There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)

Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

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]

Given a circle, point $A$ on it and point $M$ inside it. We consider the chords $BC$ passing through $M$. Prove that the circles passing through the midpoints of the sides of all the triangles $ABC$ are tangent to a fixed circle.

Let $ABC$ be an acute and scalene triangle with $AB<AC$, inscribed in a circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ intersects side $BC$ at point $E$ and circle $c$ at point $F$. $EF$ intersects for the second time circle $c$ at point $D$ and side $AC$ at point $M$. $AD$ intersects $BC$ at point $K$. Circumcircle of triangle $BKD$ intersects $AB$ at point $L$ . Prove that points $K,L,M$ lie on a line parallel to $BF$.

The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .

In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

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.

Segment $AB$ intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between $A$ and $B$. From the point $A$ tangents to the circle nearest to $A$ are drawn, and from the point $B$ tangents to the circle nearest to $B$ are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral. (P Kozhevnikov)

In the isosceles triangle $ABC$ where $AB = AC$, the point $I{}$ is the center of the inscribed circle. Through the point $A{}$ all the rays lying inside the angle $BAC$ are drawn. For each such ray, we denote by $X{}$ and $Y{}$ the points of intersection with the arc $BIC$ and the straight line $BC$ respectively. The circle $\gamma$ passing through $X{}$ and $Y{}$, which touches the arc $BIC$ at the point $X{}$ is considered. Prove that all the circles $\gamma$ pass through a fixed point.

Given are two circles which intersect at points $P$ and $Q$. Consider an arbitrary line $\ell$ through $Q$, let the second points of intersection of this line with the circles be $A$ and $B$ respectively. Let $C$ be the point of intersection of the tangents to the circles in those points. Let $D$ be the intersection of the line $AB$ and the bisector of the angle $CPQ$. Prove that all possible $D$ for any choice of $\ell$ lie on a single circle. Alexey Zaslavsky

Given a circle of radius $K$. Two other circles, the sum of the radii of which are also equal to $K$, tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.

Two circles $(U)$ and $(V)$ intersect at $A,B$. A line d meets $(U), (V)$ at $P, Q$ and $R,S$ respectively. Let $t_P, t_Q, t_R,t_S$ be the tangents at $P,Q,R, S$ of the two circles. Another circle $(W)$ passes through through $A, B$. Prove that if the circumcircle of triangle that is formed by the intersections of $t_P,t_R, AB$ is tangent to $(W)$ then the circumcircle of triangle formed by $t_Q, t_S, AB$ is also tangent to $(W)$. Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh

$ABCD$ 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]

Let $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$ be five different points which are inside $D$ or on the border of figure $D$. Let $M=min\left\{P_iP_j \mid i \neq j\right\}$ be minimal distance between different points $P_i$. For which configuration of points $P_i$, value $M$ is at maximum, if : $a)$ $D$ is unit square $b)$ $D$ is equilateral triangle with side equal $1$ $c)$ $D$ is unit circle, circle with radius $1$

A circle touches the sides $AC$ and $AB$ of the triangle $ABC $ at the points ${{B}_ {1}} $ and ${{C}_ {1}}$ respectively. The segments $B {{B} _ {1}} $ and $C {{C} _ {1}}$ are equal. Prove that the triangle $ABC $ is isosceles. (Timoshkevich Taras)

Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.

Is it possible to place some circles inside a square side length $1$, such that no two circles intersect and the sum of their radii is $2007$?

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 $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.

Let $H$ be the orthocenter of a triangle $ABC$. Given that $H$ lies on the incircle of $ABC$ , prove that three circles with centers $A, B, C$ and radii $AH, BH, CH$ have a common tangent. (Mahdi Etesami Fard)

Let $C_1$ and $C_2$ be two different circles , and let their radii be $r_1$ and $r_2$ , the two circles are passing through the two points $A$ and $B$ (i)Let $P_1$ on $C_1$ and $P_2$ on $C_2$ such that the line $P_1P_2$ passes through $A$. Prove that $P_1B \cdot r_2 = P_2B \cdot r_1$ (ii)Let $DEF$ be a triangle that it's inscribed in $C_1$ , and let $D'E'F'$ be a triangle that is inscribed in $C_2$ . The lines $EE'$,$DD'$ and $FF'$ all pass through $A$ . Prove that the triangles $DEF$ and $D'E'F'$ are similar (iii)The circle $C_3$ also passes through $A$ and $B$ . Let $l$ be a line that passes through $A$ and cuts circles $C_i$ in $M_i$ with $i = 1,2,3$ . Prove that the value of$$\frac{M_1M_2}{M_1M_3}$$is constant regardless of the position of $l$ Provided that $l$ is different from $AB$

Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended $AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y. AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\angle XOP.$

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

$ABCD$ 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]