Estonia 1996 - 2007 IX-XII 59p

geometry problems from Estonian Mathematical Olympiads, Final Round
with aops links in the names

Estonian Math Competitions 1999, 2001-21 with solutions

1996 - 2007
under construction the recent years

1996 Estonia IX p3
The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$

1996 Estonia X p4
Let $K, L, M$, and $N$ be the midpoints of $CD,DA,AB$ and $BC$ of a square $ABCD$ respectively. Find the are of the triangles $AKB, BLC, CMD$ and $DNA$ if the square $ABCD$ has area $1$.
.
1996 Estonia XI p2
Three sides of a trapezoid are equal, and a circle with the longer base as a diameter halves the two non-parallel sides. Find the angles of the trapezoid.

1996 Estonia XII p3
An equilateral triangle of side$1$ is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.

1997 Estonia IX p3
The points $A, B, M$ and $N$ are on a circle with center $O$ such that the radii $OA$ and $OB$ are perpendicular to each other, and $MN$ is parallel to $AB$ and intersects the radius $OA$ at $P$. Find the radius of the circle if $|MP|= 12$ and $|P N| = 2 \sqrt{14}$

1997 Estonia X p3
In triangle ABC, consider the sizes $\tan \angle A, \tan \angle B$, and $\tan \angle C$ into another such as the numbers $1, 2$ and $3$. Find the ratio of the sidelenghts $AC$ and $AB$ of the triangle.

1997 Estonia X p5
There are six small circles in the figure with a radius of $1$ and tangent to a large circle and the sides of the $ABC$ of an equilateral triangle, where touch points are $K, L$ and $M$ respectively with the midpoints of sides $AB, BC$ and $AC$. Find the radius of the large circle and the side of the triangle $ABC$.

1997 Estonia XΙ p3
Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the ratio of the length of each diagonal to the length of the corresponding parallel side is the same, and find this ratio.

1997 Estonia XI p5
Six small circles of radius $1$ are drawn so that they are all tangent to a larger circle, and two of them are tangent to sides $BC$ and $AD$ of a rectangle $ABCD$ at their midpoints $K$ and $L$. Each of the remaining four small circles is tangent to two sides of the rectangle. The large circle is tangent to sides $AB$ and $CD$ of the rectangle and cuts the other two sides. Find the radius of the large circle.

1997 Estonia XΙΙ p3
A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.

1997 Estonia XII p5
Find the length of the longer side of the rectangle on the picture, if the shorter side has length $1$ and the circles touch each other and the sides of the rectangle as shown.

1998 Estonia IX p2
Let $S$ be the incenter of the triangle $ABC$ and let the line $AS$ intersect the circumcircle of triangle $ABC$ at point $D$ ($D\ne A$). Prove that the segments $BD, CD$ and $SD$ are of equal length.

1998 Estonia X p2
Let $C$ and $D$ be two distinct points on a semicircle of diameter $AB$. Let $E$ be the intersection of $AC$ and $BD$, $F$ be the intersection of $AD$ and $BC$ and $X, Y$, and $Z$ are the midpoints of $AB, CD$, and $EF$, respectively. Prove that the points $X, Y,$ and $Z$ are collinear.

1998 Estonia XI p2
In a triangle $ABC, A_1,B_1,C_1$ are the midpoints of segments $BC,CA,AB, A_2,B_2,C_2$ are the midpoints of segments $B_1C_1,C_1A_1,A_1B_1$, and $A_3,B_3,C_3$ are the incenters of triangles $B_1AC_1,C_1BA_1,A_1CB_1$, respectively. Show that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrent.

1998 Estonia XII p3
n a triangle $ABC$, the bisector of the largest angle $\angle A$ meets $BC$ at point $D$. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AC$ and $AB$, respectively. Let $R$ denote the ratio between the areas of triangles $DEB$ and $DFC$.
(a) Prove that, for every real number $r > 0$, one can construct a triangle ABC for which $R$ is equal to $r$.
(b) Prove that if $R$ is irrational, then at least one side length of $\vartriangle ABC$ is irrational.
(c) Give an example of a triangle $ABC$ with exactly two sides of irrational length, but with rational $R$.

1999 Estonia IX p3
Let $E$ and $F$ be the midpoints of the lines $AB$ and $DA$ of a square $ABCD$, respectively and let $G$ be the intersection of $DE$ with $CF$. Find the aspect ratio of sidelengths of the triangle $EGC$, $| EG | : | GC | : | CE |$.

1999 Estonia X p3
The incircle of the triangle $ABC$, with the center $I$ , touches the sides $AB, AC$ and $BC$ in the points $K, L$ and $M$ respectively. Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively, such that $|AP| = |CL|$ and $|BQ| = |CM|$. Prove that the difference of areas of the figures $APIQB$ and $CPIQ$ is equal to the area of the quadrangle $CLIM$

1999 Estonia X p5
Let $C$ be an interior point of line segment $AB$. Equilateral triangles $ADC$ and $CEB$ are constructed to the same side from $AB$. Find all points which can be the midpoint of the segment $DE$.

1999 Estonia XI p3
For the given triangle $ABC$, prove that a point $X$ on the side $AB$ satisfies the condition $\overrightarrow{XA} \cdot\overrightarrow{XB} +\overrightarrow{XC} \cdot \overrightarrow{XC} = \overrightarrow{CA} \cdot \overrightarrow{CB}$, iff $X$ is the basepoint of the altitude or median of the triangle $ABC$.

1999 Estonia XII p4
Prove that the line segment, joining the orthocenter and the intersection point of the medians of the acute-angled triangle $ABC$ is parallel to the side $AB$ iff $\tan \angle A \cdot \tan \angle B = 3$.

2000 Estonia IX p4
On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.

2000 Estonia X p4
Let $E$ be the midpoint of the side $AB$ of the parallelogram $ABCD$. Let $F$ be the projection of $B$ on $AC$. Prove that the triangle $ABF$ is isosceles

2000 Estonia XI p2
Let $PQRS$ be a cyclic quadrilateral with $\angle PSR = 90^o$, and let $H,K$ be the projections of $Q$ on the lines $PR$ and $PS$, respectively. Prove that the line $HK$ passes through the midpoint of the segment $SQ$.

2000 Estonia XII p3

Let $ABC$ be an acute-angled triangle with $\angle ACB = 60^o$ , and its heights $AD$ and $BE$ intersect at point $H$. Prove that the circumcenter of triangle $ABC$ lies on a line bisecting the angles $AHE$ and $BHD$.

2001 Estonia IX p3
A circle of radius $10$ is tangent to two adjacent sides of a square and intersects its two remaining sides at the endpoints of a diameter of the circle. Find the side length of the square.

2001 Estonia X p1

A convex $n$-gon has exactly three obtuse interior angles. Find all possible values of $n$.

2001 Estonia X p3

There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$.

2001 Estonia XI p3
Points $D,E$ and $F$ are taken on the sides $BC,CA,AB$ of a triangle $ABC$ respectively so that the segments $AD, BE$ and $CF$ intersect at point $O$. Prove that $\frac{AO}{OD}= \frac{AE}{EC}+\frac{AF}{FB}$

2001 Estonia XII p3
A circle with center $I$ and radius $r$ is inscribed in a triangle $ABC$ with a right angle at $C$. Rays $AI$ and $CI$ meet the opposite sides at $D$ and $E$ respectively. Prove that $\frac{1}{AE}+\frac{1}{BD}=\frac{1}{r}$

2002 Estonia IX p1
Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$. Find $\angle KAL$.

2002 Estonia X p2
Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.

2002 Estonia XI p2
Inside an equilateral triangle there is a point whose distances from the sides of the triangle are $3, 4$ and $5$. Find the area of the triangle.

2002 Estonia XII p4
A convex quadrilateral $ABCD$ is inscribed in a circle $\omega$. The rays $AD$ and $BC$ meet in point $K$ and the rays $AB$ and $DC$ meet in $L$. Prove that the circumcircle of triangle $AKL$ is tangent to $\omega$ if and only if so is the circumcircle of triangle $CKL$.

2003 Estonia IX p1
Let $A_1, A_2, ..., A_m$ and $B_2 , B_3,..., B_n$ be the points on a circle such that $A_1A_2... A_n$ is a regular $m$-gon and $A_1B_2...B_n$ is a regular $n$-gon whereby $n > m$ and the point $B_2$ lies between $A_1$ and $A_2$. Find $\angle B_2A_1A_2$.

2003 Estonia IX p3
In the rectangle $ABCD$ with $|AB|<2 |AD|$, let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$. Prove that $FAD$ is an isosceles triangle.

2003 Estonia X p1

The picture shows $10$ equal regular pentagons where each two neighbouring pentagons have a common side. The smaller circle is tangent to one side of each pentagon and the larger circle passes through the opposite vertices of these sides. Find the area of the larger circle if the area of the smaller circle is $1$.

2003 Estonia X p3

In the acute-angled triangle $ABC$ all angles are greater than $45^o$. Let $AM$ and $BN$ be the heights of this triangle and let $X$ and $Y$ be the points on $MA$ and $NB$, respecively, such that $|MX| =|MB|$ and $|NY| =|NA|$. Prove that $MN$ and $XY$ are parallel.

2003 Estonia XI p3
Let $ABC$ be a triangle and $A_1, B_1, C_1$ points on $BC, CA, AB$, respectively, such that the lines $AA_1, BB_1, CC_1$ meet at a single point. It is known that $A, B_1, A_1, B$ are concyclic and $B, C_1, B_1, C$ are concyclic. Prove that
a) $C, A_1, C_1, A$ are concyclic,
b) $AA_1,, BB_1, CC_1$ are the heights of $ABC$.

2003 Estonia XII p3
Let $ABC$ be a triangle with $\angle C = 90^o$ and $D$ a point on the ray $CB$ such that $|AC| \cdot |CD| = |BC|^2$. A parallel line to $AB$ through $D$ intersects the ray $CA$ at $E$. Find $\angle BEC$.

2004 Estonia IX p3
On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.

2004 Estonia IX p5
Three different circles of equal radii intersect in point $Q$. The circle $C$ touches all of them. Prove that $Q$ is the center of $C$.

2004 Estonia X p2
On side, $BC, AB$ of a parallelogram $ABCD$ lie points $M,N$ respectively such that $|AM| =|CN|$. Let $P$ be the intersection of $AM$ and $CN$. Prove that the angle bisector of $\angle APC$ passes through $D$.

2004 Estonia XI p2
Draw a line passing through a point $M$ on the angle bisector of the angle $\angle AOB$, that intersects $OA$ and $OB$ at points $K$ and $L$ respectively. Prove that the valus of the sum $\frac{1}{|OK|}+\frac{1}{|OL|}$ does not depend on the choice of the straight line passing through $M$, i.e. is defined by the size of the angle AOB and the selection of the point $M$ only.

2004 Estonia XII p1
Inside a circle, point $K$ is taken such that the ray drawn from $K$ through the centre $O$ of the circle and the chord perpendicular to this ray passing through $K$ divide the circle into three pieces with equal area. Let $L$ be one of the endpoints of the chord mentioned. Does the inequality $\angle KOL < 75^o$ hold?

2004 Estonia XII p3
Let $K, L, M$ be the basepoints of the altitudes drawn from the vertices $A, B, C$ of triangle $ABC$, respectively. Prove that $\overrightarrow{AK} + \overrightarrow{BL} + \overrightarrow{CM} = \overrightarrow{O}$ if and only if $ABC$ is equilateral.

2005 Estonia IX p1
The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of $9: 1$ and two triangles of the starting triangle with a difference of areas of $48$ cm$^2$. Find the original triangle sidelengths.

2005 Estonia X p1
Seven brothers bought a round pizza and cut it $12$ piece as shown in the figure. Of the six elder brothers, each took one piece of the shape of an equilateral triangle, the remaining $6$ edge pieces by the older brothers did not want, was given to the youngest brother. Did the youngest brother get it more or less a seal than his every older brother?

2005 Estonia XI p4
In a fixed plane, consider a convex quadrilateral $ABCD$. Choose a point $O$ in the plane and let $K, L, M$, and $N$ be the circumcentres of triangles $AOB, BOC, COD$, and $DOA$, respectively. Prove that there exists exactly one point $O$ in the plane such that $KLMN$ is a parallelogram.

2005 Estonia XII p2
Consider a convex $n$-gon in the plane with $n$ being odd. Prove that if one may find a point in the plane from which all the sides of the $n$-gon are viewed at equal angles, then this point is unique. (We say that segment $AB$ is viewed at angle $\gamma$ from point $O$ iff $\angle AOB =\gamma$ .)

2006 Estonia IX p4
Triangle $ABC$ is isosceles with $AC = BC$ and $\angle{C} = 120^o$. Points $D$ and $E$ are chosen on segment $AB$ so that $|AD| = |DE| = |EB|$. Find the sizes of the angles of triangle $CDE$.

2006 Estonia X p3
Let $AG, CH$ be the angle bisectors of a triangle $ABC$. It is known that one of the intersections of the circles of triangles $ABG$ and $ACH$ lies on the side $BC$. Prove that the angle $BAC$ is $60 ^o$

2006 Estonia XI p2
In a right triangle, the length of one side is a prime and the lengths of the other side and the hypotenuse are integral. The ratio of the triangle perimeter and the incircle diameter is also an integer. Find all possible side lengths of the triangle.

2006 Estonia XI p4
In a triangle $ABC$ with circumcentre $O$ and centroid $M$, lines $OM$ and $AM$ are perpendicular. Let $AM$ intersect the circumcircle of $ABC$ again at $A'$. Let lines $BA'$ and $AC$ intersect at D and let lines $CA'S and$AB$intersect at$E$. Prove that the circumcentre of triangle$ADE$lies on the circumcircle of$ABC$

2006 Estonia XII p4
Let $O$ be the circumcentre of an acute triangle $ABC$ and let $A', B'$ and $C'$ be the circumcentres of triangles $BCO, CAO$ and $ABO$, respectively. Prove that the area of triangle $ABC$ does not exceed the area of triangle $A'B'C'$.

2007 Estonia IX p2
Two medians drawn from vertices $A$ and $B$ of triangle $ABC$ are perpendicular. Prove that side $AB$ is the shortest side of $ABC$.

2007 Estonia X p2
Two radii $OA$ and $OB$ of a circle $c$ with midpoint $O$ are perpendicular. Another circle touches $c$ in point $Q$ and the radii in points $C$ and $D$, respectively. Determine $\angle{AQC}$.

2007 Estonia XI p3
A circle passing through the endpoints of the leg $AB$ of an isosceles triangle $ABC$ intersects the base $BC$ in point $P$. A line tangent to the circle in point $B$ intersects the circumcircle of $ABC$ in point $Q$. Prove that $P$ lies on line $AQ$ if and only if $AQ$ and $BC$ are perpendicular.

2007 Estonia XII p1
Consider a cylinder and a cone with a common base such that the volume of the part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.

2007 Estonia XII p3

Does there exist an equilateral triangle

(a) on a plane

(b) in a 3-dimensional space;

such that all its three vertices have integral coordinates?

source: http://www.math.olympiaadid.ut.ee/eng/