Lusophon / Portuguese Language 2011-18 (OMCPLP) 13p

geometry problems from Lusophon Mathematical Olympiads
with aops links in the names

Mathematics Olympics Community of Portuguese Language Countries (OM CPLP)

2011 - 2018

2011 CPLP P1
Prove that the area of the circle inscribed in a regular hexagon is greater than $90\%$ of the area of the hexagon.

2011 CPLP P5
Consider two circles, tangent at $T$, both inscribed in a rectangle of height $2$ and width $4$. A point $E$ moves counterclockwise around the circle on the left, and a point $D$ moves clockwise around the circle on the right. $E$ and $D$ start moving at the same time; $E$ starts at $T$, and $D$ starts at $A$, where $A$ is the point where the circle on the right intersects the top side of the rectangle. Both points move with the same speed. Find the locus of the midpoints of the segments joining $E$ and $D$.

2012 CPLP P6
A quadrilateral $ABCD$ is inscribed in a circle of center $O$. It is known that the diagonals $AC$ and $BD$ are perpendicular. On each side we build semicircles, externally, as shown in the figure.
a) Show that the triangles $AOB$ and $COD$ have the equal areas.
b) If $AC=8$ cm and $BD= 6$ cm, determine the area of the shaede region.
2013 CPLP P2
Let $ABC$ be an acute triangle. The circumference with diameter $AB$ intersects sides $AC$ and $BC$ at $E$ and $F$ respectively. The tangent lines to the circumference at the points $E$ and $F$ meet at $P$. Show that $P$ belongs to the altitude from $C$ of triangle $ABC$.

2013 CPLP P6
Consider a triangle $ABC$. Let $S$ be a circumference in the interior of the triangle that is tangent to the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. In the exterior of the triangle we draw three circumferences $S_A$, $S_B$, $S_C$. The circumference $S_A$ is tangent to $BC$ at $L$ and to the prolongation of the lines $AB$, $AC$ at the points $M$, $N$ respectively. The circumference $S_B$ is tangent to $AC$ at $E$ and to the prolongation of the line $BC$ at $P$. The circumference $S_C$ is tangent to $AB$ at $F$ and to the prolongation of the line $BC$ at $Q$. Show that the lines $EP$, $FQ$ and $AL$ meet at a point of the circumference $S$.

2014 CPLP P3
In a convex quadrilateral $ABCD$, $P$ and $Q$ are points on sides $BC$ and $DC$ such that $B\hat{A}P = D\hat{A}Q$. If the line that passes through the orthocenters of $\triangle ABP$ and $\triangle ADQ$ is perpendicular to $AC$, prove that the area of these triangles are equals.

2014 CPLP P4
From a point $K$ of a circle, a chord $KA$ (arc $AK$ is greather than $90^{o}$) and a tangent $l$ are drawn. The line that passes through the center of the circle and that is perpendicular to the radius $OA$, intersects $KA$ at $B$ and $l$ at $C$. Show that $KC = BC$.
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2014 CPLP P6
Kilua and Ndoti play the following game in a square $ABCD$: Kilua chooses one of the sides of the square and draws a point $X$ at this side. Ndoti chooses one of the other three sides and draws a point $Y$. Kilua chooses another side that hasn't been chosen and draws a point $Z$. Finally, Ndoti chooses the last side that hasn't been chosen yet and draws a point $W$. Each one of the players can draw his point at a vertex of $ABCD$, but they have to choose the side of the square that is going to be used to do that. For example, if Kilua chooses $AB$, he can draws $X$ at the point $B$ and it doesn't impede Ndoti of choosing $BC$. A vertex cannot de chosen twice. Kilua wins if the area of the convex quadrilateral formed by $X$, $Y$, $Z$, and $W$ is greater or equal than a half of the area of $ABCD$. Otherwise, Ndoti wins. Which player has a winning strategy? How can he play?

2015 CPLP P1
In a triangle $ABC, L$ and $K$ are the points of intersections of the angle bisectors of the angles $ABC$ and $BAC$ with the segments $AC$ and $BC$ respectively. The segment $KL$ is angle bisector of the angle $AKC$, determine the angle $BAC$

2015 CPLP P5
Two circles of radius $R$ and $r$, with $R>r$, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.

2016 CPLP P2
The circle $w_1$ intersects the circle $w_2$ in the points $A$ and $B$, a tangent line to this circles intersects $w_1$ and $w_2$ in the points $E$ and $F$ respectively. Suppose that $A$ is inside of the triangle $BEF$, let $H$ be the orthocenter of $BEF$ and $M$ is the midpoint of $BH$. Prove that the centers of the circles $w_1$ and $w_2$ and the point $M$ are collinears.

2017 CPLP P2
Let $ABCD$ be a parallelogram, E the midpoint of $AD$ and $F$ the projection of $B$ on $CE$. Prove that the triangle $ABF$ is isosceles.

2017 CPLP P6
Let $ABC$ be a scalene triangle. Consider points $D, E, F$ on segments $AB, BC, CA$, respectively, such that $\overline{AF}$=$\overline{DF}$ and $\overline{BE}$=$\overline{DE}$.
Show that the circumcenter of $ABC$ lies on the circumcircle of $CEF.$

2018 CPLP P2
In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$  be the point on the extension of $BA$ such that the triangle $ADE$  is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$  and $Q$ the point of  intersection of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.