Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied: i) $2f (x) + 2f (y) \le f (x + y)$ ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

Find all functions $\mathbb R^+\to\mathbb R^+$ such that \[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d), \quad \forall a,b,c,d\in\mathbb R^+; \quad abcd=1\]

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ from $C^2$ (id est, $f$ is twice differentiable and $f''$ is continuous.) such that for every real number $t$ we have $f(t)^2=f(t \sqrt{2})$.

Do either (1) or (2): (1) Show that any solution $f(t)$ of the functional equation $$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$ for $x,y\in \mathbb{R}$ satisfies $$f''(t)= \pm c^{2} f(t)$$ for a constant $c$, assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $$ \pm \cos ct, \;\;\; \pm \cosh ct.$$ (2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by $$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$ The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of $$\prod_{i=1}^{n} (x-a_{i}).$$

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation $$ f\left( x^3+y^3 \right) =xf\left( y^2 \right) + yf\left( x^2 \right) , $$ for all real numbers $ x,y. $

Let G be an abelian group, $0\leq\varepsilon<1$ and $f : G\to\Bbb R^n$ a function that satisfies the inequality. $$||f(x+y)-f(x)-f(y)|| \leq \varepsilon ||f (y)|| \qquad (x, y)\in G^2$$ Prove that there is an additive function $A : G\to \Bbb R^n$ and a continuous function $\varphi : A (G) \to\Bbb R^n$ such that $f = \varphi\circ A$.

Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies • $f(p) = 1$ for all primes $p$, and • $f(ab) = af(b) + bf(a)$ for all $ a,b \in \mathbb{Q}^+ $. For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$?

Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$

Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$

Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.

Find all functions $f:\mathbb R\to\mathbb R$ such that \[f(x^2)+2xf(y)=yf(x)+xf(x+y).\] [i](Proposed by Yeoh Yi Shuen)[/i]

If for function $f$ holds that $$f(x)+f(x+1)+f(x+2)+...+f(x+1986)=0$$ for any $\in\mathbb{R}$, prove that $f$ is periodic and find one period of her.

Find all functions $f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}$ such that for any two rational numbers $x$ and $y$ the following conditions hold [list] [*] $f(x+y)\leq f(x)+f(y)$, [*]$f(xy)=f(x)f(y)$, [*]$f(2)=1/2$. [/list]

On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions: (i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length. (ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$ (Zero vector is considered to be perpendicular to every vector).

Find all functions $f: \mathbb{Q} \to \mathbb{R}$ such that $f(xy)=f(x)f(y)+f(x+y)-1$ for all rationals $x,y$