Let point $A$ be one of the intersections of circles $c$ and $k$. Let $X_1$ and $X_2$ be arbitrary points on circle $c$. Let $Y_i$ denote the intersection of line $AX_i$ and circle $k$ for $i=1,2$. Let $P_1$, $P_2$ and $P_3$ be arbitrary points on circle $k$, and let $O$ denote the center of circle $k$. Let $K_{ij}$ denote the center of circle $(X_iY_iP_j)$ for $i=1,2$ and $j=1,2,3$. Let $L_j$ denote the center of circle $(OK_{1j}K_{2j})$ for $j=1,2,3$. Prove that points $L_1$, $L_2$ and $L_3$ are collinear. Proposed by [i]Vilmos Molnár-Szabó[/i], Budapest

There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions? [i](Proposed by Bogdan Rublov)[/i]

Polar coordinate equation of curve $C$ is $\rho=1+\cos\theta$. Polar coordinate of point $A$ is $(2,0)$. $C$ rotate around $A$ for a whole circle, the area of the figure that $C$ swept out by is________.

Let $ABCD$ be a cyclic quadrilateral. The diagonals $AC$ and $BD$ meet at $P$, and $DA $ and $CB$ meet at $Q$. Suppose $PQ$ is perpendicular to $AC$. Let $E$ be the midpoint of $AB$. Prove that $PE$ is perpendicular to $BC$.

In triangle $ABC$, let $I$ be the incentre. Let $H$ be the orthocentre of triangle $BIC$. Show that $AH$ is parallel to $BC$ if and only if $H$ lies on the circle with diameter $AI$.

Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

A parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordinates for the $4$th vertex.

A cyclic quadrilateral $ABCD$ is given. Let $I{}$ and $J{}$ be the centers of circles inscribed in the triangles $ABC$ and $ADC$. It turns out that the points $B, I, J, D$ lie on the same circle. Prove that the quadrilateral $ABCD$ is tangential.

Which positive integers are missing in the sequence $ \left\{a_n\right\}$, with $ a_n \equal{} n \plus{} \left[\sqrt n\right] \plus{}\left[\sqrt [3]n\right]$ for all $ n \ge 1$? ($ \left[x\right]$ denotes the largest integer less than or equal to $ x$, i.e. $ g$ with $ g \le x < g \plus{} 1$.)

There are two circles with centers $O_1,O_2$ lie inside of circle $\omega$ and are tangent to it. Chord $AB$ of $\omega$ is tangent to these two circles such that they lie on opposite sides of this chord. Prove that $\angle O_1AO_2 + \angle O_1BO_2 > 90^\circ$. [i]Proposed by Iman Maghsoudi[/i]

A big $ L$ is formed as shown. What is its area? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(12pt)); draw((0,0)--(5,0)--(5,2)--(2,2)--(2,8)--(0,8)--cycle); label("5",(2.5,0),S); label("2",(5,1),E); label("2",(1,8),N); label("8",(0,4),W);[/asy]$ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

[u]Round 9[/u] [b]p25.[/b] For how many nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$ is the sum of the elements divisble by $32$? [b]p26.[/b] America declared independence in $1776$. Take the sum of the cubes of the digits of $1776$ and let that equal $S_1$. Sum the cubes of the digits of $S_1$ to get $S_2$. Repeat this process $1776$ times. What is $S_{1776}$? [b]p27.[/b] Every Golden Grahams box contains a randomly colored toy car, which is one of four colors. What is the expected number of boxes you have to buy in order to obtain one car of each color? [u]Round 10[/u] [b]p28.[/b] Let $B$ be the answer to Question $29$ and $C$ be the answer to Question $30$. What is the sum of the square roots of $B$ and $C$? [b]p29.[/b] Let $A$ be the answer to Question $28$ and $C$ be the answer to Question $30$. What is the sum of the sums of the digits of $A$ and $C$? [b]p30.[/b] Let $A$ be the answer to Question $28$ and $B$ be the answer to Question $29$. What is $A + B$? [u]Round 11[/u] [b]p31.[/b] If $x + \frac{1}{x} = 4$, find $x^6 + \frac{1}{x^6}$. [b]p32.[/b] Given a positive integer $n$ and a prime $p$, there is are unique nonnegative integers $a$ and $b$ such that $n = p^b \cdot a$ and $gcd (a, p) = 1$. Let $v_p(n)$ denote this uniquely determined $a$. Let $S$ denote the set of the first 20 primes. Find $\sum_{ p \in S} v_p \left(1 + \sum^{100}_{i=0} p^i \right)$. [b]p33. [/b] Find the maximum value of n such that $n+ \sqrt{(n - 1) +\sqrt{(n - 2) + ... +\sqrt{1}}} < 49$ (Note: there would be $n - 1$ square roots and $n$ total terms). [u]Round 12[/u] [b]p34.[/b] Give two numbers $a$ and $b$ such that $2015^a < 2015! < 2015^b$. If you are incorrect you get $-5$ points; if you do not answer you get $0$ points; otherwise you get $\max \{20-0.02(|b - a| - 1), 0\}$ points, rounded down to the nearest integer. [b]p35.[/b] Twin primes are prime numbers whose difference is $2$. Let $(a, b)$ be the $91717$-th pair of twin primes, with $a < b$. Let $k = a^b$, and suppose that $j$ is the number of digits in the base $10$ representation of $k$. What is $j^5$? If the correct answer is $n$ and you say $m$, you will receive $\max \left(20 - | \log \left(| \frac{m}{n} |\right), 0 \right)$ points, rounded down to the nearest integer. [b]p36.[/b] Write down any positive integer. Let the sum of the valid submissions (i.e. positive integer submissions) for all teams be $S$. One team will be chosen randomly, according to the following distribution: if your team's submission is $n$, you will be chosen with probability $\frac{n}{S}$ . The amount of points that the chosen team will win is the greatest integer not exceeding $\min \{K, \frac{ 10000}{S} \}$. $K$ is a predetermined secret value. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the perimeter of the right triangle whose exradius of the hypotenuse is $ 30$ ? $\textbf{(A)}\ 40 \qquad\textbf{(B)}\ 45 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$

All sides of a rectangle are odd positive integers. Prove that there does not exist a point inside the rectangle whose distance to each of the vertices is an integer.

Let $AA_1$, $BB_1$ and $CC_1$ be altitudes of triangle $ABC$ and let $A_1A_2$, $B_1B_2$ and $C_1C_2$ be diameters of Euler circle of triangle $ABC$. Prove that lines $AA_2$, $BB_2$ and $CC_2$ are concurrent

Let $T_a, T_b$ and $T_c$ be the tangent points of the incircle $\Omega$ of the triangle $ABC$ with the sides $BC, CA$ and $AB{}$ respectively. Let $X, Y$ and $Z{}$ be points on the circle $\Omega$ such that $A{}$ lies on the ray $YX$, $B{}$ lies on the ray $ZY$ and $C{}$ lies on the ray $XZ$. Let $P{}$ be the intersection point of the segments $ZX$ and $T_bT_c$, and similarly $Q=XY \cap T_cT_a$ and $R=YZ\cap T_aT_b$. Prove that the points $A, B$ and $C{}$ lie on the lines $RP, PQ$ and $QR{}$, respectively. [i]Proposed by L. Shatunov (11th grade student)[/i]

A non-isosceles triangle is given. Prove that one of the circles touching internally its incircle and circumcircle and externally one of its excircles passes through a vertex of the triangle.

Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. [i]Octavian Stanasila[/i]

Each of the diagonals $AD$, $BE$, $CF$ of a convex hexagon $ABCDEF$ bisects the area of the hexagon. Prove that these three diagonals pass through the same point.

Let $ABC$ be a triangle with incircle touching $BC, CA, AB$ at $D, E, F,$ respectively. Let $O$ and $M$ be its circumcenter and midpoint of $BC.$ Suppose that circumcircles of $AEF$ and $ABC$ intersect at $X$ for the second time. Assume $Y \neq X$ is on the circumcircle of $ABC$ such that $OMXY$ is cyclic. Prove that circumcenter of $DXY$ lies on $BC.$ [i]Proposed by tenplusten.[/i]

In right $ \triangle ABC$ with legs $ 5$ and $ 12$, arcs of circles are drawn, one with center $ A$ and radius $ 12$, the other with center $ B$ and radius $ 5$. They intersect the hypotenuse at $ M$ and $ N$. Then, $ MN$ has length: [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A); real r=degrees(B); draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270)); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N)); label("$12$", (6,0), S); label("$5$", (12,3.5), E);[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac {13}{5} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \frac {24}{5}$

Let $ S$ be a set of $ 100$ points in the plane. The distance between every pair of points in $ S$ is different, with the largest distance being $ 30$. Let $ A$ be one of the points in $ S$, let $ B$ be the point in $ S$ farthest from $ A$, and let $ C$ be the point in $ S$ farthest from $ B$. Let $ d$ be the distance between $ B$ and $ C$ rounded to the nearest integer. What is the smallest possible value of $ d$?

Let $ABC$ be a triangle with $AB < AC$ and let $D{}$ be the other intersection point of the angle bisector of $\angle A$ with the circumcircle of the triangle $ABC$. Let $E{}$ and $F{}$ be points on the sides $AB$ and $AC$ respectively, such that $AE = AF$ and let $P{}$ be the point of intersection of $AD$ and $EF$. Let $M{}$ be the midpoint of $BC{}$. Prove that $AM$ and the circumcircles of the triangles $AEF$ and $PMD$ pass through a common point.

For a point $ P $ situated in the plane determined by a triangle $ ABC, $ prove the following inequality: $$ BC\cdot PB\cdot PC+AC\cdot PC\cdot PA +AB\cdot PA\cdot PB\ge AB\cdot BC\cdot CA $$

In tetrahedron $ABCD$, $AB=1,CD=3$, the distance between $AB$ and $CD$ is $2$, the intersection angle between $AB$ and $CD$ is $\frac{\pi}{3}$, then the volume of tetrahedron $ABCD$ is $\text{(A)}\frac{\sqrt3}{2}\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}\frac{\sqrt3}{3}$