Find all functions $f: {\mathbb{R^\plus{}}}\to{\mathbb{R^\plus{}}}$ such that \[ f(1\plus{}xf(y))\equal{}yf(x\plus{}y)\] for all $x,y\in\mathbb{R^\plus{}}$.

Let's call this position of the hour and minute hands on the analog clock [i]wonderful[/i], during which the hands change places after some time. Count the total number of wonderful clockwise positions.

Find all the real numbers $x$ such that $\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}$ where $[x]$ denotes the integer part of $x$ and $\{x\}=x-[x]$. For example, $[2.5]=2, \{2.5\} = 0.5$ and $[-1.7]= -2, \{-1.7\} = 0.3$

Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.

Prove that, for every pair $ n$, $r$ of positive integers, there can be found a polynomial $ f(x)$ of degree $ n$ with integer coefficients, so that every polynomial $ g(x)$ of degree at most $ n$, for which the coefficients of the polynomial $ f(x)\minus{}g(x)$ are integers with absolute value not greater than $ r$, is irreducible over the field of rational numbers.

Find the sum of all solutions of the equation $\frac{1}{x^2-1}+\frac{2}{x^2-2}+\frac{3}{x^2-3}+\frac{4}{x^2-4}=2010x-4$

Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and \[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\] For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?

Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials. [b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant. [i]Proposed by Navid Safaie[/i]

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 $J\subsetneq I\subseteq \mathbb R$ be non-empty open intervals, and let $f_1, f_2,\ldots$ be real polynomials satisfying the following conditions: [list] [*] $f_i(x)\ge 0$ for all $i\ge 1$ and $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)$ is finite for all $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)=1$ for all $x\in J$. [/list] Do these conditions imply that $\sum\limits_{i=1}^\infty f_i(x)=1$ also for all $x\in I$? [i]Proposed by András Imolay, Budapest[/i]

Let $a,b$ be real numbers such that $a<b$. Evaluate \[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].

Suppose $n \ge 3$ is a positive integer. Let $a_1 < a_2 < ... < a_n$ be an increasing sequence of positive real numbers, and let $a_{n+1} = a_1$. Prove that $$\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}$$

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$.

If $m\geq 2$ show that there does not exist positive integers $x_1, x_2, ..., x_m,$ such that \[x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.\]

Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where \[ A = \prod_{(a,b) \in S} a , \quad B = \prod_{(a,b) \in S} b , \quad \text{and} \quad C = \sum_{(a,b) \in S} ab. \][i]Proposed by Evan Chen[/i]

Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$: \[ a_i | a_{i+1}\]

Let $f$ be a function defined on the set of pairs of nonzero rational numbers whose values are positive real numbers. Suppose that $f$ satisfies the following conditions: [b](1)[/b] $f(ab,c)=f(a,c)f(b,c),\ f(c,ab)=f(c,a)f(c,b);$ [b](2)[/b] $f(a,1-a)=1$ Prove that $f(a,a)=f(a,-a)=1,f(a,b)f(b,a)=1$.

Prove that the product of four consecutive integers plus $1$ is a perfect square.

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

Let $P(x)$ be a polynomial of degree $n$ whose roots are $i-1, i-2,\cdot\cdot\cdot, i-n$ (where $i^2=-1$), and let $R(x)$ and $S(x)$ be the polynomials with real coefficients such that $P(x)=R(x)+iS(x)$. Show that the polynomial $R$ has $n$ real roots. (R. Stanojevic)

Let $a$ and $b$ be real numbers from interval $\left[0,\frac{\pi}{2}\right]$. Prove that $$\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1$$ if and only if $a=b$

Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.

Sequences $ (x_n)$ and $ (y_n)$ are constructed as follows: $ x_0 \equal{} 365$, $ x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622$, and $ y_0 \equal{} 16$, $ y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952$, for all $ n \ge 0$. Prove that $ \left|x_n\minus{} y_k\right|\neq 0$ for any positive integers $ n$, $ k$.

Prove that for any positive integer $m$, the equation \[ \frac{n}{m}\equal{}\lfloor\sqrt[3]{n^2}\rfloor\plus{}\lfloor\sqrt{n}\rfloor\plus{}1\] has (at least) a positive integer solution $n_{m}$. [i]Cezar Lupu & Dan Schwarz[/i]