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Estonia Open 1993 - 2020 128p

geometry problems from Estonian Open Round
with aops links in the names


1993 - 2020

clarification no x stands for the school year x-1 to x, 
e.g. 1997 stands for 1996-1997 (1 and 2 stand for 1st and 2nd competition)


Junior
1995 Estonia Open Junior 1.2
Two circles of equal radius intersect at two distinct points $A$ and $B$. Let their radii $r$ and their midpoints respectively be $O_1$ and $O_2$. Find the greatest possible value of the area of the rectangle $O_1AO_2B$.

The midpoint of the hypotenuse $AB$ of the right triangle $ABC$ is $K$. The point $M$ on the side $BC$ is taken such that $BM = 2 \cdot MC$. Prove that $\angle BAM = \angle  CKM$.

A rectangle, whose one sidelength is twice the other side, is inscribed inside a triangles with sides $3$ cm, $4$ cm and $5$ cm, such that  the long sides lies entirely on the long side of the triangle. The other two remaining vertices of the rectangle lie respectively on the other two sides of the triangle. Find the lengths of the sides of this rectangle.
In a trapezoid, the two non parallel sides and a base have length $1$, while the other base and both the diagonals have length $a$. Find the value of $a$.
A pentagon (not necessarily convex) has all sides of length $1$ and its product of cosine of any four angles equal to zero. Find all possible values of the area of such a pentagon.

Juku invented an apparatus that can divide any segment into three equal segments. How can you find the midpoint of any segment, using only the Juku made, a ruler and pencil?

Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$  at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.

The points $E$ and $F$ divide the diagonal $BD$ of the convex quadrilateral $ABCD$ into three equal parts, i.e. $| BE | = | EF | = | F D |$. Line $AE$ interects side $BC$ at $X$ and line $AF$ intersects $DC$ at $Y$. Prove that:
a) if $ABCD$ is parallelogram then $X ,Y$ are the midpoints of $BC, DC$, respectively,
b) if the points $X , Y$ are the midpoints of $BC, DC$, respectively , then $ABCD$ is parallelogram

Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.

On the plane there are two non-intersecting circles with equal radii and with centres $O_1$ and $O_2$, line $s$ going through these centres, and their common tangent $t$. The third circle is tangent to these two circles in points $K$ and $L$ respectively, line $s$ in point $M$ and line $t$ in point $P$. The point of tangency of line $t$ and the first circle is $N$.
a) Find the length of the segment $O_1O_2$.
b) Prove that the points $M, K$ and $N$ lie on the same line

Consider a shape obtained from two equal squares with the same center.  Prove that the ratio of the area of this shape to the perimeter does not change when the squares are rotated around their center.
Find the total area of the shaded area in the figure if all circles have an equal radius $R$ and the centers of the outer circles divide into six equal parts of the middle circle.
In the plane, there is an acute angle $\angle AOB$ . Inside the angle points $C$ and $D$ are chosen so that $\angle AOC = \angle DOB$. From point $D$ the perpendicular on $OA$ intersects the ray $OC$ at point $G$ and from point C  the perpendicular on $OB$ intersects the ray $OD$ at point $H$. Prove that the points $C, D, G$ and $H$ are conlyclic.

Consider points $C_1, C_2$ on the side $AB$ of a triangle $ABC$, points $A_1, A_2$ on the side $BC$ and points $B_1 , B_2$ on the side $CA$ such that these points divide the corresponding sides to three equal parts. It is known that all the points $A_1, A_2, B_1, B_2 , C_1$ and $C_2$ are concyclic. Prove that triangle $ABC$ is equilateral.

In a triangle $ABC$, the lengths of the sides are consecutive integers and median drawn from $A$ is perpendicular to the bisector drawn from $B$. Find the lengths of the sides of triangle $ABC$.

A figure consisting of five equal-sized squares is placed as shown in a rectangle of size $7\times 8$ units. Find the side length of the squares.
Consider a point $M$ inside triangle $ABC$ such that triangles $ABM, BCM$ and $CAM$ have equal areas. Prove that $M$ is the intersection point of the medians of triangle $ABC$.

In a triangle $ABC$ we have $|AB| = |AC|$ and $\angle BAC = \alpha$. Let $P \ne B$ be a point on $AB$ and $Q$ a point on the altitude drawn from $A$ such that $|PQ| = |QC|$. Find $ \angle QPC$.

Circles with centres $O_1$ and $O_2$ intersect in two points, let one of which be $A$. The common tangent of these circles touches them respectively in points $P$ and $Q$. It is known that points $O_1, A$ and $Q$ are on a common straight line and points $O_2, A$ and $P$ are on a common straight line. Prove that the radii of the circles are equal.

Mari and Juri ordered a round pizza. Juri cut the pizza into four pieces by two straight cuts, none of which passed through the centre point of the pizza. Mari can choose two pieces not aside of these four, and Juri gets the rest two pieces. Prove that if Mari chooses the piece that covers the centre point of the pizza, she will get more pizza than Juri.

The shape of a dog kennel from above is an equilateral triangle with side length $1$ m and its corners in points $A, B$ and $C$, as shown in the picture. The chain of the dog is of length $6$ m and its end is fixed to the corner in point $A$. The dog himself is in point $K$ in a way that the chain is tight and points $K, A$ and $B$ are on the same straight line. The dog starts to move clockwise around the kennel, holding the chain tight all the time. How long is the walk of the dog until the moment when the chain is tied round the kennel at full?
Consider the points $A_1$ and $A_2$ on the side $AB$ of the square $ABCD$ taken in such a way that $|AB| = 3 |AA_1| $ and $|AB| = 4 |A_2B|$, similarly consider points $B_1$ and $B_2, C_1$ and $C_2, D_1$ and $D_2$ respectively on the sides $BC$, $CD$ and $DA$. The intersection point of straight lines $D_2A_1$ and $A_2B_1$ is $E$, the intersection point of straight lines $A_2B_1$ and $B_2C_1$ is $F$, the intersection point of straight lines $B_2C_1$ and $C_2D_1$ is $G$ and the intersection point of straight lines $C_2D_1$ and $D_2A_1$ is $H$. Find the area of the square $EFGH$, knowing that the area of $ABCD$ is $1$.

Diameter $AB$ is drawn to a circle with radius $1$. Two straight lines $s$ and $t$ touch the circle at points $A$ and $B$, respectively. Points $P$ and $Q$ are chosen on the lines $s$ and $t$, respectively, so that the line $PQ$ touches the circle. Find the smallest possible area of the quadrangle $APQB$.

Circles $c_1$ and $c_2$ with centres $O_1$and $O_2$, respectively, intersect at points $A$ and $B$ so that the centre of each circle lies outside the other circle. Line $O_1A$ intersects circle $c_2$ again at point $P_2$ and line $O_2A$ intersects circle $c_1$ again at point $P_1$. Prove that the points $O_1,O_2, P_1, P_2$ and $B$ are concyclic

In triangle $ABC$, the midpoints of sides $AB$ and $AC$ are $D$ and $E$, respectively. Prove that the bisectors of the angles $BDE$ and $CED$ intersect at the side $BC$ if the length of side $BC$ is the arithmetic mean of the lengths of sides $AB$ and $AC$.

The vertices of the square $ABCD$ are the centers of four circles, all of which pass through the center of the square. Prove that the intersections of the circles on the square $ABCD$ sides are vertices of a regular octagon.

Let $ABCD$ be a parallelogram, $M$ the midpoint of $AB$ and$ N$ the intersection of $CD$ with the angle bisector of $ABC$. Prove that $CM$ and $BN$ are perpendicular iff $AN$ is the angle bisector of $DAB$

Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point $P$ which lies outside each circle. The first line intersects the first circle at $A$ and $A'$ and the second circle at$ B$ and $B'$, here $A$ and $B$ are closer to $P$ than $A'$ and $B'$, respectively, and $P$ lies on segment $AB$. Analogously, the second line intersects the first circle at $C$ and $C'$ and the second circle at $D$ and $D'$. Prove that the points $A, B, C, D$ are concyclic if and only if the points $A', B', C', D'$ are concyclic.

The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.

Call a scalene triangle K disguisable if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle integral if the lengths of all its sides are integers.
(a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter.
(b) Let K be an arbitrary integral disguisable triangle for which no smaller integral
disguisable triangle similar to it exists. Prove that at least two side lengths of K are
perfect squares.

The center of square $ABCD$ is $K$. The point $P$ is chosen such that $P \ne K$ and the angle $\angle APB$ is right . Prove that the line $PK$ bisects the angle between the lines $AP$ and $BP$.
Let $M$ be the intersection of the medians $ABC$ of the triangle and the midpoint of the side $BC$. $A$ line parallel to side $BC$ and passing through point $M$ intersects sides $AB$ and $AC$ at points $X$ and $Y$ respectively. Let the point of intersection of the lines $XC$ and $MB$ be $Q$ and let $P$ intersection point of the lines $YB$ and $MC$ be $P$ . Prove that the triangles $DPQ$ and $ABC$ are similar.

In a right triangle $ABC$, $K$ is the midpoint of the hypotenuse $AB$ and $M$ such a point on the $BC$ that $| B M | = 2 | MC |$. Prove that $\angle MAB = \angle MKC$.

The feet of the altitudes drawn from vertices $A$ and $B$ of an acute triangle $ABC$ are $K$ and $L$, respectively. Prove that if $|BK| = |KL|$ then the triangle $ABC$ is isosceles.

A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed,
so that the sum of the lengths of these four cords is as small as possible?

The triangle $ABC$ is $| BC | = a$ and $| AC | = b$. On the ray starting from vertex $C$ and passing the midpoint of side $AB$ , choose any point $D$ other than vertex $C$. Let $K$ and $L$ be the projections of $D$ on the lines $AC$ and $BC$, respectively, $K$ and $L$. Find the ratio $| DK | : | DL |$.

Given a convex quadrangle $ABCD$ with $|AD| = |BD| = |CD|$ and $\angle ADB = \angle  DCA$, $\angle CBD = \angle BAC$, find the sizes of the angles of the quadrangle.

On the side $BC$ of the equilateral triangle $ABC$, choose any point $D$, and on the line $AD$, take the point $E$ such that $| B A | = | BE |$. Prove that the size of the angle $AEC$ is of does not depend on the choice of point $D$, and find its size.

Consider a parallelogram $ABCD$.
a) Prove that if the incenter of the triangle $ABC$ is located on the diagonal $BD$, then the parallelogram $ABCD$ is a rhombus.
b) Is the parallelogram $ABCD$ a rhombus whenever the circumcenter of the triangle $ABC$ is located on the diagonal $BD$?

Consider the diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_4$ and $A_6A_2$ of a convex hexagon $A_1A_2A_3A_4A_5A_6$. The hexagon whose vertices are the points of intersection of the diagonals is regular. Can we conclude that the hexagon $A_1A_2A_3A_4A_5A_6$ is also regular?

A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$, whose apex $F$ is on the leg $AC$. Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$?

A hiking club wants to hike around a lake along an exactly circular route. On the shoreline they determine two points, which are the most distant from each other, and start to walk along the circle, which has these two points as the endpoints of its diameter. Can they be sure that, independent of the shape of the lake, they do not have to swim across the lake on any part of their route?

Two circles $c$ and $c'$ with centers $O$ and $O'$ lie completely outside each other. Points $A, B$, and $C$ lie on the circle $c$ and points $A', B'$, and $C$ lie on the circle $c'$ so that segment $AB\parallel A'B'$, $BC \parallel B'C'$, and $\angle ABC = \angle A'B'C'$. The lines $AA', BB$', and $CC'$ are all different and intersect in one point $P$, which does not coincide with any of the vertices of the triangles $ABC$ or $A'B'C'$. Prove that $\angle AOB =  \angle A'O'B'$.

Is it possible that the perimeter of a triangle whose side lengths are integers, is divisible by the double of the longest side length?

Inside a circle $c$ with the center $O$ there are two circles $c_1$ and $c_2$ which go through $O$ and are tangent to the circle $c$ at points $A$ and $B$ crespectively. Prove that the circles $c_1$ and $c_2$ have a common point which lies in the segment $AB$.

In an isosceles right triangle $ABC$ the right angle is at vertex $C$. On the side $AC$ points $K, L$ and on the side $BC$ points $M, N$ are chosen so that they divide the corresponding side into three equal segments. Prove that there is exactly one point $P$ inside the triangle $ABC$ such that $\angle KPL = \angle MPN = 45^o$.

In a triangle $ABC$ the midpoints of $BC, CA$ and $AB$ are $D, E$ and $F$, respectively. Prove that the circumcircles of triangles $AEF, BFD$ and $CDE$ intersect all in one point.

In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36^o$. Find all possibilities, how big the angles of this triangle can be.

In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?

Let $ABC$ be an acute triangle. The arcs $AB$ and $AC$ of the circumcircle of the triangle are reflected over the lines AB and $AC$, respectively. Prove that the two arcs obtained intersect in another point besides $A$.

Let $ABC$ be an acute-angled triangle, $H$ the point of intersection of its altitudes , and $AA'$ the diameter of the circumcircle of triangle $ABC$. Prove that the quadrilateral $HB A'C$ is a parallelogram.

A right triangle $ABC$ has the right angle at vertex $A$. Circle $c$ passes through vertices $A$ and $B$ of the triangle $ABC$ and intersects the sides $AC$ and $BC$ correspondingly at points $D$ and $E$. The line segment $CD$ has the same length as the diameter of the circle $c$. Prove that the triangle $ABE$ is isosceles.

Let $d$ be a positive number. On the parabola, whose equation has the coefficient $1$ at the quadratic term, points $A, B$ and $C$ are chosen in such a way that the difference of the $x$-coordinates of points $A$ and $B$ is $d$ and the difference of the $x$-coordinates of points $B$ and $C$ is also $d$. Find the area of the triangle $ABC$.

On the plane three different points $P, Q$, and $R$ are chosen. It is known that however one chooses another point $X$ on the plane, the point $P$ is always either closer to $X$ than the point $Q$ or closer to $X$ than the point $R$. Prove that the point $P$ lies on the line segment $QR$.

Find all possibilities: how many acute angles can there be in a convex polygon?

Let $M$ be the intersection of the diagonals of a cyclic quadrilateral $ABCD$. Find the length of $AD$, if it is known that $AB=2$ mm , $BC = 5$ mm, $AM = 4$ mm, and $\frac{CD}{CM}= 0.6$.
Medians $AD, BE$, and $CF$ of triangle $ABC$ intersect at point $M$. Is it possible that the circles with radii $MD, ME$, and $MF$
a) all have areas smaller than the area of triangle $ABC$,
b) all have areas greater than the area of triangle $ABC$,
c) all have areas equal to the area of triangle $ABC$?

Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$. Lines $AM$ and $BC$ intersect in point $N$. What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$?

A pentagon can be divided into equilateral triangles. Find all the possibilities that the sizes of the angles of this pentagon can be.

Different points $C$ and $D$ are chosen on a circle with center $O$ and diameter $AB$ so that they are on the same side of the diameter $AB$. On the diameter $AB$ is chosen a point $P$ different from the point $O$ such that the points $P, O, D, C$ are on the same circle. Prove that $\angle APC =  \angle BPD$.

A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$. The line $BC$ intersects the circle $c$ for second time at point $F$. Prove that the lines $DE$ and $EF$ are perpendicular.

The circle $\omega_2$ passing through the center $O$ of the circle $\omega_1$, is tangent to the circle $\omega_2$ at the point $A$. On the circle $\omega_2$, the point $C$ is taken so that the ray $AC$ intersects the circle $\omega_1$ for second time at point $D$, the ray $OC$ intersects the circle $\omega_1$ at point $E$ and the lines $DE$ and $AO$ are parallel. Find the size of the angle $DAE$.


Senior
Within an equilateral triangle $ABC$, take any point $P$. Let  $L, M, N$ be the projections of $P$ on sides $AB, BC, CA$ respectively. Prove that $\frac{AP}{NL}=\frac{BP}{LM}=\frac{CP}{MN}$.

1994 Estonia Open 1.4
Prove that if $\frac{AC}{BC}=\frac{AB + BC}{AC}$ in a triangle $ABC$ , then $\angle B = 2 \angle  A$ .

1994 Estonia Open 2.2
The two sides $BC$ and $CD$ of an inscribed quadrangle $ABCD$ are of equal length. Prove that the area of this quadrangle is equal to $S =\frac12 \cdot  AC^2 \cdot \sin \angle A$

We call a tetrahedron a "trirectangular " if it has a vertex (we call this is called a "right-angled" vertex) in which the planes of the three sides of the tetrahedron intersect at right angles.
Prove the "three-dimensional Pythagorean theorem":
The square of the area of the opposite face of the "right-angled" vertex of the ""trirectangular " tetrahedron is equal to the sum of the squares of the areas of three other sides of the tetrahedron .

Find all points on the plane such that the sum of the distances of each of the four lines defined by the unit square of that plane is $4$.

A unit square has a circle of radius $r$ with center at it's midpoint. The four quarter circles are centered on the vertices of the square and are tangent to the central circle  (see figure). Find the maximum and minimum possible value of the area of the striped figure in the figure  and the corresponding values of $r$ such these, the maximum and minimum are achieved.
The figure shows a square and a circle  with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$.
Let $H, K, L$ be the feet from the altitudes from vertices $A, B, C$ of the triangle $ABC$, respectively. Prove that $| AK | \cdot | BL | \cdot| CH | = | HK | \cdot | KL | \cdot | LH | = | AL | \cdot | BH | \cdot | CK | $.

Is it possible to fill space with regular tetrahedrons so that the peak of one tetrahedron does not coincide with another tetrahedron at a point other than the top?

The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is 1.
Prove that the parallelogram $ABCD$ with relation $\angle ABD + \angle DAC = 90^o$, is either a rectangle or a rhombus.

Circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively lie on a plane such that that the circle $C_2$ passes through $O_1$. The ratio of radius of circle $C_1$ to $O_1O_2$ is $\sqrt{2+\sqrt3}$.
a) Prove that the circles $C_1$ and $C_2$ intersect at two distinct points.
b) Let $A,B$ be these  points of intersection. What proportion of the area of circle is $C_1$ is the area of the  sector $AO_1B$ ?
The plane has a semicircle with center $O$ and diameter $AB$. Chord $CD$ is parallel to the diameter $AB$ and $\angle AOC =  \angle  DOB = \frac{7}{16}$ (radians). Which of the two parts it divides into a semicircle is larger area?

On the side $BC$ of the triangle $ABC$ a point $D$ different from $B$ and $C$ is chosen so that the bisectors of the angles $ACB$ and $ADB$ intersect on the side $AB$. Let $D'$ be the symmetrical point to $D$ with respect to the line $AB$. Prove that the points $C, A$ and $D'$ are on the same line

1999 Estonia Open Senior 2.3
Two right triangles are given, of which the incircle of the first triangle is the circumcircle of the second triangle. Let the areas of the triangles be $S$ and $S'$ respectively. Prove that $\frac{S}{S'} \ge 3 +2\sqrt2$

1999 Estonia Open Senior 2.5
Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

2000 Estonia Open Senior 1.3
In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.

2000 Estonia Open Senior 2.4
The diagonals of the square $ABCD$ intersect at $P$ and the midpoint of the side $AB$ is $E$. Segment $ED$ intersects the diagonal $AC$ at point $F$ and segment $EC$ intersects the diagonal $BD$ at $G$. Inside the quadrilateral $EFPG$, draw a circle of radius $r$ tangent to all the sides of this quadrilateral. Prove that $r = | EF | - | FP |$.

Points $A, B, C, D, E$ and F are given on a circle in such a way that the three chords $AB, CD$ and $EF$ intersect in one point. Express angle $EFA$ in terms of angles ABC and CDE (find all possibilities).

Let us call a convex hexagon $ABCDEF$ boring if $\angle A+ \angle C + \angle E = \angle B + \angle D + \angle F$.
a) Is every cyclic hexagon boring?
b) Is every boring hexagon cyclic?

The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.

In a triangle $ABC$ we have $\angle B = 2 \cdot \angle C$ and the angle bisector drawn from $A$ intersects $BC$ in a point $D$ such that $|AB| = |CD|$. Find $\angle A$.

Let $ABCD$ be a rhombus with $\angle DAB = 60^o$. Let $K, L$ be points on its sides $AD$ and $DC$ and $M$ a point on the diagonal $AC$ such that $KDLM$ is a parallelogram. Prove that triangle $BKL$ is equilateral.

Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$.

Consider the points $D, E$ and $F$ on the respective sides $BC, CA$ and $AB$ of the triangle $ABC$ in a way that the segments $AD, BE$ and $CF$ have a common point $P$. Let $\frac{|AP|}{|PD|}= x,$ $\frac{|BP|}{|PE|}= y$ and $\frac{|CP|}{|PF|}= z$. Prove that $xyz - (x + y + z) = 2$.

a) Does there exist a convex quadrangle $ABCD$ satisfying the following conditions
(1) $ABCD$ is not cyclic;
(2) the sides $AB, BC, CD$ and $DA$ have pairwise different lengths;
(3) the circumradii of the triangles $ABC, ADC, BAD$ and $BCD$ are equal?
b) Does there exist such a non-convex quadrangle?

Find the smallest real number $x$ for which there exist two non-congruent triangles with integral side lengths having area $x$.

On the circumcircle of triangle $ABC$, point $P$ is chosen, such that the perpendicular drawn from point $P$ to line $AC$ intersects the circle again at a point $Q$, the perpendicular drawn from point $Q$ to line $AB$ intersects the circle again at a point $R$ and the perpendicular drawn from point $R$ to line $BC$ intersects the circle again at the initial point $P$. Let $O$ be the centre of this circle. Prove that $\angle POC = 90^o$.

Two circles $c_1$ and $c_2$ with centres $O_1$ and $O_2$, respectively, are touching externally at $P$. On their common tangent at $P$, point $A$ is chosen, rays drawn from which touch the circles $c_1$ and $c_2$ at points $P_1$ and $P_2$ both different from $P$. It is known that $\angle P_1AP_2 = 120^o$ and angles $P_1AP$ and $P_2AP$ are both acute. Rays $AP_1$ and $AP_2$ intersect line $O_1O_2$ at points $G_1$ and $G_2$, respectively. The second intersection between ray $AO_1$ and $c_1$ is $H_1$, the second intersection between ray $AO_2$ and $c_2$ is $H_2$. Lines $G_1H_1$ and $AP$ intersect at $K$. Prove that if $G_1K$ is a tangent to circle $c_1$, then line $G_2A$ is tangent to circle $c_2$ with tangency point $H_2$.
Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} >  \sin \frac{\gamma}{2}$.

Let $ABC$ be an acute triangle and choose points $A_1, B_1$ and $C_1$ on sides $BC, CA$ and $AB$, respectively. Prove that if the quadrilaterals $ABA_1B_1, BCB_1C_1$ and $CAC_1A_1$ are cyclic then their circumcentres lie on the sides of $ABC$.

Four points $A, B, C, D$ are chosen on a circle in such a way that arcs $AB, BC$, and $CD$ are of the same length and the arc $DA$ is longer than these three. Line $AD$ and the line tangent to the circle at $B$ intersect at $E$. Let $F$ be the other endpoint of the diameter starting at $C$ of the circle. Prove that triangle $DEF$ is equilateral.

Three circles with centres A, B, C touch each other pairwise externally, and touch circle c from inside. Prove that if the centre of c coincideswith the orthocentre of triangle ABC, then ABC is equilateral.

Tangents $ l_1$ and $ l_2$ common to circles $ c_1$ and $ c_2$ intersect at point $ P$, whereby tangent points remain to different sides from $ P$ on both tangent lines. Through some point $ T$, tangents $ p_1$ and $ p_2$ to circle $ c_1$ and tangents $ p_3$ and $ p_4$ to circle $ c_2$ are drawn. The intersection points of $ l_1$ with lines $ p_1, p_2, p_3, p_4$ are $ A_1, B_1, C_1, D_1$, respectively, whereby the order of points on $ l_1$ is: $ A_1, B_1, P, C_1, D_1$. Analogously, the intersection points of $ l_2$ with lines $ p_1, p_2, p_3, p_4$ are $ A_2, B_2, C_2, D_2$, respectively. Prove that if both quadrangles $ A_1A_2D_1D_2$ and $ B_1B_2C_1C_2$ are cyclic then radii of $ c_1$ and $ c_2$ are equal.

Consider triangles whose each side length squared is a rational number. Is it true
that
(a) the square of the circumradius of every such triangle is rational;
(b) the square of the inradius of every such triangle is rational?

Let $O$ be the circumcentre of triangle $ABC$. Lines $AO$ and $BC$ intersect at point $D$. Let $S$ be a point on line $BO$ such that $DS  \parallel AB$ and lines $AS$ and $BC$ intersect at point $T$. Prove that if $O, D, S$ and $T$ lie on the same circle, then $ABC$ is an isosceles triangle.

Two circles are drawn inside a parallelogram $ABCD$ so that one circle is tangent to sides $AB$ and $AD$ and the other is tangent to sides $CB$ and $CD$. The circles touch each other externally at point $K$. Prove that $K$ lies on the diagonal $AC$.

Three circles in a plane have the sides of a triangle as their diameters. Prove that there is a point that is in the interior of all three circles.

Let any point $D$ be chosen on the side $BC$ of the triangle $ABC$. Let the radii of the incircles of the triangles $ABC, ABD$ and $ACD$ be $r_1, r_2$ and $r_3$. Prove that $r_1 <r_2 + r_3$.

a) An altitude of a triangle is also a tangent to its circumcircle. Prove that some angle of the triangle is larger than $90^o$ but smaller than $135^o$.
b) Some two altitudes of the triangle are both tangents to its circumcircle. Find the angles of the triangle.

Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle ABC, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$.

The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.

Consider an acute-angled triangle $ABC$ and its circumcircle.
Let $D$ be a point on the arc $AB$ which does not include point $C$ and let $A_1$ and $B_1$ be points on the lines $DA$ and $DB$, respectively, such that $CA_1 \perp DA$ and $CB_1 \perp DB$. Prove that $|AB| \ge |A_1B_1|$.

Given a triangle $ABC$ where $|BC| = a, |CA| = b$ and $|AB| = c$, prove that the equality $\frac{1}{a + b}+\frac{1}{b + c}=\frac{3}{a + b + c}$ holds if and only if $\angle ABC = 60^o$.

A square $ABCD$ lies in the coordinate plane with its vertices $A$ and $C$ lying on different coordinate axes. Prove that one of the vertices $B$ or $D$ lies on the line $y = x$ and the other one on $y = -x$.

Let $ABC$ be a triangle with integral side lengths. The angle bisector drawn from $B$ and the altitude drawn from $C$ meet at point $P$ inside the triangle. Prove that the ratio of areas of triangles $APB$ and $APC$ is a rational number.

Let $ABC$ be a triangle with median AK. Let $O$ be the circumcenter of the triangle $ABK$.
a) Prove that if $O$ lies on a midline of the triangle $ABC$, but does not coincide with its endpoints, then $ABC$ is a right triangle.
b) Is the statement still true if $O$ can coincide with an endpoint of the midsegment?

Inside a circle $c$ there are circles $c_1, c_2$ and $c_3$ which are tangent to $c$ at points $A, B$ and $C$ correspondingly, which are all different. Circles $c_2$ and $c_3$ have a common point $K$ in the segment $BC$, circles $c_3$ and $c_1$ have a common point $L$ in the segment $CA$, and circles $c_1$ and $c_2$ have a common point $M$ in the segment $AB$. Prove that the circles $c_1, c_2$ and $c_3$ intersect in the center of the circle $c$.

Circles $c_1, c_2$ with centers $O_1, O_2$, respectively, intersect at points $P$ and $Q$ and touch circle c internally at points $A_1$ and $A_2$, respectively. Line $PQ$ intersects circle c at points $B$ and $D$. Lines $A_1B$ and $A_1D$ intersect circle $c_1$ the second time at points $E_1$ and $F_1$, respectively, and lines $A_2B$ and $A_2D$ intersect circle $c_2$ the second time at points$ E_2$ and $F_2$, respectively. Prove that $E_1, E_2, F_1, F_2$ lie on a circle whose center coincides with the midpoint of line segment $O_1O_2$.

In a plane there is a triangle $ABC$. Line $AC$ is tangent to circle $c_A$ at point $C$ and circle $c_A$ passes through point $B$. Line $BC$ is tangent to circle $c_B$ at point $C$ and circle $c_B$ passes through point $A$. The second intersection point $S$ of circles $c_A$ and $c_B$ coincides with the incenter of triangle $ABC$. Prove that the triangle $ABC$ is equilateral.

The angles of a triangle are $22.5^o, 45^o$ and $112.5^o$. Prove that inside this triangle there exists a point that is located on the median through one vertex, the angle bisector through another vertex and the altitude through the third vertex.

Let $ABC$ be a triangle. Let $K, L$ and $M$ be points on the sides $BC, AC$ and $AB$, respectively, such that $\frac{|AM|}{|MB|}\cdot  \frac{|BK|}{|KC|}\cdot  \frac{|CL|}{|LA|} = 1$. Prove that it is possible to choose two triangles out of $ALM, BMK, CKL$ whose inradii sum up to at least the inradius of triangle $ABC$.

The triangle $K_2$ has as its vertices the feet of the altitudes of a non-right triangle $K_1$. Find all possibilities for the sizes of the angles of $K_1$ for which the triangles $K_1$ and $K_2$ are similar.

The bisector of the angle $A$ of the triangle $ABC$ intersects the side $BC$ at $D$. A circle $c$ through the vertex $A$ touches the side $BC$ at $D$. Prove that the circumcircle of the triangle $ABC$ touches the circle $c$ at $A$.

The circumcentre of an acute triangle $ABC$ is $O$. Line $AC$ intersects the circumcircle of $AOB$ at a point $X$, in addition to the vertex $A$. Prove that the line $XO$ is perpendicular to the line $BC$.

On the sides $BC, CA$ and $AB$ of triangle $ABC$, respectively, points $D, E$ and $F$ are chosen. Prove that
$\frac12 (BC + CA + AB)<AD + BE + CF<\frac 32 (BC + CA + AB)$.

The bisector of the exterior angle at vertex $C$ of the triangle $ABC$ intersects the bisector of the interior angle at vertex $B$ in point $K$. Consider the diameter of the circumcircle of the triangle $BCK$ whose one endpoint is $K$. Prove that $A$ lies on this diameter.

Is there an equilateral triangle in the coordinate plane, both coordinates of each vertex of which are integers?

The midpoints of the sides $BC, CA$, and $AB$ of triangle $ABC$ are $D, E$, and $F$, respectively. The reflections of centroid $M$ of $ABC$ around points $D, E$, and $F$ are $X, Y$, and $Z$, respectively. Segments $XZ$ and $YZ$ intersect the side $AB$ in points $K$ and $L$, respectively. Prove that $AL = BK$.

The lengths of all sides of a right triangle are integers. The length of one leg is an odd prime $p$. Find the lengths of the other two sides of this triangle in terms of $p$.

Let $A'$ be the result of reflection of vertex $A$ of triangle ABC through line $BC$ and let $B'$ be the result of reflection of vertex $B$ through line $AC$. Given that $\angle BA' C = \angle BB'C$, can the largest angle of triangle $ABC$ be located:
a) At vertex $A$,
b) At vertex $B$,
c) At vertex $C$?

Juri and Mari play the following game. Juri starts by drawing a random triangle on a piece of paper. Mari then draws a line on the same paper that goes through the midpoint of one of the midsegments of the triangle. Then Juri adds another line that also goes through the midpoint of the same midsegment. These two lines divide the triangle into four pieces. Juri gets the piece with maximum area (or one of those with maximum area) and the piece with minimum area (or one of those with minimum area), while Mari gets the other two pieces. The player whose total area is bigger wins. Does either of the players have a winning strategy, and if so, who has it?

Polygon $A_0A_1...A_{n-1}$ satisfies the following:
$\bullet$ $A_0A_1 \le A_1A_2 \le  ...\le  A_{n-1}A_0$ and
$\bullet$ $\angle A_0A_1A_2 = \angle A_1A_2A_3 =  ... = \angle A_{n-2}A_{n-1}A_0$ (all angles are internal angles).
Prove that this polygon is regular.

The plane has a circle $\omega$ and a point $A$ outside it. For any point $C$, the point $B$ on the circle $\omega$ is defined such that $ABC$ is an equilateral triangle with vertices $A, B$ and $C$ listed clockwise. Prove that if point $B$ moves along the circle $\omega$, then point $C$ also moves along a circle.

A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$ . The line $BC$ intersects the circle $c$ for second time at point $F$. The point $G$ on the circle $c$ is chosen such that $| F B | = | FG |$ and $B \ne  G$. Prove that the lines $AB, EF$ and $DG$ intersect at one point.

The bisector of the interior angle at the vertex $B$ of the triangle $ABC$ and the perpendicular line on side $BC$ passing through the vertex $C$ intersects at $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $BD$, respectively, with $N$ on the side $AC$. Find all possibilities of the angles of the triangles $ABC$, if it is known that $\frac{| AM |}{| BC |}=\frac{|CD|}{|BD|}$.


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

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