Let $L$ be the midpoint of the minor arc $AC$ of the circumcircle of an acute-angled triangle $ABC$. A point $P$ is the projection of $B$ to the tangent at $L$ to the circumcircle. Prove that $P$, $L$, and the midpoints of sides $AB$, $BC$ are concyclic.

The diagonals of a rectangle $ABCD$ meet at point $E$. A circle centered at $E$ lies inside the rectangle. Let $CF$, $DG$, $AH$ be the tangents to this circle from $C$, $D$, $A$; let $CF$ meet $DG$ at point $I$, $EI$ meet $AD$ at point $J$, and $AH$ meet $CF$ at point $L$. Prove that $LJ$ is perpendicular to $AD$.

A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$, and its center lies on $AD$. Prove that the diameter of the circle is less than the medial line of the trapezoid.

Points $D$ and $E$ lie on the lateral sides $AB$ and $BC$ respectively of an isosceles triangle $ABC$ in such a way that $\angle BED = 3\angle BDE$. Let $D'$ be the reflection of $D$ about $AC$. Prove that the line $D'E$ passes through the incenter of $ABC$.

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.

Let $A_1, B_1, C_1$ be the feet of altitudes of an acute-angled triangle $ABC$. The incircle of triangle $A_1B_1C_1$ touches $A_1B_1, A_1C_1, B_1C_1$ at points $C_2, B_2, A_2$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ concur at a point lying on the Euler line of triangle $ABC$.

Let $A$ be a fixed point of a circle $\omega$. Let $BC$ be an arbitrary chord of $\omega$ passing through a fixed point $P$. Prove that the nine-points circles of triangles $ABC$ touch some fixed circle not depending on $BC$.

A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.

It is known that the reflection of the orthocenter of a triangle $ABC$ about its circumcenter lies on $BC$. Let $A_1$ be the foot of the altitude from $A$. Prove that $A_1$ lies on the circle passing through the midpoints of the altitudes of $ABC$.

Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$. The bisectors of two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$. Prove that the circumcircles of triangles $ABC$ and $PST$ are tangent.

Let $H$ be the orthocenter of an acute-angled triangle $ABC$; $E$, $F$ be points on $AB, AC$ respectively, such that $AEHF$ is a parallelogram; $X, Y$ be the common points of the line $EF$ and the circumcircle $\omega$ of triangle $ABC$; $Z$ be the point of $\omega$ opposite to $A$. Prove that $H$ is the orthocenter of triangle $XYZ$.

Let $ABC$ be a triangle with obtuse angle $B$, and $P, Q$ lie on $AC$ in such a way that $AP = PB, BQ = QC$. The circle $BPQ$ meets the sides $AB$ and $BC$ at points $N$ and $M$ respectively. $\qquad\textbf{(a)}$ (grades 8-9) Prove that the distances from the common point $R$ of $PM$ and $NQ$ to $A$ and $C$ are equal. $\qquad\textbf{(b)}$ (grades 10-11) Let $BR$ meet $AC$ at point $S$. Prove that $MN \perp OS$, where $O$ is the circumcenter of $ABC$.

The base $AD$ of a trapezoid $ABCD$ is twice greater than the base $BC$, and the angle $C$ equals one and a half of the angle $A$. The diagonal $AC$ divides angle $C$ into two angles. Which of them is greater?

Suppose that a closed oriented polygonal line $\mathcal{L}$ in the plane does not pass through a point $O$, and is symmetric with respect to $O$. Prove that the winding number of $\mathcal{L}$ around $O$ is odd. The winding number of $\mathcal{L}$ around $O$ is defined to be the following sum of the oriented angles divided by $2\pi$: $$\deg_O\mathcal{L} := \dfrac{\angle A_1OA_2+\angle A_2OA_3+\dots+\angle A_{n-1}OA_n+\angle A_nOA_1}{2\pi}.$$

Let $ABCD$ be a convex quadrilateral. Points $X$ and $Y$ lie on the extensions beyond $D$ of the sides $CD$ and $AD$ respectively in such a way that $DX = AB$ and $DY = BC$. Similarly points $Z$ and $T$ lie on the extensions beyond $B$ of the sides $CB$ and $AB$ respectively in such a way that $BZ = AD$ and $BT = DC$. Let $M_1$ be the midpoint of $XY$, and $M_2$ be the midpoint of $ZT$. Prove that the lines $DM_1, BM_2$ and $AC$ concur.

Let $AH_A$ and $BH_B$ be the altitudes of a triangle $ABC$. The line $H_AH_B$ meets the circumcircle of $ABC$ at points $P$ and $Q$. Let $A'$ be the reflection of $A$ about $BC$, and $B'$ be the reflection of $B$ about $CA$. Prove that $A',B', P,Q$ are concyclic.

A common external tangent to circles $\omega_1$ and $\omega_2$ touches them at points $T_1, T_2$ respectively. Let $A$ be an arbitrary point on the extension of $T_1T_2$ beyond $T_1$, and $B$ be a point on the extension of $T_1T_2$ beyond $T_2$ such that $AT_1 = BT_2$. The tangents from $A$ to $\omega_1$ and from $B$ to $\omega_2$ distinct from $T_1T_2$ meet at point $C$. Prove that all nagelians of triangles $ABC$ from $C$ have a common point.

Restore a bicentral quadrilateral $ABCD$ if the midpoints of the arcs $AB,BC,CD$ of its circumcircle are given.

A cyclic quadrilateral $ABCD$ is given. An arbitrary circle passing through $C$ and $D$ meets $AC,BC$ at points $X,Y$ respectively. Find the locus of common points of circles $CAY$ and $CBX$.

Let a point $D$ lie on the median $AM$ of a triangle $ABC$. The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$. Prove that $DD'$ is parallel to $AK$, where $D'$ is isogonally conjugated to $D$ with respect to $ABC$.

Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.

Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$, and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$. The tangent to the incircle at $P$ meets $BC$ at point $X$, and the tangent to the excircle at $Q$ meets $BC$ at $Y$. Prove that $MX=MY$.

An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.

A tetrahedron $ABCD$ is give. A line $\ell$ meets the planes $ABC,BCD,CDA,DAB$ at points $D_0,A_0,B_0,C_0$ respectively. Let $P$ be an arbitrary point not lying on $\ell$ and the planes of the faces, and $A_1,B_1,C_1,D_1$ be the second common points of lines $PA_0,PB_0,PC_0,PD_0$ with the spheres $PBCD,PCDA,PDAB,PABC$ respectively. Prove $P,A_1,B_1,C_1,D_1$ lie on a circle.

Let $ABC$ be an isosceles obtuse-angled triangle, and $D$ be a point on its base $AB$ such that $AD$ equals to the circumradius of triangle $BCD$. Find the value of $\angle ACD$.

The bisectors of angles $A$, $B$, and $C$ of triangle $ABC$ meet for the second time its circumcircle at points $A_1$, $B_1$, $C_1$ respectively. Let $A_2$, $B_2$, $C_2$ be the midpoints of segments $AA_1$, $BB_1$, $CC_1$ respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

The altitudes of a parallelogram are greater than $1$. Does this yield that the unit square may be covered by this parallelogram?

Let $ABC$ be an acute-angled triangle, $O$ be its circumcenter, $BM$ be a median, and $BH$ be an altitude. Circles $AOB$ and $BHC$ meet for the second time at point $E$, and circles $AHB$ and $BOC$ meet at point $F$. Prove that $ME = MF$.

The median $CM$ and the altitude $AH$ of an acute-angled triangle $ABC$ meet at point $O$. A point $D$ lies outside the triangle in such a way that $AOCD$ is a parallelogram. Find the length of $BD$, if $MO= a$, $OC = b$.

For which $n$ the plane may be paved by congruent figures bounded by $n$ arcs of circles?

The bisector of angle $A$ of triangle $ABC$ meet its circumcircle $\omega$ at point $W$. The circle $s$ with diameter $AH$ ($H$ is the orthocenter of $ABC$) meets $\omega$ for the second time at point $P$. Restore the triangle $ABC$ if the points $A$, $P$, $W$ are given.

Two circles $\omega_1$ and $\omega_2$ meeting at point $A$ and a line $a$ are given. Let $BC$ be an arbitrary chord of $\omega_2$ parallel to $a$, and $E$, $F$ be the second common points of $AB$ and $AC$ respectively with $\omega_1$. Find the locus of common points of lines $BC$ and $EF$.

The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.

Can a regular triangle be placed inside a regular hexagon in such a way that all vertices of the triangle were seen from each vertex of the hexagon? (Point $A$ is seen from $B$, if the segment $AB$ dots not contain internal points of the triangle.)

Points $A_1$, $A_2$, $B_1$, $B_2$ lie on the circumcircle of a triangle $ABC$ in such a way that $A_1B_1 \parallel AB$, $A_1A_2 \parallel BC$, $B_1B_2 \parallel AC$. The line $AA_2$ and $CA_1$ meet at point $A'$, and the lines $BB_2$ and $CB_1$ meet at point $B'$. Prove that all lines $A'B'$ concur.

The incircle $\omega$ of a triangle $ABC$ centered at $I$ touches $BC$ at point $D$. Let $P$ be the projection of the orthocenter of $ABC$ to the median from $A$. Prove that the circle $AIP$ and $\omega$ cut off equal chords on $AD$.

A point $D$ lie on the lateral side $BC$ of an isosceles triangle $ABC$. The ray $AD$ meets the line passing through $B$ and parallel to the base $AC$ at point $E$. Prove that the tangent to the circumcircle of triangle $ABD$ at $B$ bisects $EC$.

Let $ABC$ be acute-angled triangle with circumcircle $\Gamma$. Points $H$ and $M$ are the orthocenter and the midpoint of $BC$ respectively. The line $HM$ meets the circumcircle $\omega$ of triangle $BHC$ at point $N\not= H$. Point $P$ lies on the arc $BC$ of $\omega$ not containing $H$ in such a way that $\angle HMP = 90^\circ$. The segment $PM$ meets $\Gamma$ at point $Q$. Points $B'$ and $C'$ are the reflections of $A$ about $B$ and $C$ respectively. Prove that the circumcircles of triangles $AB'C'$ and $PQN$ are tangent.

Let $H$ be the orthocenter of triangle $\mathrm T$. The sidelines of triangle $\mathrm T_1$ pass through the midpoints of $\mathrm T$ and are perpendicular to the corresponding bisectors of $\mathrm T$. The vertices of triangle $\mathrm T_2$ bisect the bisectors of $\mathrm T$. Prove that the lines joining $H$ with the vertices of $\mathrm T_1$ are perpendicular to the sidelines of $\mathrm T_2$.

Let $ABC$ be a triangle with $\angle A = 120^\circ$, $I$ be the incenter, and $M$ be the midpoint of $BC$. The line passing through $M$ and parallel to $AI$ meets the circle with diameter $BC$ at points $E$ and $F$ ($A$ and $E$ lie on the same semiplane with respect to $BC$). The line passing through $E$ and perpendicular to $FI$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. Find the value of $\angle PIQ$.

Let $M$ be the midpoint of cathetus $AB$ of triangle $ABC$ with right angle $A$. Point $D$ lies on the median $AN$ of triangle $AMC$ in such a way that the angles $ACD$ and $BCM$ are equal. Prove that the angle $DBC$ is also equal to these angles.

The Euler line of a scalene triangle touches its incircle. Prove that this triangle is obtuse-angled.

Let $\omega$ be the circumcircle of triangle $ABC$, $O$ be its center, $A'$ be the point of $\omega$ opposite to $A$, and $D$ be a point on a minor arc $BC$ of $\omega$. A point $D'$ is the reflection of $D$ about $BC$. The line $A'D'$ meets for the second time at point $E$. The perpendicular bisector to $D'E$ meets $AB$ and $AC$ at points $F$ and $G$ respectively. Prove that $\angle FOG = 180^\circ - 2\angle BAC$.

Let $ABC$ be a Poncelet triangle, $A_1$ is the reflection of $A$ about the incenter $I$, $A_2$ is isogonally conjugated to $A_1$ with respect to $ABC$. Find the locus of points $A_2$.

The incircle of a triangle $ABC$ touches $BC$ at point $D$. Let $M$ be the midpoint of arc $\widehat{BAC}$ of the circumcircle, and $P$, $Q$ be the projections of $M$ to the external bisectors of angles $B$ and $C$ respectively. Prove that the line $PQ$ bisects $AD$.

Let $E$ be the projection of the vertex $C$ of a rectangle $ABCD$ to the diagonal $BD$. Prove that the common external tangents to the circles $AEB$ and $AED$ meet on the circle $AEC$.

There are $43$ points in the space: $3$ yellow and $40$ red. Any four of them are not coplanar. May the number of triangles with red vertices hooked with the triangle with yellow vertices be equal to $2023$? Yellow triangle is hooked with the red one if the boundary of the red triangle meet the part of the plane bounded by the yellow triangle at the unique point. The triangles obtained by the transpositions of vertices are identical.

A triangle $ABC$ is given. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be circles centered at points $X$, $Y$, $Z$, $T$ respectively such that each of lines $BC$, $CA$, $AB$ cuts off on them four equal chords. Prove that the centroid of $ABC$ divides the segment joining $X$ and the radical center of $\omega_2$, $\omega_3$, $\omega_4$ in the ratio $2:1$ from $X$.