Let $\Gamma_1$ and $\Gamma_2$ be circles in the plane with centers $O_1$ and $O_2$ and radii $13$ and $10$, respectively. Assume $O_1O_2=2$. Fix a circle $\Omega$ with radius $2$, internally tangent to $\Gamma_1$ at $P$ and externally tangent to $\Gamma_2$ at $Q$ . Let $\omega$ be a second variable circle internally tangent to $\Gamma_1$ at $X$ and externally tangent to $\Gamma_2$ at $Y$. Line $PQ$ meets $\Gamma_2$ again at $R$, line $XY$ meets $\Gamma_2$ again at $Z$, and lines $PZ$ and $XR$ meet at $M$. As $\omega$ varies, the locus of point $M$ encloses a region of area $\tfrac{p}{q} \pi$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$. [i]Proposed by Michael Kural[/i]

Let $ABC$ be an acute triangle and let $k_{1},k_{2},k_{3}$ be the circles with diameters $BC,CA,AB$, respectively. Let $K$ be the radical center of these circles. Segments $AK,CK,BK$ meet $k_{1},k_{2},k_{3}$ again at $D,E,F$, respectively. If the areas of triangles $ABC,DBC,ECA,FAB$ are $u,x,y,z$, respectively, prove that \[u^{2}=x^{2}+y^{2}+z^{2}.\]

Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.

The points $A, B, C, D$ lie in this order on a circle $o$. The point $S$ lies inside $o$ and has properties $\angle SAD=\angle SCB$ and $\angle SDA= \angle SBC$. Line which in which angle bisector of $\angle ASB$ in included cut the circle in points $P$ and $Q$. Prove $PS =QS$.

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

Let $ A$ and $ B$ be two Abelian groups, and define the sum of two homomorphisms $ \eta$ and $ \chi$ from $ A$ to $ B$ by \[ a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ .\] With this addition, the set of homomorphisms from $ A$ to $ B$ forms an Abelian group $ H$. Suppose now that $ A$ is a $ p$-group ( $ p$ a prime number). Prove that in this case $ H$ becomes a topological group under the topology defined by taking the subgroups $ p^kH \;(k\equal{}1,2,...)$ as a neighborhood base of $ 0$. Prove that $ H$ is complete in this topology and that every connected component of $ H$ consists of a single element. When is $ H$ compact in this topology? [L. Fuchs]

Let $ ABC$ be a triangle; $ \Gamma_A,\Gamma_B,\Gamma_C$ be three equal, disjoint circles inside $ ABC$ such that $ \Gamma_A$ touches $ AB$ and $ AC$; $ \Gamma_B$ touches $ AB$ and $ BC$; and $ \Gamma_C$ touches $ BC$ and $ CA$. Let $ \Gamma$ be a circle touching circles $ \Gamma_A, \Gamma_B, \Gamma_C$ externally. Prove that the line joining the circum-centre $ O$ and the in-centre $ I$ of triangle $ ABC$ passes through the centre of $ \Gamma$.

Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, and let $E$ be the midpoint of its side $BC$. Suppose we can inscribe a circle into the quadrilateral $ABED$, and that we can inscribe a circle into the quadrilateral $AECD$. Denote $|AB|=a$, $|BC|=b$, $|CD|=c$, $|DA|=d$. Prove that \[a+c=\frac{b}{3}+d;\] \[\frac{1}{a}+\frac{1}{c}=\frac{3}{b}\]

Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$. [b]proposed by Sergej Berlov[/b]

Triangle $ABC$ is inscribed in circle $\Omega$ with center $O$. The incircle of $ABC$ is tangent to $BC$, $AC$, $AB$ at $D$, $E$, $F$ respectively, and its center is $I$. The reflection of the tangent line to $\Omega$ at $A$ with respect to $EF$ will be denoted $\ell_A$. We similarly define $\ell_B$, $\ell_C$. Show that the orthocenter of the triangle with sides $\ell_A$, $\ell_B$, $\ell_C$ lies on $OI$.

Let $u$ and $v$ be integers satisfying $0<v<u.$ Let $A=(u,v),$ let $B$ be the reflection of $A$ across the line $y=x,$ let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is 451. Find $u+v.$

In a plane the circles $\mathcal K_1$ and $\mathcal K_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal K_1$ in $A$ intersects $\mathcal K_2$ again in $C$ and the tangent to $\mathcal K_2$ in $A$ intersects $\mathcal K_1$ again in $D$. Let $\mathcal K_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal K_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal K_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$