Let $ABC$ be a triangle with $A',B',C'$ are midpoints of $BC,CA,AB$ respectively. The circle $(\omega_A)$ of center $A$ has a big enough radius cuts $B'C'$ at $X_1,X_2$. Define circles $(\omega_B), (\omega_C)$ with $Y_1, Y_2,Z_1,Z_2$ similarly. Suppose that these circles have the same radius, prove that $X_1,X_2, Y_1, Y_2,Z_1,Z_2$ are concyclic.

Let $ABC$ be an acute triangle inscribed in a circle $(O)$ that is fixed, and two of the vertices $B$, $C$ are fixed while vertex $A$ varies on the circumference of the circle. Let $I$ be the center of the incircle, and $AD$ the angle bisector. Let $K$, $L$ be the circumcenters of $CAD$, $ABD$. A line through $O$ parallel to $DL$, $DK$ intersects the line that is through $I$ perpendicular to $IB$, $IC$ at $M$, $N$ respectively. Prove that $MN$ is tangent to a fixed circle when $A$ varies on the circle $(O)$. Tran Quang Hung, Natural Science High School, National University, Hanoi

There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?

In the picture below, the circles are tangent to each other and to the edges of the rectangle. The larger circle's radius equals 1. Determine the area of the rectangle. [img]https://i.imgur.com/g3GUg4Z.png[/img]

Two circles $\gamma_1$ and $\gamma_2$ cross one another at two points; let $A$ be one of these points. The tangent to $\gamma_1$ at $A$ meets again $\gamma_2$ at $B$, the tangent to $\gamma_2$ at $A$ meets again $\gamma_1$ at $C$, and the line $BC$ meets again $\gamma_1$ and $\gamma_2$ at $D_1$ and $D_2$, respectively. Let $E_1$ and $E_2$ be interior points of the segments $AD_1$ and $AD_2$, respectively, such that $AE_1 = AE_2$. The lines $BE_1$ and $AC$ meet at $M$, the lines $CE_2$ and $AB$ meet at $N$, and the lines $MN$ and $BC$ meet at $P$. Show that the line $PA$ is tangent to the circle $ABC$.

Let $ABC$ a triangle and $k$ a circle such that: (1) The circle $k$ passes through $A$ and $B$ and touches the line $AC.$ (2) The tangent to $k$ at $B$ intersects the line $AC$ in a point $X\ne C.$ (3) The circumcircle $\omega$ of $BXC$ intersects $k$ in a point $Q\ne B.$ (4) The tangent to $\omega$ at $X$ intersects the line $AB$ in a point $Y.$ Prove that the line $XY$ is tangent to the circumcircle of $BQY.$

$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$.\\ \\ If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.\\ \\ [i](Vlad Robu)[/i]

Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.

The two intersections of the circles $k_i$ and $k_{i + 1}$ are $P_i$ and $Q_i$ ($1 \le i \le 5, k_6 = k_1$). On the circle $k_1$ lies an arbitrary point $A$. Then the points $B, C, D, E, F, G, H, I, J, K$ lie on the circles $k_2, k_3, k_4, k_5, k_1, k_2, k_3, k_4, k_5, k_1$ respectively, such that $AP_1B, BP_2C, CP_3D, DP_4E, EP_5F, F Q_1G, GQ_2H, HQ_3I, IQ_4J, JQ_5K$ are straight line triplets. Prove that that $K = A$. [img]https://1.bp.blogspot.com/-g6rF1hcPE08/X9j1SEJT7-I/AAAAAAAAMzc/2rWIiWTHZ34zfWVeGujkCxRW1hSCw5oOwCLcBGAsYHQ/s16000/2016%2BDurer%2BC..4.png[/img] [i]Circles can have different radii, and They can be located in different ways from the figure. We assume that during editing none neither of the two points mentioned above coincide.[/i]

In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.

(a)Two circles $S_1$ and $S_2$ are placed so that they do not overlap each other, neither completely nor partially. They have centres in $O_1$ and $O_2$, respectively. Further, $L_1$ and $M_1$ are different points on $S_1$ so that $O_2L_1$ and $O_2M_1$ are tangent to $S_1$, and similarly $L_2$ and $M_2$ are different points on $S_2$ so that $O_1L_2$ and $O_1M_2$ are tangent to $S_2$. Show that there exists a unique circle which is tangent to the four line segments $O_2L_1, O_2M_1, O_1L_2$, and $O_1M_2$. (b) Four circles $S_1, S_2, S_3$ and $S_4$ are placed so that none of them overlap each other, neither completely nor partially. They have centres in $O_1, O_2, O_3$, and $O_4$, respectively. For each pair $(S_i, S_j )$ of circles, with $1 \le i < j \le 4$, we find a circle $S_{ij}$ as in part [b]a[/b]. The circle $S_{ij}$ has radius $R_{ij}$ . Show that $\frac{1}{R_{12}} + \frac{1}{R_{23}}+\frac{1}{R_{34}}+\frac{1}{R_{14}}= 2 \left(\frac{1}{R_{13}} +\frac{1}{R_{24}}\right)$

Determine the radius of the largest circle which can lie on the ellipsoid $$\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).$$

The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.

Circles $\omega_1$ and $\omega_2$ inscribed into equal angles $X_1OY$ and $Y OX_2$ touch lines $OX_1$ and $OX_2$ at points $A_1$ and $A_2$ respectively. Also they touch $OY$ at points $B_1$ and $B_2$. Let $C_1$ be the second common point of $A_1B_2$ and $\omega_1, C_2$ be the second common point of $A_2B_1$ and $\omega_2$. Prove that $C_1C_2$ is the common tangent of two circles.

A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.

$a)$ Prove that there exists $5$ nonnegative real numbers with sum equal to $1$, such that no matter how we arrange them on circle, two neighboring numbers exist with product not less than $\frac{1}{9}$ $a)$ Prove that for every $5$ nonnegative real numbers with sum equal to $1$, we can arrange them on circle, such that product of every two neighboring numbers is not greater than $\frac{1}{9}$