Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$ is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$ $a)$ Prove that points $A$, $S$ and $T$ are collinear $b)$ Prove that $AC=AE$

Let $ABC$ be a given triangle and $\ell$ be a line that meets the lines $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$ respectively. Let $A'$ be the midpoint, of the segment connecting the projections of $A_1$ onto the lines $AB$ and $AC$. Construct, analogously the points $B'$ and $C'$. (a) Show that the points $A', B'$ and $C'$ are collinear on some line $\ell'$. (b) Show that if $\ell$ contains the circumcenter of the triangle $ABC$, then $\ell' $ contains the center of it's Euler circle.

In an acute triangle $\triangle ABC$, let there be point $D$ on segment $AC, E$ on segment $AB$ such that $\angle ADE = \angle ABC$. Let the bisector of $\angle A$ hit $BC$ at $K$. Let the foot of the perpendicular from $K$ to $DE$ be $P$, and the foot of the perpendicular from $A$ to $DE$ be $L$. Let $Q$ be the midpoint of $AL$. If the incenter of $\triangle ABC$ lies on the circumcircle of $\triangle ADE$, prove that $P,Q$ and the incenter of $\triangle ADE$ are collinear.

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.

Points $M$ and $N$ are marked on the straight line containing the side $AC$ of triangle $ABC$ so that $MA = AB$ and $NC = CB$ (the order of the points on the line: $M, A, C, N$). Prove that the center of the circle inscribed in triangle $ABC$ lies on the common chord of the circles circumscribed around triangles $MCB$ and $NAB$ .

Let $ABC$ be a triangle, $H_a, H_b$ and $H_c$ the feet of its altitudes from $A, B$ and $C$, respectively, $T_a, T_b, T_c$ its touchpoints of the incircle with the sides $BC, CA$ and $AB$, respectively. The circumcircles of triangles $AH_bH_c$ and $AT_bT_c$ intersect again at $A'$. The circumcircles of triangles $BH_cH_a$ and $BT_cT_a$ intersect again at $B'$. The circumcircles of triangles $CH_aH_b$ and $CT_aT_b$ intersect again at $C'$. Prove that the points $A',B',C'$ are collinear. Malik Talbi

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. At point $A$ to $\omega_1$ and $\omega_2$ the tangents $\ell_1$ and $\ell_2$ are drawn respectively. The points $T_1$ and $T_2$ are chosen respectively on the circles $\omega_1$ and $\omega_2$ so that the angular measures of the arcs $T_1A$ and $AT_2$ are equal (the measure of the circular arc is calculated clockwise). The tangent $t_1$ at the point $ T_1$ to the circle $\omega_1$ intersects $\ell_2$ at the point $M_1$. Similarly, the tangent $t_2$ at the point $T_2$ to the circle $\omega_2$ intersects $\ell_1$ at point $M_2$. Prove that the midpoints of the segments $M_1M_2$ are on the same a straight line that does not depend on the position of points $T_1$, $T_2$.

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.

Let $ABC$ be a triangle whose incircle $(I)$ touches $BC, CA, AB$ at $D, E, F$, respectively. The line passing through $A$ and parallel to $BC$ cuts $DE, DF$ at $M, N$, respectively. The circumcircle of triangle $DMN$ cuts $(I)$ again at $L$. a) Let $K$ be the intersection of $N E$ and $M F$. Prove that $K$ is the orthocenter of the triangle $DMN$. b) Prove that $A, K, L$ are collinear.

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

Let $A,B,P$ be points on a line $\ell$, with $P$ outside the segment $AB$. Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$. A line $m$ through $P$, which is neither parallel nor perpendicular to $\ell$, intersects $a$ and $b$ at $Q$ and $R$, respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$, and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$, respectively. (a) Prove that $P,S,T$ are collinear. (b) Prove that $P,U,V$ are collinear.

Let $(O_1), (O_2)$ be given two circles intersecting at $A$ and $B$. The tangent lines of $(O_1)$ at $A, B$ intersect at $O$. Let $I$ be a point on the circle $(O_1)$ but outside the circle $(O_2)$. The lines $IA, IB$ intersect circle $(O_2)$ at $C, D$. Denote by $M$ the midpoint of $C D$. Prove that $I, M, O$ are collinear.

Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2$, $A_2A_3$,$ ...$, $A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$. Prove that it is possible a) for $k=3$, b) for every $k$ less than $(n-1)$.

Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic. (D.Prokopenko)

Given a tetrahedron $ABCD$ and inside the tetrahedron points $K, L, M, N$ that do not lie on a plane. Denote also the centroids of $P$, $Q$, $R$, $S$ of the tetrahedrons $KBCD$, $ALCD$, $ABMD$, $ABCN$ do not lie on a plane. Let $T$ be the centroid of the tetrahedron ABCD, $T_o$ be the centroid of the tetrahedron $PQRS$ and $T_1$ be the centroid of the tetrahedron $KLMN$. a) Prove that the points $T, T_0, T_1$ lie in one straight line. b) Determine the ratio $|T_0T| : |T_0 T_1|$.

Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle that external tangent to $(O)$ at $A'$ and also tangent to the lines $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B', C', B_c , B_a, C_a, C_b$ similarly. 1. Denote J as the radical center of $(O_1), (O_2), (O_3) $and suppose that $JA'$ intersects $(O_1)$ at the second point $X, JB'$ intersects $(O_2)$ at the second point Y , JC' intersects $(O_3)$ at the second point $Z$. Prove that the circle $(X Y Z)$ is tangent to $(O_1), (O_2), (O_3)$. 2. Prove that $AA', BB', CC'$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear.

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that a) points $T, \, \, X, \, \, B$ lie on one line; b) points $K, \, \, X, \, \, H$ lie on one line. (Vadym Mitrofanov)

Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear. (The [i]excircle[/i] of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Bozhidar Dimitrov, Bulgaria[/i]

Determine the poles of the inversions that transform four collienar points $A,B, C, D$, aligned in this order, at four points $A' $, $B' $, $C'$ , $D'$ that are vertices of a rectangle, and such that $A'$ and $C'$ are opposite vertices.

Consider the set of all integer points in $Z^3$. Sasha and Masha play such a game. At first, Masha marks an arbitrary point. After that, Sasha marks all the points on some a plane perpendicular to one of the coordinate axes and at no point, which Masha noted. Next, they continue to take turns (Masha can't to select previously marked points, Sasha cannot choose the planes on which there are points said Masha). Masha wants to mark $n$ consecutive points on some line that parallel to one of the coordinate axes, and Sasha seeks to interfere with it. Find all $n$, in which Masha can achieve the desired result.

Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie. (a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it. (b) Show that in all other cases the four points thus obtained lie on one circle. (Theresia Eisenkölbl)

In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$. (Anton Trygub)

Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$. (A. Zaslavsky)

On the sides $BC$ and $CD$ of the square $ABCD$, the points $M$ and $N$ are selected in such a way that $\angle MAN= 45^o$. Using the segment $MN$, as the diameter, we constructed a circle $w$, which intersects the segments $AM$ and $AN$ at points $P$ and $Q$, respectively. Prove that the points $B, P$ and $Q$ lie on the same line.

Given a quadrilateral $ABCD$, around which you can circumscribe a circle. The perpendicular bisectors of sides $AD$ and $CD$ intersect at point $Q$ and intersect sides $BC$ and $AB$ at points $P$ and $K$ resepctively. It turned out that the points $K, B, P, Q$ lie on the same circle. Prove that the points $A, Q, C$ lie on one line. (Olena Artemchuk)