geometry problems from Lusophon Mathematical Olympiads
with aops links in the names
2011 OMCPLP P1
Prove that the area of the circle inscribed in a regular hexagon is greater than $90\%$ of the area of the hexagon.
2011 OMCPLP 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 OMCPLP 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 OMCPLP 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 OMCPLP 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 OMCPLP 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 OMCPLP 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 OMCPLP 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 OMCPLP 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 $BAC$
with aops links in the names
Mathematics Olympics Community of Portuguese Language Countries (OM CPLP)
2011 - 2021
Prove that the area of the circle inscribed in a regular hexagon is greater than $90\%$ of the area of the hexagon.
2011 OMCPLP 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 OMCPLP 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.
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 OMCPLP 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 OMCPLP 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 OMCPLP 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 OMCPLP 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?
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 $BAC$
In the center of a square is a rabbit and at each vertex of this even square, a wolf. The wolves only move along the sides of the square and the rabbit moves freely in the plane. Knowing that the rabbit move at a speed of $10$ km / h and that the wolves move to a maximum speed of $14$ km / h, determine if there is a strategy for the rabbit leave the square without being caught by the wolves.
2015 OMCPLP 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 OMCPLP 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 OMCPLP 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 OMCPLP 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 OMCPLP 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.
2019 OMCPLP P3
Let $ABC$ be a triangle with $AC \ne BC$. In triangle $ABC$, let $G$ be the centroid, $I$ the incenter and O Its circumcenter. Prove that $IG$ is parallel to $AB$ if, and only if, $CI$ is perpendicular on $IO$.
2020 OMCPLP P3 (OMOP)
Let $ABC$ be a triangle and on the sides we draw, externally, the squares $BADE, CBFG$ and $ACHI$. Determine the greatest positive real constant $k$ such that, for any triangle $\triangle ABC$, the following inequality is true: $[DEFGHI]\geq k\cdot [ABC]$
Note: $[X]$ denotes the area of polygon $X$.
2020 OMCPLP P4 (OMOP)
Let $ABC$ be an acute triangle. Its incircle touches the sides $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$, respectively. Let $P$, $Q$ and $R$ be the circumcenters of triangles $AEF$, $BDF$ and $CDE$, respectively. Prove that triangles $ABC$ and $PQR$ are similar.
Let triangle $ABC$ be an acute triangle with $AB\neq AC$. the bisector of $BC$ intersects the lines
$AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$
and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$. Prove that the
line $PD$ is tangent to the circumcircle of triangle $ABC$.
2021 OMCPLP P5
There are 3 lines $r, s$ and $t$ on a plane. The lines $r$ and $s$ intersect perpendicularly at point $A$
the line $t$ intersects the line $r$ at point $B$ and the line $s$ at point $C$. There exist exactly 4
circumferences on the plane that are simultaneously tangent to all those 3 lines. Prove that the radius of
one of those circumferences is equal to the sum of the radius of the other three circumferences.
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