Ecuador 2016-19 L2 L3 (OMEC) 14p

geometry problems from final round of level 3 from Ecuadorian Mathematical Olympiad (OMEC) with aops links in the names

collected inside aops here

Seniors / Level 3 2016 - 2019

2016 Ecuador MO  L3 p3

Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.

2016 Ecuador MO L3 p4

In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$.

2017 Ecuador MO L3 p2

Let $ABC$ be a triangle with $AC = 18$ and $D$ is the point on the segment $AC$ such that $AD = 5$. Draw perpendiculars from $D$ on $AB$ and $BC$ which have lengths $4$ and $5$ respectively. Find the area of the triangle $ABC$.

2017 Ecuador MO L3 p6

Let $ABCDEF$ be a convex hexagon with sides not parallel and tangent to a circle $\Gamma$ at the midpoints $P$, $Q$, $R$ of the sides AB, $CD$, $EF$ respectively. $\Gamma$ is tangent to $BC$, $DE$ and $FA$ at the points $X, Y, Z$ respectively. Line $AB$ intersects lines $EF$ and $CD$ at points $M$ and $N$ respectively. Lines $MZ$ and $NX$ intersect at point $K$. Let $r$ be the line joining the center of $\Gamma$ and point $K$. Prove that the intersection of $PY$ and $QZ$ lies on the line $r$.

2018 Ecuador MO L3 p3

Let $ABCD$ be a convex quadrilateral with $AB\le CD$. Points $E ,F$ are chosen on segment $AB$ and points $G ,H$ are chosen on the segment $CD$, are chosen such that $AE = BF = CG = DH <\frac{AB}{2}$. Let $P, Q$, and $R$ be the midpoints of $EG$, $FH$, and $CD$, respectively. It is known that $PR$ is parallel to $AD$ and $QR$ is parallel to $BC$.

a) Show that $ABCD$ is a trapezoid.

b) Let $d$ be the difference of the lengths of the parallel sides. Show that $2PQ\le d$.

2018 Ecuador MO L3 p5

Let ABC be an acute triangle and let $M$, $N$, and $P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.

2019 Ecuador MO L3 p2

Let $ABC$ be a triangle and $D$ be a point on segment $AC$. The circumscribed circle of the triangle $BDC$ cuts $AB$ again at $E$ and the circumference circle of the triangle $ABD$ cuts $BC$ again at $F$. Prove that $AE = CF$ if and only if $BD$ is the interior bisector of $\angle ABC$.

Juniors / level 2 2016 - 2019

2016 Ecuador Juniors p3

Let $P_1P_2 . . . P_{2016 }$ be a cyclic polygon of $2016$ sides. Let $K$ be a point inside the polygon and let $M$ be the midpoint of the segment $P_{1000}P_{2000}$. Knowing that $KP_1 = KP_{2011} = 2016$ and $KM$ is perpendicular to $P_{1000}P_{2000}$, find the length of segment $KP_{2016}$.

2016 Ecuador Juniors p5

In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$.

2017 Ecuador Juniors p3

Given an isosceles triangle $ABC$ with $AB = AC$. Let $O$ be the circumcenter of $ABC$, $D$ the midpoint of $AB$ and $E$ the centroid of $ACD$. Prove that $CD \perp EO$.

2018 Ecuador Juniors p3

Let $ABCD$ be a square. Point $P, Q, R, S$ are chosen on the sides $AB$, $BC$, $CD$, $DA$, respectively, such that $AP + CR \ge AB  \ge BQ + DS$. Prove that

$$area \,\, (PQRS) \le \frac12 \,\,  area \,\,  (ABCD)$$ and determine all cases when equality holds.

2018 Ecuador Juniors p4

Given a positive integer $n > 1$ and an angle $\alpha < 90^o$, Jaime draws a spiral $OP_0P_1...P_n$ of the following form (the figure shows the first steps):

$\bullet$ First draw a triangle $OP_0P_1$ with $OP_0 = 1$, $\angle P_1OP_0 = \alpha$ and $P_1P_0O = 90^o$

$\bullet$ then for every integer $1 \le i \le n$ draw the point $P_{i+1}$ so that $\angle P_{i+1}OP_i = \alpha$, $\angle P_{i+1}P_iO = 90^o$ and $P_{i-1}$ and $P_{i+1}$ are in different half-planes with respect to the line $OP_i$

a) If $n = 6$ and $\alpha = 30^o$, find the length of $P_0P_n$.

b) If $n = 2018$ and $\alpha= 45^o$, find the length of $P_0P_n$.

2019 Ecuador Juniors p3

Let $ABC$ be a triangle and $D$ be a point on segment $AC$. The circumscribed circle of the triangle $BDC$ cuts $AB$ again at $E$ and the circumference circle of the triangle $ABD$ cuts $BC$ again at $F$. Prove that $AE = CF$ if and only if $BD$ is the interior bisector of $\angle ABC$.

2019 Ecuador Juniors p4

Let $ABCD$ be a square. On the segments $AB$, $BC$, $CD$ and $DA$, choose points $E, F, G$ and $H$, respectively, such that $AE = BF = CG = DH$. Let $P$ be the intersection point of $AF$ and $DE$, $Q$ be the intersection point of $BG$ and $AF$, $R$ the intersection point of $CH$ and $BG$, and $S$ the point of intersection of $DE$ and $CH$. Prove that $PQRS$ is a square.