In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.

Is it possible to insert numbers $1, \ldots, 16$ into a table $4 \times 4$ (each cell should have a different number) so that every two adjacent cells (i.e. cells sharing a common side) have numbers $a$ and $b$ satisfying\\ (a) $|a-b| \geq 6$\\ (b) $|a-b| \geq 7$

A magic $3 \times 5$ board can toggle its cells between black and white. Define a \textit{pattern} to be an assignment of black or white to each of the board's $15$ cells (so there are $2^{15}$ patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than $3$ cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day $1$, compute the maximum number of days it can stay alive.

Let $\alpha$, $\beta$ and $\gamma$ be the interior angles of an acute triangle. Prove that if $\alpha < \beta < \gamma$ then $$\sin 2\alpha >\ sin 2 \beta > \sin 2\gamma.$$

Equation $$x^n + a_1x^{n-1} + a_2x^{n-2} +...+ a_{n-1}x + a_n = 0$$ with integer non-zero coefficients $a_1$, $a_2$, $...$ , $a_n$ has $n$ different integer roots. Prove that if any two roots are relatively prime, then the numbers $a_{n-1}$ and $a_n$ are coprime.

Nine lines are given such that every one of them intersects given square $ABCD$ on two trapezoids, which area ratio is $2 : 3$. Prove that at least $3$ of those $9$ lines pass through the same point

Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.

Lisa and Bart are playing a game. A round table has $n$ lights evenly spaced around its circumference. Some of the lights are on and some of them off; the initial configuration is random. Lisa wins if she can get all of the lights turned on; Bart wins if he can prevent this from happening. On each turn, Lisa chooses the positions at which to flip the lights, but before the lights are flipped, Bart, knowing Lisa’s choices, can rotate the table to any position that he chooses (or he can leave the table as is). Then the lights in the positions that Lisa chose are flipped: those that are off are turned on and those that are on are turned off. Here is an example turn for $n = 5$ (a white circle indicates a light that is on, and a black circle indicates a light that is off): [asy] size(250); defaultpen(linewidth(1)); picture p = new picture; real r = 0.2; pair s1=(0,-4), s2=(0,-8); int[][] filled = {{1,2,3},{1,2,5},{2,3,4,5}}; draw(p,circle((0,0),1)); for(int i = 0; i < 5; ++i) { pair P = dir(90-72*i); filldraw(p,circle(P,r),white); label(p,string(i+1),P,2*P,fontsize(10)); } add(p); add(shift(s1)*p); add(shift(s2)*p); for(int j = 0; j < 3; ++j) for(int i = 0; i < filled[j].length; ++i) filldraw(circle(dir(90-72*(filled[j][i]-1))+j*s1,r)); label("$\parbox{15em}{Initial Position.}$", (-4.5,0)); label("$\parbox{15em}{Lisa says ``1,3,4.'' \\ Bart rotates the table one \\ position counterclockwise. }$", (-4.5,0)+s1); label("$\parbox{15em}{Lights in positions 1,3,4 are \\ flipped.}$", (-4.5,0)+s2);[/asy] Lisa can take as many turns as she needs to win, or she can give up if it becomes clear to her that Bart can prevent her from winning. (a) Show that if $n = 7$ and initially at least one light is on and at least one light is off, then Bart can always prevent Lisa from winning. (b) Show that if $n = 8$, then Lisa can always win in at most 8 turns.

Suppose that $c_n=(-1)^n(n+1)$. While the sum $\sum_{n=0}^\infty c_n$ is divergent, we can still attempt to assign a value to the sum using other methods. The Abel Summation of a sequence, $a_n$, is $\operatorname{Abel}(a_n)=\lim_{x\to1^-}\sum_{n=0}^\infty a_nx^n$. Find $\operatorname{Abel}(c_n)$.

For some positive integer $n$, consider the usual prime factorization \[n = \displaystyle \prod_{i=1}^{k} p_{i}^{e_{i}}=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k},\] where $k$ is the number of primes factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?

The angle bisector of $\angle BAC$ intersects the circumcircle of the acute-angled $\triangle ABC$ at point $D$. Let the perpendicular bisectors of $CD$ and $AD$ intersect sides $BC$ and $AB$ at points $E$ and $F$, respectively. If $O$ is the circumcenter of $\triangle ABC$, prove that the points $F, D, E$, and $O$ are concyclic. [i]Proposed by Petar Filipovski[/i]

In the pyramid $SA_1A_2 \cdots A_n$, all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$. Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.

Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0.$ Find the common divisors of $a_{1986}$ and $a_{6891}.$

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

Let $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ be a function such that for every regular $ n $-gon $ A_1A_2 \ldots A_n $ we have $ f(A_1)+f(A_2)+\cdots +f(A_n)=0 $. Prove that $ f(x)=0 $ for all reals $ x $.

For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$.