Paraguay 2001-20 (OMAPA) 38p (-16-17,-18)

geometry problems from Paraguayan Mathematical Olympiad (OMAPA) from with aops links in the names

collected inside aops here

2001 - 15, 2019 - 20

2001 Paraguay p4

In a parallelogram $ABCD$ of surface area$60$ cm$^2$ , a line is drawn by $D$ that intersects $BC$ at $P$ and the extension of $AB$ at $Q$. If the area of the quadrilateral $ABPD$ is $46$ cm$^2$ , find the area of triangle $CPQ$.

2002 Paraguay p2

In the rectangular parallelepiped in the figure, the lengths of the segments $EH$, $HG$, and $EG$ are consecutive integers. The height of the parallelepiped is $12$. Find the volume of the parallelepiped.

2002 Paraguay p5

In a trapezoid $ABCD$, the side $DA$ is perpendicular to the bases $AB$ and $CD$. Also $AB=45$, $CD =20$, $BC =65$. Let $P$ be a point on the side $BC$ such that $BP=45$ and let $M$ be the midpoint of $DA$. Calculate the length of $PM$ .

2003 Paraguay p4

Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$.

2003 Paraguay p5

In a square $ABCD$, $E$ is the midpoint of side $BC$. Line $AE$ intersects line $DC$ at $F$ and diagonal $BD$ at $G$. If the area $(EFC) = 8$, determine the area $(GBE)$.

2004 Paraguay p2

Determine for what values of $x$ the expressions $2x + 2$,$x + 4$, $x + 2$ can represent the sidelengths of a right triangle.

2004 Paraguay p3

In an equilateral triangle ABC, whose side is $4$, the line perpendicular to $AB$ is drawn through the point $A$, the line perpendicular to $BC$ through point $B$ and the line perpendicular to $CA$ through point $C$. These three lines determine another triangle. Calculate the perimeter of this triangle

2004 Paraguay p4

In a square $ABCD$, $E$ is the midpoint of $BC$ and $F$ is the midpoint of $CD$. Prove that $AF$ and $AE$ divide the diagonal $BD$ in three equal segments.

2005 Paraguay p2

If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get $5.100$. Determine the number of faces of the pyramid.

2005 Paraguay p5

Given a chord $PQ$ of a circle and $M$ the midpoint of the chord, let $AB$ and $CD$ be two chords that pass through $M$. $AC$ and $BD$ are drawn until $PQ$ is intersected at points $X$ and $Y$ respectively. Show that $X$ and $Y$ are equidistant from $M$.

2006 Paraguay p2

Consider all right triangles with integer sides such that the length of the hypotenuse and one of the two sides are consecutive. How many such triangles exist?

2006 Paraguay p3

Let $\Gamma_A$, $\Gamma_B$, $\Gamma_C$ be circles such that $\Gamma_A$ is tangent to $\Gamma_B$ and $\Gamma_B$ is tangent to $\Gamma_C$. All three circles are tangent to lines $L$ and $M$, with $A$, $B$, $C$ being the tangency points of $M$ with $\Gamma_A$, $\Gamma_B$, $\Gamma_C$, respectively. Given that $12=r_A<r_B<r_C=75$, calculate:

a) the length of $r_B$.

b) the distance between point $A$ and the point of intersection of lines $L$ and $M$.

2006 Paraguay p5

Let $ABC$ be a triangle, and let $P$ be a point on side $BC$ such that $\frac{BP}{PC}=\frac{1}{2}$. If $\measuredangle ABC$ $=$ $45^{\circ}$ and $\measuredangle APC$ $=$ $60^{\circ}$, determine $\measuredangle ACB$ without trigonometry.

2007 Paraguay p2

Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.

2007 Paraguay p3

Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$.

a) Show that $DE \perp CF$.

b) Determine the ratio $CF : PC : EP$

2007 Paraguay p5

Let $A, B, C,$ be points in the plane, such that we can draw $3$ equal circumferences in which the first one passes through $A$ and $B$, the second one passes through $B$ and $C$, the last one passes through $C$ and $A$, and all $3$ circumferences share a common point $P$.

Show that the radius of each of these circumferences is equal to the circumradius of triangle $ABC$, and that $P$ is the orthocenter of triangle $ABC$.

2008 Paraguay p3

Let $ABC$ be a triangle, where $AB = AC$ and $BC = 12$. Let $D$ be the midpoint of $BC$. Let $E$ be a point in $AC$ such that $DE \perp AC$. Let $F$ be a point in $AB$ such that $EF \parallel BC$. If $EC = 4$, determine the length of $EF$.

2008 Paraguay p4

Let $\Gamma$ be a circumference and $A$ a point outside it. Let $B$ and $C$ be points in $\Gamma$ such that $AB$ and $AC$ are tangent to $\Gamma$. Let $P$ be a point in $\Gamma$. Let $D$, $E$ and $F$ be points in $BC$, $AC$ and $AB$ respectively, such that $PD \perp BC$, $PE \perp AC$, and $PF \perp AB$. Show that $PD^2 = PE \cdot PF$

2009 Paraguay p2

In a triangle $ABC$ ($\angle{BCA} = 90^{\circ}$), let $D$ be the intersection of $AB$ with a circumference with diameter $BC$. Let $F$ be the intersection of $AC$ with a line tangent to the circumference. If $\angle{CAB} = 46^{\circ}$, find the measure of $\angle{CFD}$.

2009 Paraguay p5

In a triangle $ABC$, let $I$ be its incenter. The distance from $I$ to the segment $BC$ is $4 cm$ and the distance from that point to vertex $B$ is $12 cm$. Let $D$ be a point in the plane region between segments $AB$ and $BC$ such that $D$ is the center of a circumference that is tangent to lines $AB$ and $BC$ and passes through $I$. Find all possible values of the length $BD$.

2010 Paraguay p1

The picture below shows the way Juan wants to divide a square field in three regions, so that all three of them share a well at vertex $B$. If the side length of the field is $60$ meters, and each one of the three regions has the same area, how far must the points $M$ and $N$ be from $D$?

Note: the area of each region includes the area the well occupies.

2010 Paraguay p3

In a triangle $ABC$, let $M$ be the midpoint of $AC$. If $BC = \frac{2}{3} MC$ and $\angle{BMC}=2 \angle{ABM}$, determine $\frac{AM}{AB}$.

2010 Paraguay p5

In a triangle $ABC$, let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively. Let $D'$, $E'$ and $F'$ be the second intersection of lines $AD$, $BE$ and $CF$ with the circumcircle of $ABC$. Show that the triangles $DEF$ and $D'E'F'$ are similar.

2011 Paraguay p2

In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AC$ and $BC$ respectively. The distance from the midpoint of $BD$ to the midpoint of $AE$ is $4.5$. What is the length of side $AB$?

2011 Paraguay p5

In a rectangle triangle, let $I$ be its incenter and $G$ its geocenter. If $IG$ is parallel to one of the catheti and measures $10 cm$, find the lengths of the two catheti of the triangle.

2012 Paraguay p3

Let $ABC$ be a triangle (right in $B$) inscribed in a semi-circumference of diameter $AC=10$. Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.

2012 Paraguay p5

Let $ABC$ be an equilateral triangle. Let $Q$ be a random point on $BC$, and let $P$ be the meeting point of $AQ$ and the circumscribed circle of $\triangle ABC$. Prove that $\frac{1}{PQ}=\frac{1}{PB}+\frac{1}{PC}$.

2013 Paraguay p2

Let $ABC$ be a triangle with area $9$, and let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let $P$ be the point in side $BC$ such that $PC = \frac{1}{3}BC$. Let $O$ be the intersection point between $PN$ and $CM$. Find the area of the quadrilateral $BPOM$.

2013 Paraguay p5

Let $ABC$ be an obtuse triangle, with $AB$ being the largest side.

Draw the angle bisector of $\measuredangle BAC$. Then, draw the perpendiculars to this angle bisector from vertices $B$ and $C$, and call their feet $P$ and $Q$, respectively. $D$ is the point in the line $BC$ such that $AD \perp AP$. Prove that the lines $AD$, $BQ$ and $PC$ are concurrent.

2014 Paraguay p1

Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?

2014 Paraguay p5

Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$.

2015 Paraguay p3

A cube is divided into $8$ smaller cubes of the same size, as shown in the figure. Then, each of these small cubes is divided again into $8$ smaller cubes of the same size. This process is done $4$ more times to each resulting cube. What is the ratio between the sum of the total areas of all the small cubes resulting from the last division and the total area of the initial cube?

2015 Paraguay p4

The sidelengths of a triangle are natural numbers multiples of $7$, smaller than $40$. How many triangles satisfy these conditions?

2015 Paraguay p5

In the figure, the rectangle is formed by $4$ smaller equal rectangles.

If we count the total number of rectangles in the figure we find $10$.

How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?

2016, 2017, 2018 missing

2019 Paraguay p5

A circle of radius $4$ is inscribed in a triangle $ABC$. We call $D$ the touchpoint between the circle and side BC. Let $CD =8$, $DB= 10$. What is the length of the sides $AB$ and $AC$?

2020 Paraguay p3

In triangle $ABC$, side $AC$ is $8$ cm. Two segments are drawn parallel to $AC$ that have their ends on $AB$ and $BC$ and that divide the triangle into three parts of equal area. What is the length of the parallel segment closest to $AC$?

2020 Paraguay p4

In the square $ABCD$ the points $E$ and $F$ are marked on the sides $AB$ and $BC$ respectively, in such a way that $EB = 2AE$ and $BF = FC$. Let $G$ be the intersection between $DF$ and $EC$. If the side of the square equals $10$, what is the distance from point $G$ to side $AB$?