SHARYGIN 2021

Let $ABC$ be a triangle with $\angle C=90^\circ$ . A line joining the midpoint of its altitude $CH$ and the vertex $A$ meets $CB$ at point $K$ . Let $L$ be the midpoint of $BC$ ,and $T$ be a point of segment $AB$ such that $\angle ATK=\angle LTB$ . It is known that $BC=1$ . Find the perimeter of triangle $KTL$ .

A perpendicular bisector to the side $AC$ of triangle $ABC$ meets $BC,AB$ at points $A_1$ and $C_1$ respectively. Points $O,O_1$ are the circumcenters of triangles $ABC$ and $A_1BC_1$ respectively. Prove that $C_1O_1\perp AO$ .

Altitudes $AA_1,CC_1$ of acute-angles $ABC$ meet at point $H$ ; $B_0$ is the midpoint of $AC$ . A line passing through $B$ and parallel to $AC$ meets $B_0A_1 , B_0C_1$ at points $A',C'$ respectively. Prove that $AA',CC'$ and $BH$ concur.

Let $ABCD$ be a square with center $O$ , and $P$ be a point on the minor arc $CD$ of its circumcircle. The tangents from $P$ to the incircle of the square meet $CD$ at points $M$ and $N$ . The lines $PM$ and $PN$ meet segments $BC$ and $AD$ respectively at points $Q$ and $R$ . Prove that the median of triangle $OMN$ from $O$ is perpendicular to the segment $QR$ and equals to its half.

Five points are given in the plane. Find the maximum number of similar triangles whose vertices are among those five points.

Three circles $\Gamma_1,\Gamma_2,\Gamma_3$ are inscribed into an angle(the radius of $\Gamma_1$ is the minimal, and the radius of $\Gamma_3$ is the maximal) in such a way that $\Gamma_2$ touches $\Gamma_1$ and $\Gamma_3$ at points $A$ and $B$ respectively. Let $\ell$ be a tangent to $A$ to $\Gamma_1$ . Consider circles $\omega$ touching $\Gamma_1$ and $\ell$ . Find the locus of meeting points of common internal tangents to $\omega$ and $\Gamma_3$ .

The incircle of triangle $ABC$ centered at $I$ touches $CA,AB$ at points $E,F$ respectively. Let points $M,N$ of line $EF$ be such that $CM=CE$ and $BN=BF$ . Lines $BM$ and $CN$ meet at point $P$ . Prove that $PI$ bisects segment $MN$ .

Let $ABC$ be an isosceles triangle ( $AB=BC$ ) and $\ell$ be a ray from $B$ . Points $P$ and $Q$ of $\ell$ lie inside the triangle in such a way that $\angle BAP=\angle QCA$ . Prove that $\angle PAQ=\angle PCQ$ .

Points $E$ and $F$ lying on sides $BC$ and $AD$ respectively of a parallelogram $ABCD$ are such that $EF=ED=DC$ . Let $M$ be the midpoint of $BE$ and $MD$ meet $EF$ at $G$ . Prove that $\angle EAC=\angle GBD$ .

Prove that two isotomic lines of a triangle cannot meet inside its medial triangle. (Two lines are isotomic lines of triangle $ABC$ if their common points with $BC, CA, AB$ are symmetric with respect to the midpoints of the corresponding sides.)

The midpoints of four sides of a cyclic pentagon were marked, after this the pentagon was erased. Restore it.

Suppose we have ten coins with radii $1, 2, 3, \ldots , 10$ cm. We can put two of them on the table in such a way that they touch each other, after that we can add the coins in such a way that each new coin touches at least two of previous ones. The new coin cannot cover a previous one. Can we put several coins in such a way that the centers of some three coins are collinear?

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$ , point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$ . The excircle opposite to $A$ touches $\overline{BC}$ at point $E$ . Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$ . Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$ .

Let $\gamma_A, \gamma_B, \gamma_C$ be excircles of triangle $ABC$ , touching the sides $BC$ , $CA$ , $AB$ respectively. Let $l_A$ denote the common external tangent to $\gamma_B$ and $\gamma_C$ distinct from $BC$ . Define $l_B, l_C$ similarly. The tangent from a point $P$ of $l_A$ to $\gamma_B$ distinct from $l_A$ meets $l_C$ at point $X$ . Similarly the tangent from $P$ to $\gamma_C$ meets $l_B$ at $Y$ . Prove that $XY$ touches $\gamma_A$ .

Let $APBCQ$ be a cyclic pentagon. A point $M$ inside triangle $ABC$ is such that $\angle MAB = \angle MCA$ , $\angle MAC = \angle MBA$ and $\angle PMB = \angle QMC = 90^\circ$ . Prove that $AM$ , $BP$ , and $CQ$ concur.

Anant Mudgal and Navilarekallu Tejaswi

Let circles $\Omega$ and $\omega$ touch internally at point $A$ . A chord $BC$ of $\Omega$ touches $\omega$ at point $K$ . Let $O$ be the center of $\omega$ . Prove that the circle $BOC$ bisects segment $AK$ .

Let $ABC$ be an acute-angled triangle. Points $A_0$ and $C_0$ are the midpoints of minor arcs $BC$ and $AB$ respectively. A circle passing though $A_0$ and $C_0$ meet $AB$ and $BC$ at points $P$ and $S$ , $Q$ and $R$ respectively (all these points are distinct). It is known that $PQ\parallel AC$ . Prove that $A_0P+C_0S=C_0Q+A_0R$ .

Let $ABC$ be a scalene triangle, $AM$ be the median through $A$ , and $\omega$ be the incircle. Let $\omega$ touch $BC$ at point $T$ and segment $AT$ meet $\omega$ for the second time at point $S$ . Let $\delta$ be the triangle formed by lines $AM$ and $BC$ and the tangent to $\omega$ at $S$ . Prove that the incircle of triangle $\delta$ is tangent to the circumcircle of triangle $ABC$ .

A point $P$ lies inside a convex quadrilateral $ABCD$ . Common internal tangents to the incircles of triangles $PAB$ and $PCD$ meet at point $Q$ , and common internal tangents to the incircles of $PBC,PAD$ meet at point $R$ . Prove that $P,Q,R$ are collinear.

The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.)(a) Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$ . Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$ .(b) Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point.

Prove that for any triangle $\Delta$ , the circle $f(\Delta)$ is the Euler circle of $\Delta$ .

A trapezoid $ABCD$ is bicentral. The vertex $A$ , the incenter $I$ , the circumcircle $\omega$ and its center $O$ are given and the trapezoid is erased. Restore it using only a ruler.

A convex polyhedron and a point $K$ outside it are given. For each point $M$ of a polyhedron construct a ball with diameter $MK$ . Prove that there exists a unique point on a polyhedron which belongs to all such balls.

Six points in general position are given in the space. For each two of them color red the common points (if they exist) of the segment between these points and the surface of the tetrahedron formed by four remaining points. Prove that the number of red points is even.

A truncated trigonal pyramid is circumscribed around a sphere touching its bases at points $T_1, T_2$ . Let $h$ be the altitude of the pyramid, $R_1, R_2$ be the circumradii of its bases, and $O_1, O_2$ be the circumcenters of the bases. Prove that $$R_1R_2h^2 = (R_1^2-O_1T_1^2)(R_2^2-O_2T_2^2).$$

Let $ABCD$ be a convex quadrilateral. The circumcenter and the incenter of triangle $ABC$ coincide with the incenter and the circumcenter of triangle $ADC$ respectively. It is known that $AB = 1$ . Find the remaining sidelengths and the angles of $ABCD$ .

Three parallel lines $\ell_a, \ell_b, \ell_c$ pass through the vertices of triangle $ABC$ . A line $a$ is the reflection of altitude $AH_a$ about $\ell_a$ . Lines $b, c$ are defined similarly. Prove that $a, b, c$ are concurrent.

Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$ . Cockroach $A$ runs twice as slow than $B$ , and thee times as slow than $C$ . Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$ . The lines $AX$ and $BY$ meet at point $Z$ . Find the locus of centroids of triangles $ZAB$ .

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$ . Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$ . Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$ .

Points $A_1,A_2,A_3,A_4$ are not concyclic, the same for points $B_1,B_2,B_3,B_4$ . For all $i, j, k$ the circumradii of triangles $A_iA_jA_k$ and $B_iB_jB_k$ are equal. Can we assert that $A_iA_j=B_iB_j$ for all $i, j$ '?

Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB\parallel AC$ . Point $Q$ is such that $AQ = AC$ and $CQ\parallel AB$ . Segments $CP$ and $BQ$ meet at point $X$ . Prove that the circumcenter of triangle $ABC$ lies on the circle $(PXQ)$ .

Let $ABCDE$ be a convex pentagon such that angles $CAB$ , $BCA$ , $ECD$ , $DEC$ and $AEC$ are equal. Prove that $CE$ bisects $BD$ .

Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

Three cevians concur at a point lying inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form (in some order) a geometric progression. Prove that the lengths of the cevians also form a geometric progression.

A cyclic pentagon is given. Prove that the ratio of its area to the sum of the diagonals is not greater than the quarter of the circumradius.

Let $ABC$ be an acute-angled scalene triangle and $T$ be a point inside it such that $\angle ATB = \angle BTC = 120^o$ . A circle centered at point $E$ passes through the midpoints of the sides of $ABC$ . For $B, T, E$ collinear, find angle $ABC$ .

Define the distance between two triangles to be the closest distance between two vertices, one from each triangle. Is it possible to draw five triangles in the plane such that for any two of them, their distance equals the sum of their circumradii?

Let $O$ be the clrcumcenter of triangle $ABC$ . Points $X$ and $Y$ on side $BC$ are such that $AX = BX$ and $AY = CY$ . Prove that the circumcircle of triangle $AXY$ passes through the circumceuters of triangles $AOB$ and $AOC$ .

The diagonals of trapezoid $ABCD$ ( $BC\parallel AD$ ) meet at point $O$ . Points $M$ and $N$ lie on the segments $BC$ and $AD$ respectively. The tangent to the circle $AMC$ at $C$ meets the ray $NB$ at point $P$ ; the tangent to the circle $BND$ at $D$ meets the ray $MA$ at point $R$ . Prove that $\angle BOP =\angle AOR$ .

Three sidelines of on acute-angled triangle are drawn on the plane. Fyodor wants to draw the altitudes of this triangle using a ruler and a compass. Ivan obstructs him using an eraser. For each move Fyodor may draw one line through two markeed points or one circle centered at a marked point and passing through another marked point. After this Fyodor may mark an arbitrary number of points (the common points of drawn lines, arbitrary points on the drawn lines or arbitrary points on the plane). For each move Ivan erases at most three of marked point. (Fyodor may not use the erased points in his constructions but he may mark them for the second time). They move by turns, Fydors begins. Initially no points are marked. Can Fyodor draw the altitudes?

A quadrilateral $ABCD$ is circumscribed around a circle $\omega$ centered at $I$ . Lines $AC$ and $BD$ meet at point $P$ , lines $AB$ and $CD$ meet at point $£$ , lines $AD$ and $BC$ meet at point $F$ . Point $K$ on the circumcircle of triangle $E1F$ is such that $\angle IKP = 90^o$ . The ray $PK$ meets $\omega$ at point $Q$ . Prove that the circumcircle of triangle $EQF$ touches $\omega$ .