Three circles lie inside a square. Each of them touches externally two remaining circles. Also each circle touches two sides of the square. Prove that two of these circles are congruent.

A cyclic quadrilateral $ABCD$ is given. The lines $AB$ and $DC$ meet at point $E$, and the lines $BC$ and $AD$ meet at point $F$. Let $I$ be the incenter of triangle $AED$, and a ray with origin $F$ be perpendicular to the bisector of angle AID. In which ratio this ray dissects the angle $AFB$?

Let $AL$ be the bisector of triangle $ABC$, $D$ be its midpoint, and $E$ be the projection of $D$ to $AB$. It is known that $AC = 3AE$. Prove that $CEL$ is an isosceles triangle.

Let $ABCD$ be a cyclic quadrilateral. A point $P$ moves along the arc $AD$ which does not contain $B$ and $C$. A fixed line $l$, perpendicular to $BC$, meets the rays $BP$, $CP$ at points $B_0$, $C_0$ respectively. Prove that the tangent at $P$ to the circumcircle of triangle $PB_0C_0$ passes through some fixed point.

The vertex $C$ of equilateral triangles $ABC$ and $CDE$ lies on the segment $AE$, and the vertices $B$ and $D$ lie on the same side with respect to this segment. The circumcircles of these triangles centered at $O_1$ and $O_2$ meet for the second time at point $F$. The lines $O_1O_2$ and $AD$ meet at point $K$. Prove that $AK = BF$.

Let $CH$ be the altitude of a right-angled triangle $ABC$ ($\angle C = 90^{\circ}$) with $BC = 2AC$. Let $O_1$, $O_2$ and $O$ be the incenters of triangles $ACH$, $BCH$ and $ABC$ respectively, and $H_1$, $H_2$, $H_0$ be the projections of $O_1$, $O_2$, $O$ respectively to $AB$. Prove that $H_1H = HH_0 = H_0H_2$.

Let $E$ be a common point of circles $\omega _1$ and $\omega _2$. Let $AB$ be a common tangent to these circles, and $CD$ be a line parallel to $AB$, such that $A$ and $C$ lie on $\omega _1$, $B$ and $D$ lie on $\omega _2$. The circles $ABE$ and $CDE$ meet for the second time at point $F$. Prove that $F$ bisects one of arcs $CD$ of circle $CDE$.

Restore a triangle $ABC$ by the Nagel point, the vertex $B$ and the foot of the altitude from this vertex.

A square is inscribed into an acute-angled triangle: two vertices of this square lie on the same side of the triangle and two remaining vertices lies on two remaining sides. Two similar squares are constructed for the remaining sides. Prove that three segments congruent to the sides of these squares can be the sides of an acute-angled triangle.

In the plane, $2018$ points are given such that all distances between them are different. For each point, mark the closest one of the remaining points. What is the minimal number of marked points?

Let $I$ be the incenter of a nonisosceles triangle $ABC$. Prove that there exists a unique pair of points $M$, $N$ lying on the sides $AC$, $BC$ respectively, such that $\angle AIM = \angle BIN$ and $MN|| AB$.

Let $BD$ be the external bisector of a triangle $ABC$ with $AB > BC$; $K$ and $K_1$ be the touching points of side $AC$ with the incircle and the excircle centered at $I$ and $I_1$ respectively. The lines $BK$ and $DI_1$ meet at point $X$, and the lines $BK_1$ and $DI$ meet at point $Y$. Prove that $XY \perp AC$.

Let $ABCD$ be a cyclic quadrilateral, and $M$, $N$ be the midpoints of arcs $AB$ and $CD$ respectively. Prove that $MN$ bisects the segment between the incenters of triangles $ABC$ and $ADC$.

Let $ABC$ be a right-angled triangle with $\angle C = 90^{\circ}$, $K$, $L$, $M$ be the midpoints of sides $AB$, $BC$, $CA$ respectively, and $N$ be a point of side $AB$. The line $CN$ meets $KM$ and $KL$ at points $P$ and $Q$ respectively. Points $S$, $T$ lying on $AC$ and $BC$ respectively are such that $APQS$ and $BPQT$ are cyclic quadrilaterals. Prove that a) if $CN$ is a bisector, then $CN$, $ML$ and $ST$ concur; b) if $CN$ is an altitude, then $ST$ bisects $ML$.

The altitudes $AH_1,BH_2,CH_3$ of an acute-angled triangle $ABC$ meet at point $H$. Points $P$ and $Q$ are the reflections of $H_2$ and $H_3$ with respect to $H$. The circumcircle of triangle $PH_1Q$ meets for the second time $BH_2$ and $CH_3$ at points $R$ and $S$. Prove that $RS$ is a medial line of triangle $ABC$.

Let $ABC$ be a triangle with $AB < BC$. The bisector of angle $C$ meets the line parallel to $AC$ and passing through $B$, at point $P$. The tangent at $B$ to the circumcircle of $ABC$ meets this bisector at point $R$. Let $R'$ be the reflection of $R$ with respect to $AB$. Prove that $\angle R'P B = \angle RPA$.

Let each of circles $\alpha, \beta, \gamma$ touches two remaining circles externally, and all of them touche a circle $\Omega$ internally at points $A_1, B_1, C_1$ respectively. The common internal tangent to $\alpha$ and $\beta$ meets the arc $A_1B_1$ not containing $C_1$ at point $C_2$. Points $A_2$, $B_2$ are defined similarly. Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ concur.

Let $C_1, A_1, B_1$ be points on sides $AB, BC, CA$ of triangle $ABC$, such that $AA_1, BB_1, CC_1$ concur. The rays $B_1A_1$ and $B_1C_1$ meet the circumcircle of the triangle at points $A_2$ and $C_2$ respectively. Prove that $A, C$, the common point of $A_2C_2$ and $BB_1$ and the midpoint of $A_2C_2$ are concyclic.

Let a triangle $ABC$ be given. On a ruler three segment congruent to the sides of this triangle are marked. Using this ruler construct the orthocenter of the triangle formed by the tangency points of the sides of $ABC$ with its incircle.

Let the incircle of a nonisosceles triangle $ABC$ touch $AB$, $AC$ and $BC$ at points $D$, $E$ and $F$ respectively. The corresponding excircle touches the side $BC$ at point $N$. Let $T$ be the common point of $AN$ and the incircle, closest to $N$, and $K$ be the common point of $DE$ and $FT$. Prove that $AK||BC$.

In the plane a line $l$ and a point $A$ outside it are given. Find the locus of the incenters of acute-angled triangles having a vertex $A$ and an opposite side lying on $l$.

Six circles of unit radius lie in the plane so that the distance between the centers of any two of them is greater than $d$. What is the least value of $d$ such that there always exists a straight line which does not intersect any of the circles and separates the circles into two groups of three?

The plane is divided into convex heptagons with diameters less than 1. Prove that an arbitrary disc with radius 200 intersects most than a billion of them.

A crystal of pyrite is a parallelepiped with dashed faces. The dashes on any two adjacent faces are perpendicular. Does there exist a convex polytope with the number of faces not equal to 6, such that its faces can be dashed in such a manner?

The incircle of a right-angled triangle $ABC$ ($\angle C = 90^\circ$) touches $BC$ at point $K$. Prove that the chord of the incircle cut by line $AK$ is twice as large as the distance from $C$ to that line.

A rectangle $ABCD$ and its circumcircle are given. Let $E$ be an arbitrary point on the minor arc $BC$. The tangent to the circle at $B$ meets $CE$ at point $G$. The segments $AE$ and $BD$ meet at point $K$. Prove that $GK$ and $AD$ are perpendicular.

Let $ABC$ be a triangle with $\angle A = 60^\circ$, and $AA', BB', CC'$ be its internal angle bisectors. Prove that $\angle B'A'C' \le 60^\circ$.

Find all sets of six points in the plane, no three collinear, such that if we partition the set into two sets, then the obtained triangles are congruent.

The side $AB$ of a square $ABCD$ is the base of an isosceles triangle $ABE$ such that $AE=BE$ lying outside the square. Let $M$ be the midpoint of $AE$, $O$ be the intersection of $AC$ and $BD$. $K$ is the intersection of $OM$ and $ED$. Prove that $EK=KO$.

Suppose $ABCD$ and $A_1B_1C_1D_1$ be quadrilaterals with corresponding angles equal. Also $AB=A_1B_1$, $AC=A_1C_1$, $BD=B_1D_1$. Are the quadrilaterals necessarily congruent?

Let $\omega_1,\omega_2$ be two circles centered at $O_1$ and $O_2$ and lying outside each other. Points $C_1$ and $C_2$ lie on these circles in the same semi plane with respect to $O_1O_2$. The ray $O_1C_1$ meets $\omega _2$ at $A_2,B_2$ and $O_2C_2$ meets $\omega_1$ at $A_1,B_1$. Prove that $\angle A_1O_1B_1=\angle A_2O_2B_2$ if and only if $C_1C_2||O_1O_2$.

Let $I$ be the incenter of fixed triangle $ABC$, and $D$ be an arbitrary point on $BC$. The perpendicular bisector of $AD$ meets $BI,CI$ at $F$ and $E$ respectively. Find the locus of orthocenters of $\triangle IEF$ as $D$ varies.

Let $M$ be the midpoint of $AB$ in a right angled triangle $ABC$ with $\angle C = 90^\circ$. A circle passing through $C$ and $M$ meets segments $BC, AC$ at $P, Q$ respectively. Let $c_1, c_2$ be the circles with centers $P, Q$ and radii $BP, AQ$ respectively. Prove that $c_1, c_2$ and the circumcircle of $ABC$ are concurrent.

A triangle $ABC$ is given. A circle $\gamma$ centered at $A$ meets segments $AB$ and $AC$. The common chord of $\gamma$ and the circumcircle of $ABC$ meets $AB$ and $AC$ at $X$ and $Y$, respectively. The segments $CX$ and $BY$ meet $\gamma$ at point $S$ and $T$, respectively. The circumcircles of triangles $ACT$ and $BAS$ meet at points $A$ and $P$. Prove that $CX, BY$ and $AP$ concur.

The vertices of a triangle $DEF$ lie on different sides of a triangle $ABC$. The lengths of the tangents from the incenter of $DEF$ to the excircles of $ABC$ are equal. Prove that $4S_{DEF} \ge S_{ABC}$. Note: By $S_{XYZ}$ we denote the area of triangle $XYZ$.

Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.

Let $ABCD$ be a cyclic quadrilateral, $BL$ and $CN$ be the internal angle bisectors in triangles $ABD$ and $ACD$ respectively. The circumcircles of triangles $ABL$ and $CDN$ meet at points $P$ and $Q$. Prove that the line $PQ$ passes through the midpoint of the arc $AD$ not containing $B$.

Let $ABCD$ be a circumscribed quadrilateral. Prove that the common point of the diagonals, the incenter of triangle $ABC$ and the centre of excircle of triangle $CDA$ touching the side $AC$ are collinear.

Let $B_1,C_1$ be the midpoints of sides $AC,AB$ of a triangle $ABC$ respectively. The tangents to the circumcircle at $B$ and $C$ meet the rays $CC_1,BB_1$ at points $K$ and $L$ respectively. Prove that $\angle BAK = \angle CAL$.

Consider a fixed regular $n$-gon of unit side. When a second regular $n$-gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure. [asy][asy] int n=9; draw(polygon(n)); for (int i = 0; i<n;++i) { draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4)); draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2)); } [/asy][/asy] Let $A$ be the area of a regular $n$-gon of unit side, and let $B$ be the area of a regular $n$-gon of unit circumradius. Prove that the area enclosed by $\kappa$ equals $6A-2B$.

The altitudes $AH, CH$ of an acute-angled triangle $ABC$ meet the internal bisector of angle $B$ at points $L_1, P_1$, and the external bisector of this angle at points $L_2, P_2$. Prove that the orthocenters of triangles $HL_1P_1, HL_2P_2$ and the vertex $B$ are collinear.

A fixed circle $\omega$ is inscribed into an angle with vertex $C$. An arbitrary circle passing through $C$, touches $\omega$ externally and meets the sides of the angle at points $A$ and $B$. Prove that the perimeters of all triangles $ABC$ are equal.

A cyclic $n$-gon is given. The midpoints of all its sides are concyclic. The sides of the $n$-gon cut $n$ arcs of this circle lying outside the $n$-gon. Prove that these arcs can be coloured red and blue in such a way that the sum of the lengths of the red arcs is equal to the sum of the lengths of the blue arcs.

We say that a finite set $S$ of red and green points in the plane is separable if there exists a triangle $\delta$ such that all points of one colour lie strictly inside $\delta$ and all points of the other colour lie strictly outside of $\delta$. Let $A$ be a finite set of red and green points in the plane, in general position. Is it always true that if every $1000$ points in $A$ form a separable set then $A$ is also separable?

Let $\omega$ be the incircle of a triangle $ABC$. The line passing though the incenter $I$ and parallel to $BC$ meets $\omega$ at $A_b$ and $A_c$ ($A_b$ lies in the same semi plane with respect to $AI$ as $B$). The lines $BA_b$ and $CA_c$ meet at $A_1$. The points $B_1$ and $C_1$ are defined similarly. prove that $AA_1,BB_1,CC_1$ concur.

Let $\omega$ be the circumcircle of $ABC$, and $KL$ be the diameter of $\omega$ passing through $M$ midpoint of $AB$ ($K,C$ lies on different sides of $AB$). A circle passing through $L$ and $M$ meets $CK$ at points $P$ and $Q$ ($Q$ lies on $KP$). Let $LQ$ meet the circumcircle of $KMQ$ again at $R$. Prove that $APBR$ is cyclic.

A convex quadrilateral $ABCD$ is circumscribed about a circle of radius $r$. What is the maximum value of $\frac{1}{AC^2}+\frac{1}{BD^2}$?

Two triangles $ABC$ and $A'B'C'$ are given. The lines $AB$ and $A'B'$ meet at $C_1$ and the lines parallel to them and passing through $C$ and $C'$ meet at $C_2$. The points $A_1,A_2$, $B_1,B_2$ are defined similarly. Prove that $A_1A_2,B_1B_2,C_1C_1$ are either parallel or concurrent.