line $AP$ and the segment $BC$ by $D$, and the intersection of the line $BP$ with the segment $AC$ by $E$. The circumcircles of the triangles $ADC$ and $BEC$ meet at $C$ and $F$. Show that the line $CP$ bisects the angle $DFE$.

2017 Puerto Rico Ibero TST p3

In triangle $ABC$, the altitude through $B$ intersects $AC$ at $E$ and the altitude through $C$ intersects $AB$ at $F$. Point $T$ is such that $AETF$ is a parallelogram and points $A$ ,$T$ lie on different half-planes wrt the line $EF$. Point $D$ is such that $ABDC$ is a parallelogram and points $A$ ,$D$ lie in different half-planes wrt line $BC$. Prove that $T, D$ and the orthocenter of $ABC$ are collinear.

2018 Puerto Rico Ibero TST p2

Let $ABC$ be an acute triangle and let $P,Q$ be points on $BC$ such that $\angle QAC =\angle ABC$ and $\angle PAB = \angle ACB$. We extend $AP$ to $M$ so that $P$ is the midpoint of $AM$ and we extend $AQ$ to $N$ so that $Q$ is the midpoint of $AN$. If T is the intersection point of $BM$ and $CN$, show that quadrilateral $ABTC$ is cyclic.

2018 Puerto Rico Ibero TST p5

In the square shown in the figure, find the value of $x$.

2018 Puerto Rico Ibero TST p6

Starting from an equilateral triangle with perimeter $P_0$, we carry out the following iterations: the first iteration consists of dividing each side of the triangle into three segments of equal length, construct an exterior equilateral triangle on each of the middle segments, and then remove these segments (bases of each new equilateral triangle formed). The second iteration consists of apply the same process of the first iteration on each segment of the resulting figure after the first iteration. Successively, follow the other iterations. Let $A_n$ be the area of the figure after the $n$ th iteration, and let $P_n$ the perimeter of the same figure. If $A_n = P_n$, find the value of $P_0$ (in its simplest form).