The number $10!$ $(10$ is written in base $10)$, when written in the base $12$ system, ends in exactly $k$ zeroes. The value of $k$ is $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) } 5$

Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$. [i]Proposed by David Altizio[/i]

In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.

Let $Z,W,\lambda$ be complex numbers, $|\lambda|\neq1$. Which statements are correct about the equation $\overline{Z}-\lambda Z=W$? I. $Z=\frac{\overline{\lambda}W+\overline{W}}{1-|\lambda|^2}$ is a solution to the equation. II. The equation has only one solution. III. The equation has two solutions. IV. The equation has infinitely many solutions. $\text{(A)}$ Only I and II. $\text{(B)}$ Only I and III. $\text{(C)}$ Only I and IV. $\text{(D)}$ None of $\text{(A)(B)(C)}$.

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

If $4^{4^4} =\sqrt[128]{2^{2^{2^n}}}$ , find $n$.

For real numbers $a_0,a_1,\cdots,a_n(a_0\neq a_1)$, we have$a_{i-1}+a_{i+1}=2a_i$ for $i=1,2,\cdots,n-1$. Prove that $P(x)=a_0\text{C}_n^0(1-x)^n+a_1\text{C}_n^1x(1-x)^{n-1}+\cdots+a_n\text{C}_n^nx^n$ is a linear polynomial.

Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\sqrt[3]{\left (\frac{1+a}{b+c}\right )^{\frac{1-a}{bc}}\left (\frac{1+b}{c+a}\right )^{\frac{1-b}{ca}}\left (\frac{1+c}{a+b}\right )^{\frac{1-c}{ab}}} \geq 64 $$

Find the union of all projections of a given line segment $AB$ to all lines passing through a given point $O$.

Let a, b, c be natural numbers , so that c> b> a> 0. Show that, among any 2c consecutive natural numbers, there are three distinct numbers x, y, z so abc divides xyz.

Let $D$ be an arbitrary point on the side $BC$ of triangle $ABC$. Points $E$ and $F$ are on $CA$ and $BA$ are such that $CD=CE$ and $BD=BF$. Lines $BE$ and $CF$ intersect at point $P$. Prove that when point $D$ varies along the line $BC$, $PD$ passes through a fixed point.

Let $n$ be a positive integer. For a finite set $M$ of positive integers and each $i \in \{0,1,..., n-1\}$, we denote $s_i$ the number of non-empty subsets of $M$ whose sum of elements gives remainder $i$ after division by $n$. We say that $M$ is "$n$-balanced" if $s_0 = s_1 =....= s_{n-1}$. Prove that for every odd number $n$ there exists a non-empty $n$-balanced subset of $\{0,1,..., n\}$. For example if $n = 5$ and $M = \{1,3,4\}$, we have $s_0 = s_1 = s_2 = 1, s_3 = s_4 = 2$ so $M$ is not $5$-balanced.(Czech Republic)

Let $ABC$ be an acute, non isosceles triangle with $O,H$ are circumcenter and orthocenter, respectively. Prove that the nine-point circles of $AHO,BHO,CHO$ has two common points.

A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012$.

Prove that there exists a positive real number $N$ such that for arbitrary real numbers $a,b>N$ it is possible to cover the perimeter of a rectangle with side lengths $a$ and $b$ using non-overlapping unit disks (the unit disks can be tangent to each other). [i]Submitted by Benedek Váli, Budapest[/i]

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn$?$ $\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$

Let $ABC$ be a triangle. A point $D$ lies on line $BC$ and points $E,F$ are taken on $AC,AB$ such that $DE \parallel AB$ and $DF\parallel AC$. Let $G = (AEF) \cap (ABC) \neq A$ and $I = (DEF) \cap BC\neq D$. Let $H$ and $O$ denote the orthocenter and the circumcenter of triangle $DEF$. Prove that $A,O,I$ are collinear if and only if $G,H,I$ are collinear. [i]Proposed by Kaan Bilge[/i]

$g(x):\mathbb{Z}\rightarrow\mathbb{Z}$ is a function that satisfies $$g(x)+g(y)=g(x+y)-xy.$$ If $g(23)=0$, what is the sum of all possible values of $g(35)$?

In an $8\times 8$ chessboard, the rows are numbers from $1$ to $8$ and the columns are labelled from $a$ to $h$. In a two-player game on this chessboard, the first player has a White Rook which starts on the square $b2$, and the second player has a Black Rook which starts on the square $c4$. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?

[u]Round 9[/u] [b]p25.[/b] Let $a$, $b$, and $c$ be positive numbers with $a +b +c = 4$. If $a,b,c \le 2$ and $$M =\frac{a^3 +5a}{4a^2 +2}+\frac{b^3 +5b}{4b^2 +2}+\frac{c^3 +5c}{4c^2 +2},$$ then find the maximum possible value of $\lfloor 100M \rfloor$. [b]p26.[/b] In $\vartriangle ABC$, $AB = 15$, $AC = 16$, and $BC = 17$. Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $CE = 1$ and $BF = 3$. A point $D$ is chosen on side $BC$, and let the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect at point $P \ne D$. Given that $\angle PEF = 30^o$, the length of segment $PF$ can be expressed as $\frac{m}{n}$ . Find $m+n$. [b]p27.[/b] Arnold and Barnold are playing a game with a pile of sticks with Arnold starting first. Each turn, a player can either remove $7$ sticks or $13$ sticks. If there are fewer than $7$ sticks at the start of a player’s turn, then they lose. Both players play optimally. Find the largest number of sticks under $200$ where Barnold has a winning strategy [u]Round 10[/u] [b]p28.[/b] Let $a$, $b$, and $c$ be positive real numbers such that $\log_2(a)-2 = \log_3(b) =\log_5(c)$ and $a +b = c$. What is $a +b +c$? [b]p29.[/b] Two points, $P(x, y)$ and $Q(-x, y)$ are selected on parabola $y = x^2$ such that $x > 0$ and the triangle formed by points $P$, $Q$, and the origin has equal area and perimeter. Find $y$. [b]p30.[/b] $5$ families are attending a wedding. $2$ families consist of $4$ people, $2$ families consist of $3$ people, and $1$ family consists of $2$ people. A very long row of $25$ chairs is set up for the families to sit in. Given that all members of the same family sit next to each other, let the number of ways all the people can sit in the chairs such that no two members of different families sit next to each other be $n$. Find the number of factors of $n$. [u]Round 11[/u] [b]p31.[/b] Let polynomial $P(x) = x^3 +ax^2 +bx +c$ have (not neccessarily real) roots $r_1$, $r_2$, and $r_3$. If $2ab = a^3 -20 = 6c -21$, then the value of $|r^3_1+r^3_2+r^3_3|$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. [b]p32.[/b] In acute $\vartriangle ABC$, let $H$, $I$ , $O$, and $G$ be the orthocenter, incenter, circumcenter, and centroid of $\vartriangle ABC$, respectively. Suppose that there exists a circle $\omega$ passing through $B$, $I$ , $H$, and $C$, the circumradius of $\vartriangle ABC$ is $312$, and $OG = 80$. Let $H'$, distinct from $H$, be the point on $\omega$ such that $\overline{HH'}$ is a diameter of $\omega$. Given that lines $H'O$ and $BC$ meet at a point $P$, find the length $OP$. [b]p33.[/b] Find the number of ordered quadruples $(x, y, z,w)$ such that $0 \le x, y, z,w \le 1000$ are integers and $$x!+ y! =2^z \cdot w!$$ holds (Note: $0! = 1$). [u]Round 12[/u] [b]p34.[/b] Let $Z$ be the product of all the answers from the teams for this question. Estimate the number of digits of $Z$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- |A-E| \rceil \right).$$ Your answer must be a positive integer. [b]p35.[/b] Let $N$ be number of ordered pairs of positive integers $(x, y)$ such that $3x^2 -y^2 = 2$ and $x < 2^{75}$. Estimate $N$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- 2|A-E| \rceil \right).$$ [b]p36.[/b] $30$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \left \lceil 15- \ln \frac{A}{E} \right \rceil \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that [b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$; [b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.

The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=3*dir(180), B=3*dir(140), C=3*dir(0); dot(O); draw(Arc(origin,1,0,360)); draw(Arc(origin,3,0,360)); draw(A--B--C--A); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); [/asy] $ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 26 $

Find all natural numbers $a$ such that for every positive integer $n$ the number $n(a+n)$ is not a perfect square.

Let $x,y,$ and $z$ be nonnegative real numbers with $x+y+z=120$. Compute the largest possible value of the median of the three numbers $2x+y,2y+z,$ and $2z+x$. [i]Proposed by Ankit Bisain[/i]