We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

A function $f$ from the set of positive real numbers to itself satisfies $$f(x + f(y) + xy) = xf(y) + f(x + y)$$ for all positive real numbers $x$ and $y$. Prove that $f(x) = x$ for all positive real numbers $x$.

Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational. [i]Proposed by A. Golovanov[/i]

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property: [i] for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer.[/i] Prove that this polynomial is divisible by $x$.

Suppose $a,b,c$ are real numbers so that $a+b+c=15$ and $ab+ac+bc=27$. Find the range of values that may be obtained by the expression $abc$.

Let $p(x) = x^{2} + bx + c$, where $b$ and $c$ are integers. If $p(x)$ is a factor of both \[x^{4} + 6x^{2} + 25\quad\text{and}\quad 3x^{4} + 4x^{2} + 28x + 5,\] what is $p(1)$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 8 $

Let ${a}$ be the fractional part of the number $a$, that is, $\{a\} = a - [a]$, where$ [a]$ is the integer part of $ a$. (For example, $\{1.7\} = 1.7 -1 = 0.7$,$\{-\sqrt2 \}= -\sqrt2 -(-3) = 3-\sqrt2$.) a) How many solutions does the equation have $$ \{5\{4\{3\{2\{x\}\}\}\}\}=1\,\, ?$$ b) Find its greatest solution.

Prove that there exists a positive integer $N$ such that for every polynomial $P(x)$ of degree $2019$, there exist $N$ linear polynomials $p_1,p_2, \ldots p_N$ such that $P(x)=p_1(x)^{2019}+p_2(x)^{2019}+ \ldots + p_N(x)^{2019}$. (Assume all polynomials in this problem have real coefficients, and leading coefficients cannot be zero.)

Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$, let $a_i = a_k$ and $b_i = b_k$, where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$. We say $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] if each $a_i$ equals the following arithmetic mean: \[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\] Suppose that neither $\mathbf a $ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] and $\mathbf b$ is $\mathbf a$-[i]harmonic[/i]. Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.

Let $ a$, $ b$, $ c$ be three integers. Prove that there exist six integers $ x$, $ y$, $ z$, $ x^{\prime}$, $ y^{\prime}$, $ z^{\prime}$ such that $ a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}$.

Find the smallest positive integer $ n$ with the property that the polynomial $ x^4 \minus{} nx \plus{} 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

Find the maximum possible value of the product of distinct positive integers whose sum is $2004$.

Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression: (b-a)(b^3+3a^3)

Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that: (a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$; (b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$. [hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]

Let $x_1,x_2,\ldots x_n$ be positive integers such that no one of them is an initial fragment of any other (for example, $12$ is an initial fragment of $\underline{12},\underline{12}5$ and $\underline{12}405$). Prove that \[\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}<3. \]

For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\] [i]Proposed by A. Antropov[/i]

Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$. $\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$

An equation $P(x)=Q(y)$ is called [b]Interesting[/b] if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in $\mathbb{N}$. An interesting equation $P(x)=Q(y)$ [b]yields in[/b] interesting equation $F(x)=G(y)$ if there exists polynomial $R(x) \in \mathbb{Q} [x]$ such that $F(x) \equiv R(P(x))$ and $G(x) \equiv R(Q(x))$. (a) Suppose that $S$ is an infinite subset of $\mathbb{N} \times \mathbb{N}$.$S$ [i]is an answer[/i] of interesting equation $P(x)=Q(y)$ if each element of $S$ is an answer of this equation. Prove that for each $S$ there's an interesting equation $P_0(x)=Q_0(y)$ such that if there exists any interesting equation that $S$ is an answer of it, $P_0(x)=Q_0(y)$ yields in that equation. (b) Define the degree of an interesting equation $P(x)=Q(y)$ by $max\{deg(P),deg(Q)\}$. An interesting equation is called [b]primary[/b] if there's no other interesting equation with lower degree that yields in it. Prove that if $P(x)=Q(y)$ is a primary interesting equation and $P$ and $Q$ are monic then $(deg(P),deg(Q))=1$. Time allowed for this question was 2 hours.

Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.

Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.

Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$

Proove that in every five positive numbers there is a pair, say $a,b$, for which $$\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.$$

Let $P(x)$ be a non-constant polynomial with integer coefficients and let $n$ be a positive integer. The sequence $a_0,a_1,\ldots$ is defined as follows: $a_0=n$ and $a_k=P(a_{k-1})$ for all positive integers $k.$ Assume that for every positive integer $b$ the sequence contains a $b$th power of an integer greater than $1.$ Show that $P(x)$ is linear.