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Postoji li funkcija $f : \mathbb{N} \to \mathbb{N}$ za koju vrijedi

$$f(f(n)) = f(n + 1) - f(n)$$

za svaki $n \in \mathbb{N}$?

Odredi sve funkcije $f: \mathbb{Z} \to \mathbb{Z}$ takve da za sve $a, b \in \mathbb{Z}$ vrijedi: $$f(f(a) + f(b)) = a + b - 1.$$

Odredi sve funkcije $f : \mathbb{R} \to \mathbb{R}$ takve da za sve realne brojeve $x$ i $y$ vrijedi

$$f(x^2 + f(y)) = (f(x) + y^2)^2.$$

Odredi sve funkcije $f : \mathbb{R} \to \mathbb{R}$ takve da za sve realne brojeve $x$ i $y$ vrijedi

$$f(y + f(x)) - f(x + f(y)) = f(x - y)(f(x + y) - 1).$$

Odredi sve funkcije $f: \mathbb{R} \to \mathbb{R}$ takve da je $f(1) \geqslant 0$ i da za sve realne brojeve $x$ i $y$ vrijedi

$$f(x) - f(y) \geqslant (x - y)f(x - y).$$

Neka je $\alpha$ realni broj. Nađi sve funkcije $f\colon \mathbb{R}\to \mathbb{R}$ takve da za sve $x,y\in \mathbb{R}$ vrijedi: $$f(x + \alpha + f(y)) = f(f(x)) + f(\alpha) + y.$$

Odredi sve funkcije $f\colon \mathbb{R}\to \mathbb{R}$ za koje vrijedi $$f(f(x))(x - f(y)) + 2xy = f(x)f(x + y), \quad \text{za sve } x, y \in \mathbb{R}.$$

Odredi sve funkcije $f\colon \mathbb{R}\to \mathbb{R}$ za koje vrijedi $$f (x f (x) + f (x y)) = f \left(x ^ {2}\right) + y f (x), \quad \text{za sve } x, y \in \mathbb {R}.$$

Odredi sve funkcije $f\colon \mathbb{R}\to \mathbb{R}$ takve da za sve realne brojeve $x$ i $y$ vrijedi

$$f(x^2) + x f(y) = f(x) f(x + f(y)).$$

Dana je funkcija $f\colon \mathbb{R}\to \mathbb{R}$ takva da za sve realne brojeve $x$ i $y$ vrijedi

$$f (x f (y)) + f (x ^ {2}) = f (x) (x + f (y))$$

te da je $f(2) = 3$. Odredi $f(2^{2016})$.

Odredi sve funkcije $f\colon \mathbb{R}\to \mathbb{R}$ takve da za sve realne brojeve $x$ i $y$ vrijedi

$$xf(x) - yf(y) = (x - y)(f(x + y) - xy).$$

Odredi sve funkcije $f\colon \mathbb{N}\to \mathbb{N}$ takve da za sve prirodne brojeve $a$ i $b$ vrijedi

$$f(a) + f(b) - ab \mid af(a) + bf(b).$$

Odredi sve funkcije $f\colon \mathbb{R}\to \mathbb{R}$ takve da za sve realne brojeve $x$, $y$ vrijedi

$$f(xf(y)) = (1 - y)f(xy) + x^2y^2f(y).$$

Odredi sve funkcije $f: \mathbb{R}^+ \to \mathbb{R}$ takve da vrijedi $$\left(x + \frac{1}{x}\right) f(y) = f(xy) + f\left(\frac{y}{x}\right), \quad \text{za sve } x, y \in \mathbb{R}^+.$$

($\mathbb{R}^+$ je oznaka za skup svih pozitivnih realnih brojeva.)

Odredi sve funkcije $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ takve da vrijedi $$f(x^2 (f(y))^2) = (f(x))^2 f(y), \quad \text{za sve } x, y \in \mathbb{Q}^+.$$

($\mathbb{Q}^+$ je oznaka za skup svih pozitivnih racionalnih brojeva.)

Odredi sve funkcije $f: \mathbb{R} \to \mathbb{R}$ takve da za sve $x, y \in \mathbb{R}$ vrijedi $$f(x + yf(x)) = f(xf(y)) - x + f(y + f(x)).$$

Odredi sve polinome $P$ s realnim koeficijentima takve da izraz $$P(x + 3y) + P(3x - y)$$ ima istu vrijednost za sve realne brojeve $x$, $y$ za koje je $x^2 + y^2 = 1$.

Odredi sve realne brojeve $a$ za koje postoji funkcija $f: \mathbb{R} \to \mathbb{R}$ takva da je $$f(x + f(y)) = f(x) + a\lfloor y\rfloor,$$ za sve realne brojeve $x$ i $y$.

Napomena: $\lfloor y\rfloor$ je najveći cijeli broj koji nije veći od $y$. Npr. $\lfloor 1.7\rfloor = 1$, $\lfloor -\pi \rfloor = -4$, $\lfloor 0\rfloor = 0$.

Neka je $\mathbb{R}^+$ skup svih pozitivnih, a $\mathbb{R}_0^+$ skup svih nenegativnih realnih brojeva.

Odredi sve funkcije $f: \mathbb{R}^+ \to \mathbb{R}_0^+$ takve da za sve pozitivne realne brojeve $x$ i $y$ vrijedi

$$f(x) - f(x + y) = f(x^2 f(y) + x).$$

Neka je $\mathbb{Q}_0^+$ skup svih nenegativnih racionalnih brojeva.

Odredi sve funkcije $f: \mathbb{Q}_0^+ \to \mathbb{Q}_0^+$ takve da za sve nenegativne racionalne brojeve $x$, $y$ vrijedi

$$yf(x + y) + (y - 1)f(xy) = f(y^2)f(x + 1).$$

Odredi sve funkcije $f : \mathbb{R} \to \mathbb{R}$ takve da za sve realne brojeve $x, y$ vrijedi

$$f(x + yf(x + 1)) = f(x) + f(xy) + 1.$$

Neka je $S = \{n \in \mathbb{N} : n \geq 2024\}$.

Odredi sve funkcije $f: S \to \mathbb{N}$ takve da za sve $m, n \in S$ vrijedi

$$2m(f(m) + f(n)) = \sum_{k=0}^{f(m)} f(n + k).$$

Odredi sve funkcije $f: \mathbb{R} \to \mathbb{R}$ takve da za sve $x, y \in \mathbb{R}$ vrijedi

$$f(y)f(1 + xf(y)) + xf(y^2) = 2yf(xy) + f(f(y)).$$

Odredi sve funkcije $f: \mathbb{R} \to \mathbb{R}$ takve da za sve $x, y \in \mathbb{R}$ vrijedi

$$f(xy) = \max\{f(x), f(y)\} \cdot \min\{x, y\}.$$

Odredi sve polinome $P$ s realnim koeficijentima takve da za svaki prirodan broj $n$ postoji prirodan broj $m$ takav da vrijedi

$$P(n + 1) = P(n) + P(m).$$

Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$, the equation $$f(x + a) = \frac{1}{2} + \sqrt{f(x) - [f(x)]^2}$$ holds for all $x$.

(a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that $f(x + b) = f(x)$ for all $x$).

(b) For $a = 1$, give an example of a non-constant function with the required properties.

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $$f(x + y) + f(x - y) = 2f(x)g(y)$$ for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)| \leq 1$ for all $y$.

$G$ is a set of non-constant functions of the real variable $x$ of the form $$f(x) = ax + b, \text{ } a \text{ and } b \text{ are real numbers,}$$ and $G$ has the following properties:

(a) If $f$ and $g$ are in $G$, then $g \circ f$ is in $G$; here $(g \circ f)(x) = g[f(x)]$.

(b) If $f$ is in $G$, then its inverse $f^{-1}$ is in $G$; here the inverse of $f(x) = ax + b$ is $f^{-1}(x) = (x - b)/a$.

(c) For every $f$ in $G$, there exists a real number $x_f$ such that $f(x_f) = x_f$.

Prove that there exists a real number $k$ such that $f(k) = k$ for all $f$ in $G$.

Find all polynomials $P$ in two variables, with the following properties: (i) for a positive integer $n$ and all real $t, x, y$

$$P(tx, ty) = t^n P(x, y)$$

(that is, $P$ is homogeneous of degree $n$), (ii) for all real $a, b, c$,

$$P(b + c, a) + P(c + a, b) + P(a + b, c) = 0,$$

(iii) $P(1, 0) = 1$.

Let $f(n)$ be a function defined on the set of all positive integers and having all its values in the same set. Prove that if $$f(n + 1) > f(f(n))$$ for each positive integer $n$, then $$f(n) = n \text{ for each } n.$$

The function $f(x, y)$ satisfies

(1) $f(0, y) = y + 1$,

(2) $f(x + 1, 0) = f(x, 1)$,

(3) $f(x + 1, y + 1) = f(x, f(x + 1, y))$,

for all non-negative integers $x, y$. Determine $f(4, 1981)$.

The function $f(n)$ is defined for all positive integers $n$ and takes on non-negative integer values. Also, for all $m, n$

$$f(m + n) - f(m) - f(n) = 0 \text{ or } 1$$

$$f(2) = 0, f(3) > 0, \text{ and } f(9999) = 3333.$$

Determine $f(1982)$.

Find all functions $f$ defined on the set of positive real numbers which take positive real values and satisfy the conditions:

(i) $f(xf(y)) = yf(x)$ for all positive $x, y$;

(ii) $f(x) \rightarrow 0$ as $x \rightarrow \infty$.

Find all functions $f$, defined on the non-negative real numbers and taking non-negative real values, such that:

(i) $f(xf(y))f(y) = f(x + y)$ for all $x, y \geq 0$,

(ii) $f(2) = 0$,

(iii) $f(x) \neq 0$ for $0 \leq x < 2$.

Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$.

Let $\mathbb{Q}^+$ be the set of positive rational numbers. Construct a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that

$$f(xf(y)) = \frac{f(x)}{y}$$

for all $x, y$ in $\mathbb{Q}^+$.

Let $\mathbf{R}$ denote the set of all real numbers. Find all functions $f: \mathbf{R} \to \mathbf{R}$ such that

$$f \left(x^2 + f(y)\right) = y + \left(f(x)\right)^2 \quad \text{for all } x, y \in R.$$

Does there exist a function $f: \mathbf{N} \to \mathbf{N}$ such that $f(1) = 2$, $f(f(n)) = f(n) + n$ for all $n \in \mathbf{N}$, and $f(n) < f(n + 1)$ for all $n \in \mathbf{N}$?

Let $S$ be the set of real numbers strictly greater than $-1$. Find all functions $f: S \to S$ satisfying the two conditions:

1. $f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ for all $x$ and $y$ in $S$;
2. $\frac{f(x)}{x}$ is strictly increasing on each of the intervals $-1 < x < 0$ and $0 < x$.

Let $S$ denote the set of nonnegative integers. Find all functions $f$ from $S$ to itself such that

$$f(m + f(n)) = f(f(m)) + f(n) \qquad \forall m, n \in S.$$

Consider all functions $f$ from the set $N$ of all positive integers into itself satisfying $f(t^2f(s)) = s(f(t))^2$ for all $s$ and $t$ in $N$. Determine the least possible value of $f(1998)$.

Determine all functions $f: \mathbf{R} \longrightarrow \mathbf{R}$ such that

$$f(x - f(y)) = f(f(y)) + x f(y) + f(x) - 1$$

for all real numbers $x, y$.

Find all real-valued functions on the reals such that $(f(x)+f(y))(f(u)+f(v))=f(xu-yv)+f(xv+yu)$ for all $x,y,u,v$.

Find all polynomials $f$ with real coefficients such that for all reals $a$, $b$, $c$ such that $ab + bc + ca = 0$ we have the following relations $$f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c).$$

Find all functions $f: (0, \infty) \to (0, \infty)$ (so, $f$ is a function from the positive real numbers to the positive real numbers) such that $$\frac{(f(w))^2 + (f(x))^2}{f(y^2) + f(z^2)} = \frac{w^2 + x^2}{y^2 + z^2}$$ for all positive real numbers $w, x, y, z$, satisfying $wx = yz$.

Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths

$$a, f(b) \text{ and } f(b + f(a) - 1).$$

(A triangle is non-degenerate if its vertices are not collinear.)

Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that the equality $$f \big (\lfloor x \rfloor y \big) = f (x) \big \lfloor f (y) \rfloor$$ holds for all $x, y \in \mathbb{R}$. (Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)

Let $\mathbb{N}$ be the set of positive integers. Determine all functions $g\colon \mathbb{N}\to \mathbb{N}$ such that $$\left(g (m) + n\right) \left(m + g (n)\right)$$ is a perfect square for all $m, n \in \mathbb{N}$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$.

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a, b, c$ that satisfy $a + b + c = 0$, the following equality holds: $$f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).$$

(Here $\mathbb{Z}$ denotes the set of integers.)