Two players are playing game. Players alternately write down one natural number greater than $1$, but it is not allowed to write linear combination previously written numbers with nonnegative integer coefficients. Player lose a game if he can't write a new number. Does any of players can have wiining strategy, if yes, then which one of them? [i]Journal "Kvant" / Aleksandar Ilic[/i]

Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probablility that the selected number is prime? $\textbf{(A) } \dfrac{8}{99} \qquad\textbf{(B) } \dfrac{2}{5} \qquad\textbf{(C) } \dfrac{9}{20} \qquad\textbf{(D) } \dfrac{1}{2} \qquad\textbf{(E) } \dfrac{9}{16} $

When a company releases a new social media software, the marketing development of the company researches and analyses the characteristics of the customer group apart from paying attention to the active customer depending on the change of the time. We use $n(t, x)$ to express the customer density (which will be abbreviated as density). Here $t$ is the time and $x$ is the time of the customer spent on the social media software. In the instant time $t$, for $0<x_1<x_2$, the number of customers of spending time between $x_1$ and $x_2$ is $\int_{x_1}^{x_2}n(t,x)dx$. We assume the density $n(t,x)$ depends on the time and the following factors: Assumption 1. When the customer keeps using that social media software, their time spent on social media increases linearly. Assumption 2. During the time that the customer uses the social media software, they may stop using it. We assumption the speed of stopping using it $d(x)>0$ only depends on $x$. Assumption 3. There are two sources of new customer. (i) The promotion from the company: A function of time that expresses the increase of number of people in a time unit, expressed by $c(t)$. (ii) The promotion from previous customer: Previous customer actively promotes this social media software to their colleagues and friends actively. The speed of promoting sucessfully depends on $x$, denoted as $b(x)$. Assume if in an instant time, denoted as $t=0$, the density function is known and $n(0,x)=n_0(x)$. We can derive. The change of time $n(t,x)$ can satisfy the equation: $\begin{cases} \frac{\partial}{\partial t}n(t,x)+\frac{\partial}{\partial x}n(t,x)+d(x)n(t,x)=0, t\ge 0, x\ge 0 \\ N(t):=n(t,x=0)=c(t)+\int_{0}^{\infty}b(y)n(t,y)dy \end{cases}\,$ where $N(t)$ iis the speed of the increase of new customers. We assume $b, d \in L^\infty_-(0, \infty)$. $b(x)$ and $d(x)$ is bounded in essence. The following, we first make a simplified assumption: $c(t)\equiv 0$, i.e. the increase of new customer depends only on the promotion of previous customer. (a) According to assumption 1 and 2, formally derive the PDE that $n(t, x)$ satisfies in the two simtaneous equation above. You are required to show the assumption of model and the relationship between the Maths expression. Furthermore, according to assumption 3, explain the definition and meaning of $N(t)$ in the simtaneous equation above. (b) We want to research the relationship of the speed of the increase of the new customers $N(t)$ and the speed of promoting sucessfully $b(x)$. Derive an equation that $N(t)$ satisfies in terms of $N(t), n_0(x), b(x), d(x)$ only and does not include $n(t, x)$. Prove that $N(t)$ satifies the estimation $|N(t)|\le ||b||_\infty e^{||b||_\infty t}\int_{0}^{\infty}|n_0(x)|dx$, where $||\cdot||_\infty$ is the norm of $L^\infty$. (c) Finally, we want to research, after sufficiently long time, what trend of number density function $n(t, x) $\frac{d} has. As the total number of customers may keep increasing so it is not comfortable for us to research the number density function $n(t, x)$. We should try to find a density function which is renormalized. Hence, we first assume there is one only solution $(\lambda_0,\varphi(x))$ of the following eigenvalue problem: $\begin{cases} \varphi'(x)+(\lambda_0+d(x))\varphi(x)=0, x\ge 0 \\ \varphi(x)>0,\varphi(0)=\int_{0}^{\infty}b(x)\varphi(x)dx=1 \end{cases}\, $ and its dual problem has only solution $\psi(x)$: $\begin{cases} -\varphi'(x)+(\lambda_0+d(x))\psi(x)=\psi(0)b(x), x\ge 0 \\ \psi(x)>0,\int_{0}^{\infty}\psi(x)\varphi(x)dx=1 \end{cases}\,$ Prove that for any convex function $H:\mathbb{R}^+\to \mathbb{R}^+$ which satisfies $H(0)=0$. We have $\frac{d}{dx}\int_{0}^{\infty}\psi(x)\varphi(x)H(\frac{\tilde{n}(t,x)}{\varphi(x)})dx\le 0, \forall t\ge 0$. Furthermore, prove that $\int_{0}^{\infty}\psi(x)n(t,x)dx=e^{\lambda_0t}\int_{0}^{\infty}\psi(x)n_0(x)dx$ To simplify the proof, the contribution of boundary terms in $\infty$ is negligible.

Merlin's castle has 100 rooms and 1000 corridors. Each corridor links some two rooms. Each pair of rooms is linked by one corridor at most. Merlin has given out the plan of the castle to the wise men and declared the rules of the challenge. The wise men need to scatter across the rooms in a manner they wish. Each minute Merlin will choose a corridor and one of the wise men will have to pass along it from one of the rooms at its ends to the other one. Merlin wins when in both rooms on the ends of the chosen corridor there are no wise men. Let us call a number $m$ the magic number of the castle if $m$ wise men can pre-agree before the challenge and act in such a way that Merlin never wins, $m$ being the minimal possible number. What are the possible values of the magic number of the castle? (Merlin and all the wise men always know the location of all the wise men). Timofey Vasilyev

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 $ABC$ be an acute triangle, $ M $ the foot of the height from $A$ and point $P\in(MA)$ different from the orthocenter of $ABC.$ Prove that the feet of perpendiculars from $ M $ to $AC, AB, BP$ and $CP$ lie on a circle.

Find all functions $f:\mathbb R\to\mathbb R$ such that (i) $f(0)=1$; (ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$; (iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.

Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$ [i]Proposed by Stijn Cambie, Belgium [/i]

Given seven points on a plane, with no three points collinear. Prove that it is always possible to divide these points into the vertices of a triangle and a convex quadrilateral, with no shared parts between the two shapes.

[b]i)[/b]If $X,Y,Z$ be the angles of a triangle then show that \[\tan {\frac{X}{2}}\tan {\frac{Y}{2}}+\tan {\frac{Y}{2}}\tan {\frac{Z}{2}}+\tan {\frac{Z}{2}}\tan {\frac{X}{2}}=1\] [b]ii)[/b] Prove using [b](i)[/b] or otherwise that \[\tan {\frac{X}{2}}\tan {\frac{Y}{2}}\tan {\frac{Z}{2}}\leq\frac {1}{3\sqrt{3}}\]

A point in the plane is called [b]integral[/b] if both its $x$ and $y$ coordinates are integers. We are given a triangle whose vertices are integral. Its sides do not contain any other integral points. Inside the triangle, there are exactly 4 integral points. Must those 4 points lie on one line?

$a)$ Prove that for all positive integers $n$ exists a set $M_n$ of positive integers with exactly $n$ elements and: $i)$ Arithmetic mean of arbitrary non-empty subset of $M_n$ is integer $ii)$ Geometric mean of arbitrary non-empty subset of $M_n$ is integer $iii)$ Both arithmetic mean and geometry mean of arbitrary non-empty subset of $M_n$ is integer $b)$ Does there exist infinite set $M$ of positive integers such that arithmetic mean of arbitrary non-empty subset of $M$ is integer

Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

Suppose point $P$ is inside quadrilateral $ABCD$ such that $\angle{PAB} = \angle{PDA}, \angle{PAD} = \angle{PDC}, \angle{PBA} = \angle{PCB}, \angle{PBC} = \angle{PCD}.$ If $PA = 4, PB = 5,$ and $PC = 10$, compute the perimeter of $ABCD$.

How many positive divisors does $200$ have?

In how many ways can six marbles be placed in the squares of a $6$-by-$6$ grid such that no two marbles lie in the same row or column?

Solve the system of equations in integers: \[ ab + 1 = (c+1)(d+1), \quad cd + 1 = (a-1)(b-1). \]

Prove that there is no function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(0) >0$, and such that \[f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}. \]

The edges of a complete graph on $1000$ vertices are colored in three colors. Prove that this graph contains a non-self-intersecting single-color cycle whose length is odd and not less than $41$.

Let $a_1, a_2, a_3, a_4, a_5$ be nonzero real numbers. Prove that the polynomial $$P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}$$, where $a_{5+i} = a_i$ for $i = 1,2, 3,4$, has a root with negative real part.

Let $b,m,n\in\mathbb{N}$ with $b>1$ and $m\not=n$. Suppose that $b^{m}-1$ and $b^{n}-1$ have the same set of prime divisors. Show that $b+1$ must be a power of $2$.

Compute the number of ordered pairs $(a, b)$ of positive integers such that $a$ and $b$ divide $5040$ but share no common factors greater than $1$.

Determine functions $f : \mathbb{R} \rightarrow \mathbb{R},$ with property that \[ f(f(x)) + y \cdot f(x) \le x + x \cdot f(f(y)), \] for every $x$ and $y$ are real numbers.

Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.

For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that \[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\]