The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights. [i]Warut Suksompong, Thailand[/i]

Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$

A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$. [asy] import three; size(250); defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2"); currentprojection = orthographic(0,-15,5); draw(circle((0,0,0), 15),dashes); draw(circle((0,0,80), 15)); draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80)); draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes); draw("48", (-24,0,-20)--(24,0,-20)); draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17)); label("30", (0,0,-15)); draw("60", (50,0,0)--(50,0,60)); draw("20", (50,0,60)--(50,0,80)); draw((50,0,60)--(47,0,60));[/asy]

Points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ of a convex quadrilateral $ABCD$ such that $\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1$. Points $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ of a convex quadrilateral $E_1F_1G_1H_1$ such that $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that $\frac{E_1A}{AH_1}=a$, express $\frac{F_1C}{CG_1}$ in terms of $a$. 

Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is $10\%$. She leaves a $15\%$ tip on the prices of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50$ for dinner. What is the cost of here dinner without tax or tip? $ \textbf{(A)}\ \$18\qquad\textbf{(B)}\ \$20\qquad\textbf{(C)}\ \$21\qquad\textbf{(D)}\ \$22\qquad\textbf{(E)}\ \$24$

$p > 3$ is a prime. Find all integers $a$, $b$, such that $a^2 + 3ab + 2p(a+b) + p^2 = 0$.

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

Solve in the set of real numbers: \[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1, \] \[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3. \]

Let $n$ be a positive integer and let $a_n$ be the number of ways to write $n$ as a sum of positive integers, such that any two summands differ by at least $2$. Also, let $b_n$ be the number of ways to write $n$ as a sum of positive integers of the form $5k\pm 1$, $k \in Z$. Prove that $\frac{a_n}{b_n}$ is a constant for all positive integers $n$.

Find the angles of triangle $ABC$. [asy] import graph; size(9.115122858763474cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273; /* image dimensions */ /* draw figures */ draw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6)); draw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6)); draw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); draw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6)); draw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6)); draw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6)); draw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6)); draw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); draw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6)); draw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6)); draw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6)); draw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6)); draw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6)); draw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); draw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6)); draw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); draw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6)); label("$A$",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14)); label("$B$",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14)); label("$C$",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((3.0842,-3.6348),linewidth(3.pt) + dotstyle); dot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle); dot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle); dot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle); dot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle); dot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle); dot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] [i]Proposed by Morteza Saghafian[/i]

$ABC$ is an acute triangle. (angle $C$ is bigger than angle $B$) Let $O$ be a center of the circle which passes $B$ and tangents to $AC$ at $C$. $O$ meets the segment $AB$ at $D$. $CO$ meets the circle $(O)$ again at $P$, a line, which passes $P$ and parallel to $AO$, meets $AC$ at $E$, and $EB$ meets the circle $(O)$ again at $L$. A perpendicular bisector of $BD$ meets $AC$ at $F$ and $LF$ meets $CD$ at $K$. Prove that two lines $EK$ and $CL$ are parallel.

It is known that every student of the class for Sunday once visited the rink, and every boy met there with every girl. Prove that there was a point in time when all the boys, or all the girls of the class were simultaneously on the rink.

Let $n$, $i$, and $j$ be integers, for which $0<i<j<n$. Is it always true that the binomial coefficients $\binom{n}{i}$ and $\binom{n}{j}$ have a common divisor greater than 1?

A sequence $(x_n)_{n\ge 0}$ is defined as follows: $x_0=a,x_1=2$ and $x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1$ for all $n>1$. Find all integers $a$ such that $2x_{3n}-1$ is a perfect square for all $n\ge 1$.

For positive real numbers $a,b,c$, $\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}$ is true. Prove that: $$ \frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2$$

Let $a_n$ be the number of unordered sets of three distinct bijections $f, g, h : \{1, 2, ..., n\} \to \{1, 2, ..., n\}$ such that the composition of any two of the bijections equals the third. What is the largest value in the sequence $a_1, a_2, ...$ which is less than $2021$?

Find the remainder after division of $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ by $7$.

Let $a_1$, $a_2$, ..., $a_{2004}$ be non-negative real numbers such that $a_1+...+ a_{2004} \le 25$. Prove that among them there exist at least two numbers $a_i$ and $a_j$ ($i \neq j$) such that $|\sqrt{a_i}-\sqrt{a_j}| \le \frac{5}{2003}$.

Bianca has a rectangle whose length and width are distinct primes less than 100. Let $P$ be the perimeter of her rectangle, and let $A$ be the area of her rectangle. What is the least possible value of $\frac{P^2}{A}$?

A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$.

You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of $10$ m/s throughout the race. Meanwhile, Al starts running at a constant speed of $ 1$ m/s. Thereafter, when exactly $ 1$ second has passed from when Al last changed its speed, Al’s speed instantaneously becomes $1$ m/s faster, so that Al runs at a constant speed of $k$ m/s in the kth second of the race. (Start counting seconds at $ 1$). Suppose Al beats you by exactly $1$ second. How many meters was the race?

There is a ruble coin in each cell of the board with $2\times 200$. Dasha and Sonya play, taking turns making moves, Dasha starts. In one move, it is allowed to select any coin and move it: Dasha moves the coin to a diagonally adjacent cell, Sonya is to the side adjacent. If two coins end up in the same cell, one of them is immediately removed from the board and goes to Sonya. Sonya can stop the game at any time and take all the coins she has received. What is the biggest win she can get, no matter how she plays Dasha? [i]A. Kuznetsov[/i]

There exists a function $G(x)$ such that $G(x)+G\left(\frac{x-\sqrt{3}}{x\sqrt{3}+1}\right)=\sqrt{2}+x$. Find $G(x)$. [i]Proposed by beanielove2.[/i]

It is said that a set of three different numbers is an [i]arithmetical set[/i] if one of the three numbers is the average of the other two. Consider the set $A_n = \{1, 2,..., n\}$, where $n $ is a positive integer, $n\ge 3$. a) How many [i]arithmetical sets[/i] are in $A_{10}$? b) Find the smallest $n$, such that the number of [i]arithmetical sets[/i] in $A_n$ is greater than $2004$.