Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$. Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.

Four points are given $A, B, C, D$. Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD, CDA, DAB, ABC$ and $A_2, B_2, C_2,D_2$ are orthocenters of the triangles $B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1$ etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.

Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the centre $O$ lies between these chords. Suppose $AB = 6, CD = 8$. Suppose further that the area of the part of the circle lying between the chords $AB$ and $CD$ is $(m\pi + n) / k$, where $m, n, k$ are positive integers with gcd$(m, n, k) = 1$. What is the value of $m + n + k$ ?

There are four circles. The chord$ AB$ is drawn in the first one, and the distance from the midpoint of the smaller of the two formed arcs to $AB$ is equal to $1$. The second, third and fourth circles are located inside the larger segment and touch the chord $AB$. The second and fourth circles touch internally the first and externally the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle if it is known that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.

Two circles intersect in points $A$ and $B$. Their common tangent nearer $B$ touches the circles at points $E$ and $F$, and intersects the extension of $AB$ at the point $M$. The point $K$ is chosen on the extention of $AM$ so that $KM = MA$. The line $KE$ intersects the circle containing $E$ again at the point $C$. The line $KF$ intersects the circle containing $F$ again at the point $D$. Prove that the points $A, C$ and $D$ are collinear.

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let the internal angle bisectors at $A$ and $B$ meet at $X$, the internal angle bisectors at $B$ and $C$ meet at $Y$, the internal angle bisectors at $C$ and $D$ meet at $Z$, and the internal angle bisectors at $D$ and $A$ meet at $W$. Further, let $AC$ and $BD$ meet at $P$. Suppose that the points $X$, $Y$, $Z$, $W$, $O$, and $P$ are distinct. Prove that $O$, $X$, $Y$, $Z$, $W$ lie on the same circle if and only if $P$, $X$, $Y$, $Z$, and $W$ lie on the same circle.

A pentagon $ABCDE$ is inscribed in a circle $O$, and satises $AB = BC , AE = DE$. The circle that is tangent to $DE$ at $E$ and passing $A$ hits $EC$ at $F$ and $BF$ at $G (\ne F)$. Let $DG\cap O = H (\ne D)$. Prove that the tangent to $O$ at $E$ is perpendicular to $HA$.

In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.

Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.

Determine the smallest real number $a$ such that there is a square of side $a$ such that contains $5$ unit circles inside it without common interior points in pairs.

A polyline $AMB$ is inscribed in the arc $AB{}$, consisting of two segments, and $AM>MB$. Let $K$ be the midpoint of the arc $AB{}$. Prove that the foot $H{}$ of the perpendicular from $K$ onto $AM$ divides the polyline in two equal segments: \[AH=HM+MB.\][i]Discovered by Archimedes[/i]

Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.

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.

Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$

Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$. (Oleksii Masalitin)

a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$. Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$. b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$. Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.

Let $P$ be an arbitrary point in the plane of triangle $ABC$. Lines $PA, PB, PC$ meets the perpendicular bisectors of $BC,CA,AB$ at $O_a,O_b,O_c$ respectively. Let $(O_a)$ be the circle with center $O_a$ passing through two points $B,C$, two circles $(O_b), (O_c)$ are defined in the same manner. Two circles $(O_b), (O_c)$ meets at $A_1$, distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Let $Q$ be an arbitrary point in the plane of $ABC$ and $QB,QC$ meets $(O_c)$ and $(O_b)$ at $A_2,A_3$ distinct from $B,C$. Similarly, we have points $B_2,B_3,C_2,C_3$. Let $(K_a), (K_b), (K_c)$ be the circumcircles of triangles $A_1A_2A_3, B_1B_2B_3, C_1C_2C_3$. Prove that (a) three circles $(K_a), (K_b), (K_c)$ have a common point. (b) two triangles $K_aK_bK_c, ABC$ are similar. Trần Quang Hùng

Let $ABCD$ be a trapezium in which $AB$ is parallel to $CD$. The circles on $AD$ and $BC$ as diameters intersect at two distinct points $P$ and $Q$. Prove that the lines $PQ,AC,BD$ are concurrent.

Let $ABCD$ be a convex quadrilateral. Prove that the circles inscribed in the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a common point if and only if $ABCD$ is a rhombus.

Given are two concentric circles $\Omega$ and $\omega$. Each of the circles $b_1$ and $b_2$ is externally tangent to $\omega$ and internally tangent to $\Omega$, and $\omega$ each of the circles $c_1$ and $c_2$ is internally tangent to both $\Omega$ and $\omega$. Mark each point where one of the circles $b_1, b_2$ intersects one of the circles $c_1$ and $c_2$. Prove that there exist two circles distinct from $b_1, b_2, c_1, c_2$ which contain all $8$ marked points. (Some of these new circles may appear to be lines.)

Two circles $k_1,k_2$ touch each other at $P$, and touch a line $\ell$ at $A$ and $B$ respectively. Line $AP$ meets $k_2$ at $C$. Prove that $BC$ is perpendicular to $\ell$.

Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.

A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.

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

Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$