Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

Find the complex numbers $ z$ for which the series \[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\] converges and find its sum.

An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. [quote]For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1[/quote] The number of partitions of n is given by the partition function $p\left ( n \right )$. So $p\left ( 4 \right ) = 5$ . Determine all the positive integers so that $p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right )$.

Let $ H$ be an arbitrary subgroup of the diffeomorphism group $ \mathsf{Diff}^\infty(M)$ of a differentiable manifold $ M$. We say that an $ \mathcal C^\infty$-vector field $ X$ is [i]weakly tangent[/i] to the group $ H$, if there exists a positive integer $ k$ and a $ \mathcal C^\infty$-differentiable map $ \varphi \mathrel{: } \mathord{]} \minus{} \varepsilon,\varepsilon\mathord{[}^k\times M\to M$ such that (i) for fixed $ t_1,\dots,t_k$ the map \[ \varphi_{t_1,\dots,t_k} : x\in M\mapsto \varphi(t_1,\dots,t_k,x)\] is a diffeomorphism of $ M$, and $ \varphi_{t_1,\dots,t_k}\in H$; (ii) $ \varphi_{t_1,\dots,t_k}\in H \equal{} \mathsf{Id}$ whenever $ t_j \equal{} 0$ for some $ 1\leq j\leq k$; (iii) for any $ \mathcal C^\infty$-function $ f: M\to \mathbb R$ \[ X f \equal{} \left.\frac {\partial^k(f\circ\varphi_{t_1,\dots,t_k})}{\partial t_1\dots\partial t_k}\right|_{(t_1,\dots,t_k) \equal{} (0,\dots,0)}.\] Prove, that the commutators of $ \mathcal C^\infty$-vector fields that are weakly tangent to $ H\subset \textsf{Diff}^\infty(M)$ are also weakly tangent to $ H$.

For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove $\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$. [i]proposed by Mohammad Ahmadi[/i]

Let a real function $f$ defined on the real numbers satisfy the following conditions: (i) $f(10+x) = f(10- x)$ (ii) $f(20+x) = - f(20- x)$ for all $x$. Prove that f is odd and periodic.

Determine whether there exists a differentiable function $f:[0,1]\to\mathbb R$ such that $$f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.$$

Given two positive real numbers $a$ and $b$, suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation \[f(f(x)) + af(x) = b(a + b)x.\] Prove that there exists a unique solution of this equation.

Let $f:[0,1]\to\mathbb R$ be an increasing function. Prove that if $\int_0^1f(x)dx=\int_0^1\left(\int_0^xf(t)dt\right)dx=0$ then $f(x)=0,\forall x\in(0,1)$. [i]Mihai Piticari[/i]

Let $\mathbb{R}^+$ the set of positive real numbers. Determine all the functions $f, g: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that, for all positive real numbers $x, y$ we have that $$f(x + g(y)) = f(x + y) + g(y) \text{ and } g(x + f(y)) = g(x + y) + f(y)$$

Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $

Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$