Find all functions $f: (1,\infty)\text{to R}$ satisfying $f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$. [hide="hint"]you may try to find $f(x^5)$ by two ways and then continue the solution. I have also solved by using this method.By finding $f(x^5)$ in two ways I found that $f(x)=xf(x^2)$ for all $x>1$.[/hide]

If $a$, $b$, $c>0$ and $abc=1$, $\alpha = max\{a,b,c\}$; $f,g : (0, +\infty )\to \mathbb{R}$, where $f(x)=\frac{2(x+1)^2}{x}$ and $g(x)= (x+1)\left (\frac{1}{\sqrt{x}}+1\right )^2$, then $$(a+1)(b+1)(c+1)\geq min\{ \{f(x),g(x) \}\ |\ x\in\{a,b,c\} \backslash \{ \alpha \}\} $$

Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.

What is the value of $2-(-2)^{-2}$? $ \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6 $

Let $X$ be a bounded, nonempty set of points in the Cartesian plane. Let $f(X)$ be the set of all points that are at a distance of at most $1$ from some point in $X$. Let $f_n(X) = f(f(\cdots(f(X))\cdots))$ ($n$ times). Show that $f_n(X)$ becomes “more circular” as $n$ gets larger. In other words, if $r_n = \sup\{\text{radii of circles contained in } f_n(X) \}$ and $R_n = \inf \{\text{radii of circles containing } f_n(X)\}$, then show that $R_n/r_n$ gets arbitrarily close to $1$ as $n$ becomes arbitrarily large. [hide]I'm not sure that I'm posting this in a right forum. If it's in a wrong forum, please mods move it.[/hide]

Find all functions $ f: Z \rightarrow R$ that verify the folowing two conditions: (i) for each pair of integers $ (m,n)$ with $ m<n$ one has $ f(m)<f(n)$; (ii) for each pair of integers $ (m,n)$ there exists an integer $ k$ such that $ f(m)\minus{}f(n)\equal{}f(k)$.

Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality: \[\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).\]

Let be two real numbers $ a<b, $ a nonempty and non-maximal subset $ K $ of the interval $ (a,b) $ and three functions $$ f:(a,b)\longrightarrow\mathbb{R}, g,h:\mathbb{R}\longrightarrow\mathbb{R} $$ satisfying the following relations. $ \text{(i)} g $ and $ h $ are primitivable. $ \text{(ii)} g-h $ hasn't any root in $ (a,b). $ $ \text{(iii)} $ The restrictions of $ f $ at $ K $ and $ (a,b)\setminus K $ are equal to $ g,h, $ respectively. Prove that $ f $ is not primitivable.

The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: $\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 100$

Let $(A,+,\cdot)$ be a finite unit ring, with $n\geq 3$ elements in which there exist [b]exactly[/b] $\dfrac {n+1}2$ perfect squares (e.g. a number $b\in A$ is called a perfect square if and only if there exists an $a\in A$ such that $b=a^2$). Prove that a) $1+1$ is invertible; b) $(A,+,\cdot)$ is a field. [i]Proposed by Marian Andronache[/i]

Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ : $$(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)$$

Find all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ for which $f(0)=0$ and \[f(x^2-y^2)=f(x)f(y) \] for all $x,y\in\mathbb{N}_0$ with $x>y$.

Put $\mathbb{A}=\{ \mathrm{yes}, \mathrm{no} \}$. A function $f\colon \mathbb{A}^n\rightarrow \mathbb{A}$ is called a [i]decision function[/i] if (a) the value of the function changes if we change all of its arguments; and (b) the values does not change if we replace any of the arguments by the function value. A function $d\colon \mathbb{A}^n \rightarrow \mathbb{A}$ is called a [i]dictatoric function[/i], if there is an index $i$ such that the value of the function equals its $i$th argument. The [i]democratic function[/i] is the function $m\colon \mathbb{A}^3 \rightarrow \mathbb{A}$ that outputs the majority of its arguments. Prove that any decision function is a composition of dictatoric and democratic functions.

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

Let $ f:[0,\infty )\longrightarrow\mathbb{R} $ a bounded and periodic function with the property that $$ |f(x)-f(y)|\le |\sin x-\sin y|,\quad\forall x,y\in[0,\infty ) . $$ Show that the function $ [0,\infty ) \ni x\mapsto x+f(x) $ is monotone.

Assume that the following generating function equation is correct, prove the following statement: $\Pi_{i=1}^{\infty} (1+x^{3i})\Pi_{j=1}^{\infty} (1-x^{6j+3})=1$ Statement: The number of partitions of $n$ to numbers not of the form $6k+1$ or $6k-1$ is equal to the number of partitions of $n$ in which each summand appears at least twice. (10 points) [i]Proposed by Morteza Saghafian[/i]

Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$. We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?

Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that: $a)$ $f(1)+2>0$ $b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$ $c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$

Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $ Show that $ f $ and $ g $ are the same function.

What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$?

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times. Let $b_n$ be the number of words of length $n$, formed with letters from $B$, in which appear all the letters from $B$, each an odd number of times. Compute $\frac{b_n}{a_n}$.

Let $f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R$ and $\varphi: \mathbb R \to \mathbb R$ be three ascendant functions such that \[f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.\] Prove that \[f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.\] [i]Note. The function is $k(x)$ ascendant if for every $ x,y \in D_k, x \leq {y}$ we have $g(x)\leq{g(y)}$.[/i]

Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with the properties: [list=a] [*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $ [*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]

Find out the number of real solutions of $x^2e^{\sin x}=1$ [list=1] [*] 0 [*] 1 [*] 2 [*] 3 [/list]