There are $2010$ cities in country, and $3$ roads go from every city. President and Prime Minister play next game. They sell roads by turn to one of $3$ companies( one road is one turn). President will win, if three roads from some city are sold to different companies. Who will win?

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Determine all naturals $n$ such that $2^n + 5$ is a perfect square.

Let \( \triangle ABC \) be an equilateral triangle, and let \( M \) be the midpoint of \( BC \). Let \( C_1 \) be the circumcircle of triangle \( \triangle ABC \) and \( C_2 \) the circumcircle of triangle \( \triangle ABM \). Determine the ratio between the areas of the circles \( C_1 \) and \( C_2 \).

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers in which $\lim_{n\to\infty} a_n = 0$ such that there is a constant $c >0$ so that for all $n \geq 1$, $|a_{n+1}-a_n| \leq c\cdot a_n^2$. Show that exists $d>0$ with $na_n \geq d, \forall n \geq 1$.

Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation $$ a^2 + b^2 = c^2,$$ then: 1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $, 2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $, 3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.

13. The triangle with vertices $(0,0), (a,b)$, and $(a,-b)$ has area $10$. Find the sum of all possible positive integer values of $a$, given that $b$ is a positive integer. 14. Elsa is venturing into the unknown. She stands on $(0,0)$ in the coordinate plane, and each second, she moves to one of the four lattice points nearest her, chosen at random and with equal probability. If she ever moves to a lattice point she has stood on before, she has ventured back into the known, and thus stops venturing into the unknown from then on. After four seconds have passed, the probability that Elsa is still venturing into the unknown can be expressed as $\frac{a}{b}$ in simplest terms. Find $a+b$. (A lattice point is a point with integer coordinates.) 15. Let $ABCD$ be a square with side length $4$. Points $X, Y,$ and $Z$, distinct from points $A, B, C,$ and $D$, are selected on sides $AD, AB,$ and $CD$, respectively, such that $XY = 3, XZ = 4$, and $\angle YXZ = 90^{\circ}$. If $AX = \frac{a}{b}$ in simplest terms, then find $a + b$.

The sum $$\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\cdots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}$$ is a root of the quadratic $x^2+x+c$. What is $c$ in terms of $n$? $\text{(A) }-\frac{n}{2}\qquad\text{(B) }2n\qquad\text{(C) }-2n\qquad\text{(D) }n+\frac{1}{2}\qquad\text{(E) }n-2$

Let $\mathcal{H}$ be the set of all lines in the plane. Call a function $f:\mathbb{R}^2\to\mathcal{H}$ [i]polarising[/i], if $P\in f(Q)$ implies $Q\in f(P)$ for any pair of points $P,Q\in\mathbb{R}^2.$ [list=a] [*]Show that there is no surjective polarising function. [*]Give an example of an injective polarising function. [*]Prove that for every injective polarising function there exists a point $P$ in the plane for which $P\in f(P).$ [/list]

Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition \[y^2 \equiv  x^3 - x \pmod p\] is exactly $p.$

Let $ \{f_n \}_{n=0}^{\infty}$ be a uniformly bounded sequence of real-valued measurable functions defined on $ [0,1]$ satisfying \[ \int_0^1 f_n^2=1.\] Further, let $ \{ c_n \}$ be a sequence of real numbers with \[ \sum_{n=0}^{\infty} c_n^2= +\infty.\] Prove that some re-arrangement of the series $ \sum_{n=0}^{\infty} c_nf_n$ is divergent on a set of positive measure. [i]J. Komlos[/i]

Given right triangle $ABC$ with right angle at $C$, construct three external squares $ABDE$, $BCFG$, and $ACHI$. If $DG = 19$ and $EI = 22$, compute the length of $FH$.

The sum of the digits of the time $19$ minutes ago is two less than the sum of the digits of the time right now. Find the sum of the digits of the time in $19$ minutes. (Here, we use a standard $12$-hour clock of the form $hh:mm$.)

Let $n$ be a positive integer. Aavid has a card deck consisting of $ 2n$ cards, each colored with one of $n$ colors such that every color is on exactly two of the cards. The $2n$ cards are randomly ordered in a stack. Every second, he removes the top card from the stack and places the card into an area called the pit. If the other card of that color also happens to be in the pit, Aavid collects both cards of that color and discards them from the pit. Of the $(2n)!$ possible original orderings of the deck, determine how many have the following property: at every point, the pit contains cards of at most two distinct colors.

An acute triangle ABC is inscribed in a circle C. Tangents in A and C to circle C intersect at F. Segment bisector of AB intersects the line BC at E. Show that the lines FE and AB are parallel.

There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$.