Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18.$ Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega$. Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A$, $\omega_B$, and $\omega_C$ meet in six points$-$two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC$, and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC$. The side length of the smaller equilateral triangle can be written as $\sqrt{a}-\sqrt{b}$, where $a$ and $b$ are positive integers. Find $a+b$.

Regular hexagon $P_1P_2P_3P_4P_5P_6$ has side length $2$. For $1 \le i \le 6$, let $C_i$ be a unit circle centered at $P_i$ and $\ell_i$ be one of the internal common tangents of $C_i$ and $C_{i+2}$, where $C_7 = C_1$ and $C_8 = C_2$. Assume that the lines $\{\ell_1, \ell_2, \ell_3, \ell_4, \ell_5,\ell_6\}$ bound a regular hexagon. The area of this hexagon can be expressed as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Let $\Gamma$ and $\Gamma'$ be two concentric circles. Let $ABC$ and $A'B'C'$ be any two equilateral triangles inscribed in $\Gamma$ and $\Gamma'$ respectively. If $P$ and $P'$ are any two points on $\Gamma$ and $\Gamma'$ respectively, show that \[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]

Let the medians of the triangle $ABC$ intersect at point $M$. A line $t$ through $M$ intersects the circumcircle of $ABC$ at $X$ and $Y$ so that $A$ and $C$ lie on the same side of $t$. Prove that $BX\cdot BY=AX\cdot AY+CX\cdot CY$.

Let $ABC$ be a triangle with centroid $G$. Let $D, E, F$ be the circumcenters of triangles $BCG, CAG, ABG$. Let $X$ be the intersection of the perpendiculars from $E$ to $AB$ and from $F$ to $AC$. Prove that $DX$ bisects $EF$.

A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share 2-D square faces with neighbors ($\textit{one}$ 2-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many 2-D squares are faces of one or more of the unit $6$-cubes?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many parts can space be divided into by : a) three half-plane? b) four half-planes?

A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? $ \textbf{(A)}\ 2\text{ by }4 \qquad\textbf{(B)}\ 2\text{ by }6 \qquad\textbf{(C)}\ 2\text{ by }8 \qquad\textbf{(D)}\ 4\text{ by }4 \qquad\textbf{(E)}\ 4\text{ by }8 $

Let $P$ be an interior point of the triangle $ABC$ which is not on the median belonging to $BC$ and satisfying $\angle CAP = \angle BCP. \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}$ and $Q$ is the second point of intersection of $AP$ and the circumcircle of $ABC. \: B'Q$ intersects $CC'$ at $R$ and $B'Q$ intersects the line through $P$ parallel to $AC$ at $S.$ Let $T$ be the point of intersection of lines $B'C'$ and $QB$ and $T$ be on the other side of $AB$ with respect to $C.$ Prove that \[\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'| \]

A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is squarefree. Find $m + n + p$.

Inside the parallelogram $ABCD$ is a point $X$. Make a line that passes through point $X$ and divides the parallelogram into two parts whose areas differ from each other the most.

Let $S_1$ be a square with side $s$ and $C_1$ be the circle inscribed in it. Let $C_2$ be a circle with radius $r$ and $S_2$ be a square inscribed in it. We are told that the area of $S_1 - C_1$ is the same as the area of $C_2 - S_2.$ Which of the following numbers is closest to $s/r?$ $$ \mathrm a. ~ 1\qquad \mathrm b.~2\qquad \mathrm c. ~3 \qquad \mathrm d. ~4 \qquad \mathrm e. ~5 $$

In a trapezoid $ABCD$ with $AB\parallel CD$, the diagonals $AC$ and $BD$ meet at $O$. Let $M$ and $N$ be the centers of the regular hexagons constructed on the sides $AB$ and $CD$ in the exterior of the trapezoid. Prove that $M,O$ and $N$ are collinear.

A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.

On the diagonal $AC$ of the convex quadrilateral $ABCD$ points $P$,$Q$ are chosen such that triangles $ABD$,$PCD$ and $QBD$ are similar to each other in this order. Prove that $AQ=PC$ [i]M. Zorka[/i]

Dürer explains art history to his students. The following gothic window is examined. Where the center of the arc of $BC$ is $A$, and similarly the center of the arc of $AC$ is $B$. The question is how much is the radius of the circle (radius marked $r$ in the figure).[img]https://cdn.artofproblemsolving.com/attachments/5/c/28e5ee47005bfde7f925908b519099d5e28d91.png[/img]

Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$. (a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s. (b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.

In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.

Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$, we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$, respectively. Prove that, as point $D$ moves along the ray $AC$, the line $EF$ passes through a fixed point.

A convex quadrilateral $ ABCD$ with area $ 2002$ contains a point $ P$ in its interior such that $ PA \equal{} 24$, $ PB \equal{} 32$, $ PC \equal{} 28$, and $ PD \equal{} 45$. FInd the perimeter of $ ABCD$. $ \textbf{(A)}\ 4\sqrt {2002}\qquad \textbf{(B)}\ 2\sqrt {8465}\qquad \textbf{(C)}\ 2\left(48 \plus{} \sqrt {2002}\right)$ $ \textbf{(D)}\ 2\sqrt {8633}\qquad \textbf{(E)}\ 4\left(36 \plus{} \sqrt {113}\right)$

Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.

A circular disk is divided by $2n$ equally spaced radii($n>0$) and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is $\textbf{(A) }2n+1\qquad\textbf{(B) }2n+2\qquad\textbf{(C) }3n-1\qquad\textbf{(D) }3n\qquad \textbf{(E) }3n+1$

In the triangle $ABC$ with sides $AC = b$ and $AB = c$, the extension of the bisector of angle $A$ intersects it's circumcircle at point with $W$. Circle $\omega$ with center at $W$ and radius $WA$ intersects lines $AC$ and $AB$ at points $D$ and $F$, respectively. Calculate the lengths of segments $CD$ and $BF$. (Evgeny Svistunov) [img]https://cdn.artofproblemsolving.com/attachments/7/e/3b340afc4b94649992eb2dccda50ca8f3f7d1d.png[/img]

A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line).