Found problems: 313
Find all functions $f:R \to R$ such that $$f(x^2+f(y)-y) =(f(x))^2-f(y)$$ for all $x,y \in R$
Determine all polynomials $P(x)$ with real coefficients such that
$P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$.
Find all functions $f : R \to R$ such that $$2f(x) =f(x+y)+f(x+2y)$$, for all $x \in R$ and for all $y \ge 0$.
Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.
Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions:
1) $f(f(x)) =xf(x)-ax$ for all real $x$
2) $f$ is not constant
3) $f$ takes the value $a$
Find all functions $f : R \to R$ that satisfy $f((x-y)^2)= f(x)^2 -2x f(y)+y^2$ for all $x,y \in R$
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$.
I.Voronovich
Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$.
Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.
Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.
Given $P$ a real polynomial with degree greater than $ 1$.
Find all pairs $(f,Q)$ with function $f : R \to R$ and the real polynomial $Q$ satisfying the following two conditions:
i) for all $x, y \in R$, we have $f(P(x) + f(y)) = y + Q(f(x))$.
ii) there exists $x_0 \in R$ such that $f(P(x_0)) = Q(f(x_0))$.
Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$
Find all functions $f : \{1, 2, ... , n,...\} \to Z$ with the following properties
(i) if $a, b$ are positive integers and $a | b$, then $f(a) \ge f(b)$;
(ii) if $a, b$ are positive integers then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.
Determine if there exists a function f: $N^*\to N^*$ that satisfies that for all $n \in N^*$, $$10^{f (n)} <10n + 1 <10^{f (n) +1}.$$ Justify your answer.
Note: $N^*$ denotes the set of positive integers.
Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$
Determine all functions $f : R \to R$ satisfying $f (f(x)f(y) + f(y)f(z) + f(z)f(x))= f(x) + f(y) + f(z)$ for all real numbers $x, y, z$.
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 $f(2548)$.
Consider the function $ h$, defined for all positive real numbers, such that:
$$10x -6h(x) = 4h \left(\frac{2020}{x}\right) $$
for all $x > 0$. Find $h(x)$ and the value of $h(4)$.
Let $f: N^+ \to N^+$ be a function that satisfies that
$$kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+$$
Prove that
$$f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+$$
Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$
A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.
Prove that there is a function $f$ from the set of all natural numbers to itself such that for any natural number $n$, $f(f(n)) = n^2$.
Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.
Find all functions $f : R \to R$ such that for all $x, y \in R$ holds $$yf(2x) - xf(2y) = 8xy(x^2 - y^2).$$
Find all functions $f: \mathbb{R}^+\to\mathbb{R}$ such that $f(x+y)=f(x^2)+f(y^2)$ for any $x,y \in\mathbb{R}^+$