A positive rational number is written on the blackboard. Every minute Vasya replaces the number $ r $ written on the board with $ \sqrt {r + 1} $. Prove that someday he will get an irrational number.

Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).

Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $. Problem was post earlier [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road]here[/url] , but solution not gives and olympiad doesn't indicate, so I post it again :blush: Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

Is there a real number $\alpha$ such that $\cos\alpha$ is irrational but $\cos 2\alpha$, $\cos 3\alpha$, $\cos 4\alpha$, $\cos 5\alpha$ are all rational? (Author: V. Senderov)

Are there such $x,y$ that $\lg{(x+y)}=\lg x \lg y$ and $\lg{(x-y)}=\frac{\lg x}{\lg y}$ ?

For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$: (I) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$, (II) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$ ? Justify your answer.

Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]

Let $ R$ be a finite commutative ring. Prove that $ R$ has a multiplicative identity element $ (1)$ if and only if the annihilator of $ R$ is $ 0$ (that is, $ aR\equal{}0, \;a\in R $ imply $ a\equal{}0$).

Assume that real numbers $x$, $y$ and $z$ satisfy $x + y + z = 3$. Prove that $xy + xz + yz \leq 3$.

Show that $x^4 + y^4 + z^2\ge xyz \sqrt8$ for all positive reals $x, y, z$.

We know that $R^3 = \{(x_1, x_2, x_3) | x_i \in R, i = 1, 2, 3\}$ is a vector space regarding the laws of composition $(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3)$, $\lambda (x_1, x_2, x_3) = (\lambda x_1, \lambda x_2, \lambda x_3)$, $\lambda \in R$. We consider the following subset of $R^3$ : $L =\{(x_1, x2, x_3) \in R^3 | x_1 + x_2 + x_3 = 0\}$. a) Prove that $L$ is a vector subspace of $R^3$ . b) In $R^3$ the following relation is defined $\overline{x} R \overline{y} \Leftrightarrow \overline{x} -\overline{y} \in L, \overline{x} , \overline{y} \in R^3$. Prove that it is an equivalence relation. c) Find two vectors of $R^3$ that belong to the same class as the vector $(-1, 3, 2)$.

Initially, a pair of numbers $(1,1)$ is written on the board. If for some $x$ and $y$ one of the pairs $(x, y-1)$ and $(x+y, y+1)$ is written on the board, then you can add the other one. Similarly for $(x, xy)$ and $(\frac {1} {x}, y)$. Prove that for each pair that appears on the board, its first number will be positive.

Let $S$ be the sum of all positive integers that can be expressed in the form $2^a \cdot 3^b \cdot 5^c$, where $a$, $b$, $c$ are positive integers that satisfy $a+b+c=10$. Find the remainder when $S$ is divided by $1001$. [i]Proposed by Michael Ren[/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 $g(35)$?

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ a function having the property that $$ \left| f(x+y)+\sin x+\sin y \right|\le 2, $$ for all real numbers $ x,y. $ [b]a)[/b] Prove that $ \left| f(x) \right|\le 1+\cos x, $ for all real numbers $ x. $ [b]b)[/b] Give an example of what $ f $ may be, if the interval $ \left( -\pi ,\pi \right) $ is included in its [url=https://en.wikipedia.org/wiki/Support_(mathematics)]support.[/url]

The numbers $x_1,x_2,x_3, ..., x_n$ satisfy the two conditions $$\sum^n_{i=1}x_i=0 \,\, , \,\,\,\,\sum^n_{i=1}x_i^2=1$$ Prove that there are two numbers among them whose product is no greater than $- 1/n$. (Stolov, Kharkov)

Suppose that $ f$ is a real-valued function for which \[ f(xy)+f(y-x)\geq f(y+x)\] for all real numbers $ x$ and $ y$. a) Give a non-constant polynomial that satisfies the condition. b) Prove that $ f(x)\geq 0$ for all real $ x$.

If $P$ is a function such that $P(2x) = 2^{-3}P(x) + 1$, find $P(0)$.

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

For integer $n > 1$, consider $n$ cube polynomials $P_1(x), ..., P_n(x)$ such that each polynomial has $3$ distinct real roots. Denote $S$ as the set of roots of following equation $P_1(x)P_2(x)P_3(x)... P_n(x) = 0$. It is also known that for each $1 \le i < j \le n, P_i(x)P_j(x) = 0$ has $5$ distinct real roots. 1. Prove that if for each $a, b \in S$, there is exactly one $i \in\{1,2, 3,..., n\}$ such that $P_i(a) = P_i(b) = 0$ then $n = 7$. 2. Prove that if $n > 7$ then $|S| = 2n + 1$.

Solve the equation $$\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\frac{10\cdot 9\cdot 8}{(x+10)(x+9)(x+8)}+ ...+\frac{10\cdot 9\cdot ... \cdot 2 \cdot 1}{(x+10)(x+9)\cdot ... \cdot(x+1)}=11$$

Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that for any real $x, y$ holds the following: $$f(x+yf(x+y)) = f(y^2) + xf(y) + f(x)$$ [i]Proposed by Vadym Koval[/i]

Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$.