In the accompanying figure, semicircles with centers$ A$ and $B$ have radii $4$ and $2$, respectively. Furthermore, they are internally tangent to the circle of diameter $PQ$. Also the semicircles with centers $ A$ and $ B$ are externally tangent to each other. The circle with center $C$ is internally tangent to the semicircle with diameter $PQ$ and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center $C$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/281b335f6a2d6230a5b79060e6d85d6ca6f06c.png[/img]

Square ABCD has side length 7. Let $A_1$, $B_1$, $C_1$, and $D_1$ be points on rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, where each point is $3$ units from the end of the ray so that $A_1B_1C_1D_1$ forms a second square as shown. SImilarly, let $A_2$, $B_2$, $C_2$, and $D_2$ be points on segments $A_1B_1$, $B_1C_1$, $C_1D_1$, and $D_1A_1$, respectively, forming another square where $A_2$ divides segment $A_1B_1$ into two pieces whose lengths are in the same ratio as $AA_1$ is to $A_1B$. Continue this process to construct square $A_nB_nC_nD_n$ for each positive integer $n$. Find the total of all the perimeters of all the squares. [asy] size(180); pair[] A={(-1,-1),(-1,1),(1,1),(1,-1),(-1,-1)}; string[] X={"A","B","C","D"}; for(int k=0;k<10;++k) { for(int m=0;m<4;++m) { if(k==0) label("$"+X[m]+"$",A[m],A[m]); if(k==1) label("$"+X[m]+"_1$",A[m],A[m]); draw(A[m]--A[m+1]); A[m]+=3/7*(A[m+1]-A[m]); } A[4]=A[0]; }[/asy]

The angle bisectors $AA_1$ and $BB_1$ of the triangle ABC intersect at point $O$. Prove that when the angle $C$ is equal to $60^0$, then $OA_1=OB_1$

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.

Points $A_1$ and $C_1$ are on $BC$ and $AB$ of acute-angled triangle $ABC$ . $AA_1$ and $CC_1$ intersect in $K$. Circumcircles of $AA_1B,CC_1B$ intersect in $P$ - incenter of $AKC$. Prove, that $P$ - orthocenter of $ABC$

Which convex domains (figures) on a plane can contain an entire straight line? It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then the area of $\triangle AEF$ is $\textbf{(A) }16\qquad\textbf{(B) }24\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad \textbf{(E) }48$

Circles $\Omega$ and $\Gamma$ meet at points $A$ and $B$. The line containing their centres intersects $\Omega$ and $\Gamma$ at point $P$ and $Q$, respectively, such that these points lie on same side of the line $AB$ and point $Q$ is closer to $AB$ than point $P$. The circle $\delta$ lies on the same side of the line $AB$ as $P$ and $Q$, touches the segment $AB$ at point $D$ and touches $\Gamma$ at point $T$. The line $PD$ meets $\delta$ and $\Omega$ again at points $K$ and $L$, respectively. Prove that $\angle QTK=\angle DTL$

Let $O$ be the circumcenter of triangle $ABC$: Points $D$ and $E$ are chosen at sides $AB$ and $AC$ respectively such that $\angle ADO = \angle AEO = 60^o$ and $BDEC$ is inscribed quadrangle. Prove or disprove that $ABC$ is isosceles triangle.

2. Let [i]ABCD [/i]be a square and let $\Gamma$ denote the circle with diameter [i]CD[/i]. A tangent line is drawn to the circle $\Gamma$ from [b][i]B[/i][/b], meeting the circle $\Gamma$ at [i]E[/i] and intersecting the segment [i]AD[/i] at [i]K[/i]. Prove that |[i]AD[/i]| = 4 |[i]KD[/i]|.

The dimensions of a rectangle $R$ are $a$ and $b$, $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, $x < a$, $y < a$, so that its perimeter is one-third that of $R$, and its area is one-third that of $R$. The number of such (different) rectangles is: $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{infinitely many}$

For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$? (NB Vassiliev)

Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.

Let $ABC$ be a triangle and $A_1$, $B_1$, $C_1$ the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Prove that if $M$ is a point in the plane of the triangle such that \[ \frac{MA}{MA_1} = \frac{MB}{MB_1} = \frac{MC}{MC_1} = 2 , \] then $M$ is the centroid of the triangle.

Median $AM$ and the angle bisector $CD$ of a right triangle $ABC$ ($\angle B=90^o$) intersect at the point $O$. Find the area of the triangle $ABC$ if $CO=9, OD=5$.

What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.) $ \textbf{(A) }2\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }5\qquad \textbf{(E) }6\qquad $

Let $\Gamma_1, \Gamma_2$ be circles that touch at a point $A$, and $\Gamma_2$ is inside $\Gamma_1$. Let $B$ be on $\Gamma_2$, and let $AB$ intersect $\Gamma_1$ on $C$. Let $D$ be on $\Gamma_1$ and $P$ be on the line $CD$ (may be outside of the segment $CD$). $BP$ intersects $\Gamma_2$ at $Q$. Prove that $A,D,P,Q$ lie on a circle.

Among the $5$-sided polygons, as many vertices as possible collinear , that is, belonging to a single line, is three, as shown below. What is the largest number of collinear vertices a $12$-sided polygon can have? [img]https://cdn.artofproblemsolving.com/attachments/1/1/53d419efa4fc4110730a857ae6988fc923eb13.png[/img] Attention: In addition to drawing a $12$-sided polygon with the maximum number of vertices collinear , remember to show that there is no other $12$-sided polygon with more vertices collinear than this one.

In an acute triangle $ABC$, point $H$ is the intersection point of altitude $CE$ to $AB$ and altitude $BD$ to $AC$. A circle with $DE$ as its diameter intersects $AB$ and $AC$ at $F$ and $G$, respectively. $FG$ and $AH$ intersect at point $K$. If $BC=25$, $BD=20$, and $BE=7$, find the length of $AK$.

Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)

Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and has diameter $k$ ($1 < k < 3$); $C_3$ has center $A$ and diameter $2k$. We regard $k$ as fixed. Now consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$, one endpoint $Y$ on $C_3$, and contain the point $B$. For what ratio $XB/BY$ will the segment $XY$ have minimal length?

Triangle $ABC$ has incenter $I$ and circumcenter $O$. $AI, BI, CI$ intersect the circumcircle of $ABC$ again at $M_A, M_B, M_C$, respectively. Show that the Euler line of $BIC$ passes through the circumcenter of $OM_BM_C$. (houkai)

What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ?

Let $ABCD$ be a cyclic quadrilateral so that arcs $AB$ and $BC$ are equal. Given that $AD = 6, BD = 4$ and $CD = 1$, compute $AB$.