Circle $\omega_1$ has radius $7$ and center $C_1$. Circle $\omega_2$ has radius $23$ and center $C_2$ with $C_1C_2 = 34$. Let a common internal tangent of $\omega_1$ and $\omega_2$ pass through $A_1$ on $\omega_1$ and $A_2$ on $\omega_2$, and let a common external tangent of $\omega_1$ and $\omega_2$ pass through $B_1$ on $\omega_1$ and $B_2$ on $\omega_2$ such that $A_1$ and $B_1$ lie on the same side of the line $C_1C_2$. Let $P$ be the intersection of lines $A_1A_2$ and $B_1B_2$. Find the area of quadrilateral $PC_1A_2C_2$.

A regular $n$-gon has $135$ diagonals. What is the measure of its exterior angle, in degrees? (An exterior angle is the supplement of an interior angle.)

The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$. [asy] size(150); defaultpen(linewidth(0.8)); int i; for(i=5;i>=1;i=i-1) { if (floor(i/2)==i/2) { filldraw(circle(origin,4*i),white); } else { filldraw(circle(origin,4*i),red); } } [/asy]

Prove that for any convex quadrilateral there exist an inellipse and circumellipse which are homothetic. Proposed by [i]Benny Wang + Oron Wang[/i]

Let $ABC$ be an isosceles triangle right at $C$ and $P$ any point on $CB$. Let also $Q$ be the midpoint of $AB$ and $R, S$ be the points on $AP$ such that $CR$ is perpendicular to $AP$ and $|AS|=|CR|$. Prove that the $|RS| = \sqrt2 |SQ|$.

There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.

The following is the construction of the [I]twindragon[/I] fractal. $I_0$ is the solid square region with vertices $(0,0),$ $ (\tfrac{1}{2}, \tfrac{1}{2}),$ $(1,0), (\tfrac{1}{2}, -\tfrac{1}{2}).$ Recursively, the region $I_{n+1}$ is consists of two copies of $I_n:$ one copy which is rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\tfrac{1}{\sqrt{2}},$ and another copy which is also rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\tfrac{1}{\sqrt{2}}$ and translated by $(\tfrac{1}{2}, -\tfrac{1}{2}).$ We have displayed $I_0$ and $I_1$ below. Let $I_{\infty}$ be the limiting region of $I_0, I_1, \ldots.$ The area of the smallest convex polygon which encloses $I_{\infty}$ can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b.$ Find $a+b.$ [center] [img]https://cdn.artofproblemsolving.com/attachments/b/c/50b5d5a70c293329cf693bfaef823fb2813b07.png[/img] [/center]

Lines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto $m$ into two line segments of length $a$ and $b$ respectively. Determine the maximum value of $\left\lfloor \frac{1000a}{b} \right\rfloor$. (The floor notation $\lfloor x \rfloor$ denotes largest integer not exceeding $x$)

A point $(x,y)$ is a [i]lattice point[/i] if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a non-degenerate convex quadrilateral region.

In an isosceles trapezoid, the diagonals are perpendicular. Find the distance from the center of the circle described around the trapezoid to the point of intersection of its diagonals, if the lengths of the bases are equal to $a$ and $b$.

On a side of the triangle, take four points $K$, $P$, $H$ and $M$, which are respectively the midpoint of this side, the foot of the bisector with the opposite angle of the triangle, the touchpoint of this side of the circle inscribed in the triangle and the foot of the corresponding altitude. Find $KH$ if $KP = a$, $KM =b$.

Let $ t(n)$ for $ n \equal{} 3, 4, 5, \ldots,$ represent the number of distinct, incongruent, integer-sided triangles whose perimeter is $ n;$ e.g., $ t(3) \equal{} 1.$ Prove that \[ t(2n\minus{}1) \minus{} t(2n) \equal{} \left[ \frac{6}{n} \right] \text{ or } \left[ \frac{6}{n} \plus{} 1 \right].\]

There are four circles $k_1 , k_2 , k_3$ and $k_4$ of equal radius inside the triangle $ABC$. The circle $k_1$ touches the sides $AB, CA$ and the circle $k_4 $, $k_2$ touches the sides $AB,BC$ and $k_4$, and $k_3$ touches the sides $AC, BC$ and $k_4.$ Prove that the center of $k_4$ lies on the line connecting the incenter and circumcenter of $ABC.$

Three right-angled triangles have been placed in a halfplane determined by a line $\ell$, each with one leg lying on $\ell$. Assume that there is a line parallel to $\ell$ cutting the triangles in three congruent segments. Show that, if each of the triangles is rotated so that its other leg lies on $\ell$, then there still exists a line parallel to $\ell$ cutting them in three congruent segments.

Let $A,B,C,D$ be points in the space such that $|AB|=|AC|=3$, $|DB|=|DC|=5$, $|AD|=6$, and $|BC|=2$. Let $P$ be the nearest point of $BC$ to the point $D$, and $Q$ be the nearest point of the plane $ABC$ to the point $D$. What is $|PQ|$? $ \textbf{(A)}\ \frac{1}{\sqrt 2} \qquad\textbf{(B)}\ \frac{3\sqrt 7}{2} \qquad\textbf{(C)}\ \frac{57}{2\sqrt{11}} \qquad\textbf{(D)}\ \frac{9}{2\sqrt 2} \qquad\textbf{(E)}\ 2\sqrt 2 $

Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.

Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.

The incircle of a non-isosceles triangle $ABC$ touches $AB$ at point $C'$. The circle with diameter $BC'$ meets the incircle and the bisector of angle $B$ again at points $A_1$ and $A_2$ respectively. The circle with diameter $AC'$ meets the incircle and the bisector of angle $A$ again at points $B_1$ and $B_2$ respectively. Prove that lines $AB, A_1B_1, A_2B_2$ concur. (E. H. Garsia)

Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$ and perpendicular to $BC$. Let $M$ be a point on $BC$ such that $\angle AMB = \angle DMC$. If $AB = 3$, $BC = 24$, and $CD = 4$, what is the value of $AM + MD$?

Catherine lowers five matching wooden discs over bars placed on the vertices of a regular pentagon. Then she leaves five smaller congruent checkers these rods drop. Then she stretches a ribbon around the large discs and a second ribbon around the small discs. The first ribbon has a length of $56$ centimeters and the second one of $50$ centimeters. Catherine looks at her construction from above and sees an area demarcated by the two ribbons. What is the area of that area? [img]https://cdn.artofproblemsolving.com/attachments/1/0/68e80530742f1f0775aff5a265e0c9928fa66c.png[/img]

Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\). [i]Proposed by Elliott Liu[/i]

[u]Round 5[/u] [b]p13.[/b] In coordinate space, a lattice point is a point all of whose coordinates are integers. The lattice points $(x, y, z)$ in three-dimensional space satisfying $0 \le x, y, z \le 5$ are colored in n colors such that any two points that are $\sqrt3$ units apart have different colors. Determine the minimum possible value of $n$. [b]p14.[/b] Determine the number of ways to express $121$ as a sum of strictly increasing positive Fibonacci numbers. [b]p15.[/b] Let $ABCD$ be a rectangle with $AB = 7$ and $BC = 15$. Equilateral triangles $ABP$, $BCQ$, $CDR$, and $DAS$ are constructed outside the rectangle. Compute the area of quadrilateral $P QRS$. [u] Round 6[/u] Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [b]p16.[/b] Let $C$ be the answer to problem $18$. Suppose that $x$ and $y$ are real numbers with $y > 0$ and $$x + y = C$$ $$x +\frac{1}{y} = -2.$$ Compute $y +\frac{1}{y}$. [b]p17.[/b] Let $A$ be the answer to problem $16$. Let $P QR$ be a triangle with $\angle P QR = 90^o$, and let $X$ be the foot of the perpendicular from point $Q$ to segment $P R$. Given that $QX = A$, determine the minimum possible area of triangle $PQR$. [b]p18.[/b] Let $B$ be the answer to problem $17$ and let $K = 36B$. Alice, Betty, and Charlize are identical triplets, only distinguishable by their hats. Every day, two of them decide to exchange hats. Given that they each have their own hat today, compute the probability that Alice will have her own hat in $K$ days. [u]Round 7[/u] [b]p19.[/b] Find the number of positive integers a such that all roots of $x^2 + ax + 100$ are real and the sum of their squares is at most $2013$. [b]p20.[/b] Determine all values of $k$ such that the system of equations $$y = x^2 - kx + 1$$ $$x = y^2 - ky + 1$$ has a real solution. [b]p21.[/b] Determine the minimum number of cuts needed to divide an $11 \times 5 \times 3$ block of chocolate into $1\times 1\times 1$ pieces. (When a block is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.) [u]Round 8[/u] [b]p22.[/b] A sequence that contains the numbers $1, 2, 3, ... , n$ exactly once each is said to be a permutation of length $n$. A permutation $w_1w_2w_3... w_n$ is said to be sad if there are indices $i < j < k$ such that $w_j > w_k$ and $w_j > w_i$. For example, the permutation $3142756$ is sad because $7 > 6$ and $7 > 1$. Compute the number of permutations of length $11$ that are not sad. [b]p23.[/b] Let $ABC$ be a triangle with $AB = 39$, $BC = 56$, and $CA = 35$. Compute $\angle CAB - \angle ABC$ in degrees. [b]p24.[/b] On a strange planet, there are $n$ cities. Between any pair of cities, there can either be a one-way road, two one-way roads in different directions, or no road at all. Every city has a name, and at the source of every one-way road, there is a signpost with the name of the destination city. In addition, the one-way roads only intersect at cities, but there can be bridges to prevent intersections at non-cities. Fresh Mann has been abducted by one of the aliens, but Sophy Moore knows that he is in Rome, a city that has no roads leading out of it. Also, there is a direct one-way road leading from each other city to Rome. However, Rome is the secret police’s name for the so-described city; its official name, the name appearing on the labels of the one-way roads, is unknown to Sophy Moore. Sophy Moore is currently in Athens and she wants to head to Rome in order to rescue Fresh Mann, but she does not know the value of $n$. Assuming that she tries to minimize the number of roads on which she needs to travel, determine the maximum possible number of roads that she could be forced to travel in order to find Rome. Express your answer as a function of $n$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2809419p24782489]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$. Circles with diameters $AB$ and $CD$ intersect at point $P$. Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.

What can be the angle between the hour and minute hands of a clock if it is known that its value has not changed after $30$ minutes?

A convex polyhedron has two triangle and three quadrilateral faces. Connect every vertex of one of the triangle faces with the intersection point of the diagonals in the quadrilateral face opposite to it. Show that the resulting three lines are concurrent.