Construct a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following properties: $ \text{(i)} f $ is not monotonic on any real interval. $ \text{(ii)} f $ has Darboux property (intermediate value property) on any real interval. $ \text{(iii)} f(x)\leqslant f\left( x+1/n \right) ,\quad \forall x\in\mathbb{R} ,\quad \forall n\in\mathbb{N} $ [i]Alexandru Cioba[/i]

Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$ for all positive reals $x$ and $y$.

The sequence $ (x_n)_{n \geq 1}$ is defined by: $ x_1\equal{}1$ $ x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n}$ Prove that $ (x_n)$ increases and $ [x_n^2]\equal{}n$.

Let $a < b$ be real numbers and let $f : (a, b) \to \mathbb{R}$ be a function such that the functions $g : (a, b) \to \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \to \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing. Show that the function $f$ is continuous on $(a, b)$.

Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that: $ f(x)+f(\frac1x)=1$ $ 2f(f(x))=f(2x)$

Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$

$f(x)$ is a continuous function defined in $x>0$. For all $a,\ b\ (a>0,\ b>0)$, if $\int_a^b f(x)\ dx$ is determined by only $\frac{b}{a}$, then prove that $f(x)=\frac{c}{x}\ (c: constant).$

The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$. Show that $|f(x)| \le 1$ for all $x$.

Let $P (x) = x^{3}-3x+1.$ Find the polynomial $Q$ whose roots are the fifth powers of the roots of $P$.

[color=darkred]Let $f,g\colon [0,1]\to [0,1]$ be two functions such that $g$ is monotonic, surjective and $|f(x)-f(y)|\le |g(x)-g(y)|$ , for any $x,y\in [0,1]$ . [list] [b]a)[/b] Prove that $f$ is continuous and that there exists some $x_0\in [0,1]$ with $f(x_0)=g(x_0)$ . [b]b)[/b] Prove that the set $\{x\in [0,1]\, |\, f(x)=g(x)\}$ is a closed interval. [/list][/color]

Let consider the following function set $$F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}$$ [list=1] [*] Find $|F|$ [*] For $n=2k$ prove that $|F|< e{(4k)}^{k}$ [*] Find $n$, if $|F|=540$ and $n=2k$ [/list]

For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?

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]

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$, we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

For any two positive integers $n$ and $p$, prove that there are exactly ${{(p+1)}^{n+1}}-{{p}^{n+1}}$ functions $f:\left\{ 1,2,...,n \right\}\to \left\{ -p,-p+1,-p+2,....,p-1,p \right\}$ such that $\left| f(i)-f(j) \right|\le p$ for all $i,j\in \left\{ 1,2,...,n \right\}$.

Each vertex of a regular $11$-gon is colored black or gold. All possible triangles are formed using these vertices. Prove that there are either two congruent triangles with three black vertices or two congruent triangles with three gold vertices.

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(n))+f(n)=2n+2001 \text{ or }2n+2002.\]

Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.

Let $g$ be the multiplicative function given by $$g(p^{\alpha}) = \alpha p^{\alpha-1},$$ for all $\alpha\in\mathbb Z^+$ and primes $p$. Prove that there exist infinitely many integers $n$ such that $$g(n+1) = g(n) + g(1).$$

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$? $\textbf{(A) } 9 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 13 \qquad\textbf{(E) } 15 $

Let $Q^+$ denote the set of all positive rational number and let $\alpha\in Q^+.$ Determine all functions $f:Q^+ \to (\alpha,+\infty )$ satisfying $$f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}$$ for all $x,y\in Q^+ .$

Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations: $$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$ Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$ doesn't have limit at $ \infty . $ [i]Romeo Ilie[/i]

Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.