Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$

Find all functions $f: R \to R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$

Find all functions $f:R \rightarrow R$ such that for any reals $x,y$, $f(x^2+y)=f(x)f(x+1)+f(y)+2x^2y$.

Find all functions $f:R^{*} \rightarrow R$ such that $f(x)\not = x$ and $$ f(y(f(x)-x))=\frac{f(x)}{y}-\frac{f(y)}{x} $$ for any $x,y \not = 0$.

Find all non-constant functions $f : Q^+ \to Q^+$ satisfying the equation $$f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a)$$ for all $a, b,c \in Q^+$ .

Find all functions $f: R \to R$ such that $f (f (x) f (y)) = xy$ and there is no $k \in R -\{0,1,-1\}$ such that $f (k) = k$.

Find all functions $f : N \to N$ such that $$\frac{f(x+y)+f(x)}{2x+f(y)}= \frac{2y+f(x)}{f(x+y)+f(y)}$$ , for all $x, y$ in $N$.

A function $f : R \to R$ satisfy the functional equation $f(x + 2y) + 2f(y - 2x) = 3x -4y + 6$ for all reals $x, y$. Compute $f(2548)$.

The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$. You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$. (a) Find $g(2021)$. (b) Find $g(1/2021)$.

Let $Oxy$ be a fixed rectangular coordinate system in the plane. Each ordered pair of points $A_1, A_2$ from the same plane which are different from O and have coordinates $x_1, y_1$ and $x_2, y_2$ respectively is associated with real number $f(A_1,A_2)$ in such a way that the following conditions are satisfied: (a) If $OA_1 = OB_1$, $OA_2 = OB_2$ and $A_1A_2 = B_1B_2$ then $f(A_1,A_2) = f(B_1,B_2)$. (b) There exists a polynomial of second degree $F(u,v,w,z)$ such that $f(A_1,A_2)=F(x_1,y_1,x_2,y_2)$. (c) There exists such a number $\phi \in (0,\pi)$ that for every two points $A_1, A_2$ for which $\angle A_1OA_2 = \phi$ is satisfied $f(A_1,A_2) = 0$. (d) If the points $A_1, A_2$ are such that the triangle $OA_1A_2$ is equilateral with side $1$ then$ f(A_1,A_2) = \frac12$. Prove that $f(A_1,A_2) = \overrightarrow{OA_1} \cdot \overrightarrow{OA_2}$ for each ordered pair of points $A_1, A_2$.

Find all functions $f : Q \to Q$, such that $$f(x + f (y + f(z))) = y + f(x + z)$$ for all $x ,y ,z \in Q$ . (I. Voronovich)

A function $f : R \to R$ has $f(1) < 0$, and satisfy the functional equation $$f(\cos (x + y)) = (\cos x)f(\cos y) + 2f(\sin x)f(\sin y)$$ for all reals $x, y$. Compute $f \left(\frac{2006}{2549 }\right)$

Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$. (I. Voronovich)

Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

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

Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$

Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$. Find all possible values of $f(1995)$.

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$ (Vadym Koval)

Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.

Let $f:] 0. \infty [ \to R$ be a strictly increasing function, such that $$f (x) f\left(f (x) +\frac{1}{x} \right)= 1.$$ Find $f (1)$.

Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies: $\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$. $\bullet$ $f (1998) = 2$ Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is satisfied,

Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$

Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.

Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers