Found problems: 313
Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$
Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies
(i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$.
(ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.
An increasing function $f : N \to R$ satisfies
$$f(kl) = f(k)+ f(l)\,\,\, for \,\,\, all \,\,\, k,l \in N.$$
Show that there is a real number $p > 1$ such that $f(n) =\ log_pn$ for all $n$.
Find all functions $f : R \to R$ that satisfy $f(x^2 + f(y) - y) = (f(x))^2$ for all $x,y \in R$.
Find all functions $f$ from the positive integers to the positive integers such that such that for all integers $x, y$ we have $$2yf(f(x^2)+x)=f(x+1)f(2xy).$$
Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$, and for which holds:
$\bullet$ $x \le y \Rightarrow f(x) \le f(y)$, for all $x, y \in R$, and
$\bullet$ $f(f(x)) = x$, for all $x \in R$.
Find all real functions of a real numbers, such that for any $x$, $y$ and $z$ holds the equality $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$
Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$.
(Here $\sqrt[3]{x}$ is defined all over $R$.)
Consider the function $f : R^+ \to R^+$ and satisfying
$$f(x + 2y + f(x + y)) = f(2x) + f(3y), \,\, \forall \,\, x, y > 0.$$
1. Find all functions $f(x)$ that satisfy the given condition.
2. Suppose that $f(4\sin^4x)f(4\cos^4x) \ge f^2(1)$ for all $x \in \left(0\frac{\pi}{2}\right) $. Find the minimum value of $f(2022)$.
Let $f(n)$ be defined on the positive integers and satisfy:
$f(prime) = 1$, $f(ab) = a f(b) + f(a) b$.
Show that $f$ is unique and find all $n$ such that $n = f(n)$.
For which $a$, there is a function $f: R \to R$, different from a constant, such that
$$f(a(x + y)) = f(x) + f(y) ?$$
Find all functions $f : R_{>0} \to R_{>0}$ with the following property:
$$f \left( \frac{x}{y + 1}\right) = 1 - xf(x + y)$$ for all $x > y > 0$ .
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)
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.
Find all functions $f : R- \{-1\} \to R$ such that
$$f(x)f \left( f \left(\frac{1 - y}{1 + y} \right)\right) = f\left(\frac{x + y}{xy + 1}\right) $$
for all $x, y \in R$ that satisfy $(x + 1)(y + 1)(xy + 1) \ne 0$.
Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.
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 functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.
Find all functions $f : R_{>0} \to R_{>0}$, which for all $x > y > z > 0$ is the following equation holds
$$f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz.$$
Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.
Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ .
Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.
Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.
Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.
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 $N$.