Kosovo 2011-21 IX-XII 26p (-12,-14,-15,-18)

geometry problems from Kosovo National Mathematical Olympiad with aops links

2011 - 2021 

missing 2012, 2014, 2015, 2018

2011 Kosovo 9.4

In triangle $ABC$ medians of triangle $BE$ and $AD$ are perpendicular to each other. Find the length of $\overline{AB}$, if $\overline{BC}=6$ and $\overline{AC}=8$

2011 Kosovo 10.4

Let $a$, $b$, $c$ be the sides of a triangle, and $S$ its area. Prove $a^{2} + b^{2} + c^{2}\geq 4S \sqrt {3}$. In what case does equality hold?

2011 Kosovo 11.4

A point $P$ is given in the square $ABCD$ such that $\overline{PA}=3$, $\overline{PB}=7$ and $\overline{PD}=5$. Find the area of the square.

2011 Kosovo 12.4

It is given a convex hexagon $A_1A_2 \cdots A_6$ such that all its interior angles are same valued (congruent). Denote by $a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.$

a) Prove that holds: $a_1-a_4=a_2-a_5=a_3-a_6$

b) Prove that if $a_1,a_2,a_3,...,a_6$ satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent) interior angles.

2013 Kosovo 9.5

Let $ABC$ be an equilateral triangle, with sidelength equal to $a$. Let $P$ be a point in the interior of triangle $ABC$, and let $D,E$ and $F$ be the feet of the altitudes from $P$ on $AB, BC$ and $CA$, respectively. Prove that $\frac{|PD|+|PE|+|PF|}{3a}=\frac{\sqrt{3}}{6}$

2013 Kosovo 10.5

Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle

2013 Kosovo 11.5

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. . Assume that $ABCD$ has been inscribed in the circle with center $O$. Prove that $AOC$ separates $ABCD$ into two quadrilaterals of equal area

2013 Kosovo 12.5

A trapezium has parallel sides of length equal to $a$ and $b$ ($a <b$), and the distance between the parallel sides is the altitude $h$. The extensions of the non-parallel lines intersect at a point that is a vertex of two triangles that have as sides the parallel sides of the trapezium. Express the areas of the triangles as functions of $a,b$ and $h$.

2016 Kosovo 9.5

It is given rectangle $ABCD$ with length $|AB|=15cm$ and with length of altitude $|BE|=12cm$ where $BC$ is altitude of triangle $ABC$ . Find perimeter and area of rectangle $ABCD$ .

2016 Kosovo 10.5

If $a,b,c$ are sides of right triangle with $c$ hypothenuse then show that for every positive integer $n>2$ we have $c^n>a^n+b^n$ .

2016 Kosovo 11.5

In angle $\angle AOB=60^{\circ}$ are two circle which circumscribed and tangjent to each other . If we write with $r$ and $R$ the radius of smaller and bigger circle respectively and if $r=1$ find $R$ .

2016 Kosovo 12.5

In trapezoid $ABCD$ with $AB$ parallel to $CD$ show that :

$\frac{|AB|^2-|BC|^2+|AC|^2}{|CD|^2-|AD|^2+|AC|^2}=\frac{|AB|}{|CD|}=\frac{|AB|^2-|AD|^2+|BD|^2}{|CD|^2-|BC|^2+|BD|^2}$

2017 Kosovo 9.5

Given the point $T$ in rectangle $ABCD$, the distances from $T$ to $A,B,C$ is $15,20,25$.

Find the distance from $T$ to $D$.

2017 Kosovo 10 missing from aops

2017 Kosovo 11.5

A sphere with ray $R$ is cut by two parallel planes. such that the center of the sphere is outside the region determined by these planes. Let $S_{1}$ and $S_{2}$ be the areas of the intersections, and $d$ the distance between these planes. Find the area of the intersection of the sphere with the plane parallel with these two planes, with equal distance from them.

2017 Kosovo 12.5

Lines determined by sides $AB$ and $CD$ of the convex quadrilateral $ABCD$ intersect at point $P$. Prove that $\alpha +\gamma =\beta +\delta$ if and only if $PA\cdot PB=PC\cdot PD$, where $\alpha ,\beta ,\gamma ,\delta$ are the measures of the internal angles of vertices $A, B, C, D$ respectively.

2019 Kosovo 9.3

Let $ABCD$ be a rectangle with $AB>BC$. Let points $E,F$ be on side $CD$ such that $CE=ED$ and $BC=CF$. Show that if $AC$ is prependicular to $BE$ then $AB=BF$.

2019 Kosovo 10.5

Let $ABCDE$ be a regular pentagon. Let point $F$ be intersection of segments $AC$ and $BD$. Let point $G$ be in segment $AD$ such that $2AD=3AG$. Let point $H$ be the midpoint of side $DE$. Show that the points $F,G,H$ lie on a line.

2019 Kosovo 11.3

Let $ABC$ be a triangle with $\angle CAB=60^{\circ}$ and with incenter $I$. Let points $D,E$ be on sides $AC,AB$, respectively, such that $BD$ and $CE$ are angle bisectors of angles $\angle ABC$ and $\angle BCA$, respectively. Show that $ID=IE$.

2019 Kosovo 12.4

Let $ABC$ be an acute triagnle with its circumcircle $\omega$. Let point $D$ be the foot of triangle $ABC$ from point $A$. Let points $E,F$ be midpoints of sides $AB,AC$, respectively. Let points $P$ and $Q$ be the second intersections of of circle $\omega$ with circumcircle of triangles $BDE$ and $CDF$, respectively. Suppose that $A,P,B,Q$ and $C$ be on a circle in this order. Show that the lines $EF,BQ$ and $CP$ are concurrent.

2020 Kosovo 9.3 

Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.

2020 Kosovo 10.4 

Let $B'$ and $C'$ be points in the circumcircle of triangle $\triangle ABC$ such that $AB=AB'$ and $AC=AC'$. Let $E$ and $F$ be the foot of altitudes from $B$ and $C$ to $AC$ and $AB$, respectively. Show that $B'E$ and $C'F$ intersect on the circumcircle of triangle $\triangle ABC$.

2020 Kosovo 11.4 

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The exterior angle bisector of $\angle BAC$ intersects $\omega$ at point $D$. Let $X$ be the foot of the altitude from $C$ to $AD$ and let $F$ be the intersection of the internal angle bisector of $\angle BAC$ and $BC$. Show that $BX$ bisects segment $AF$.

2020 Kosovo 12.3 (2019 JBMO SL G3)

Let $ABC$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $CA$ and $BC$ respectively, such that $CD = CE$. Let $F$ be a point on the segment $CD$. Prove that the quadrilateral $ABEF$ is circumscribable if and only if the quadrilateral $DIEF$ is cyclic.

by Dorlir Ahmeti, Albania

2021 Kosovo 9.4

Let $ABCDE$ be a convex pentagon such that:

$\angle ABC=90^o$, $\angle BCD=135^o$, $\angle DEA=60^o$ and $AB=BC=CD=DE$. Find angle $\angle DAE$.

2021 Kosovo 10.4

Let $M$ be the midpoint of segment $BC$ of $\triangle ABC$. Let $D$ be a point such that $AD=AB$, $AD\perp AB$ and points $C$ and $D$ are on different sides of $AB$. Prove that: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$

2021 Kosovo 10.4

Let $ABC$ be a triangle with $AB<AC$. Let $D$ be the point where the bisector of angle $\angle BAC$ touches $BC$ and let $D'$ be the reflection of $D$ in the midpoint of $BC$. Let $X$ be the intersection of the bisector of angle $\angle BAC$ with the line parallel to $AB$ that passes through $D'$. Prove that the line $AC$ is tangent with the circumscribed circle of triangle $XCD'$

2021 Kosovo 11.4

Let $ABC$ be a triangle and let $O$ be the centre of its circumscribed circle. Points $X, Y$ which are neither of the points $A, B$ or $C$, lie on the circumscribed circle and are so that the angles $XOY$ and $BAC$ are equal (with the same orientation). Show that the orthocentre of the triangle that is formed by the lines $BY, CX$ and $XY$ is a fixed point.