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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 $n \in \mathbb{N}$?

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 $f\colon \mathbb{R}\to \mathbb{R}$ 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 $f(2^{2016})$.

Neka je $f: \mathbb{N} \to \mathbb{N}$ multiplikativna funkcija takva da je $f(4) = 4$ i vrijedi $$f(m^2 + n^2) = f(m^2) + f(n^2) \quad \text{za sve } m, n \in \mathbb{N}.$$

Dokaži da je $f(m^2) = m^2$ za sve $m \in \mathbb{N}$.

Za funkciju $f$ kažemo da je multiplikativna ako za svaki izbor relativno prostih prirodnih brojeva $m$ i $n$ vrijedi $f(mn) = f(m)f(n)$.

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.)

Dan je prirodni broj $C$. Odredi sve funkcije $f: \mathbb{N} \to \mathbb{N}$ sa sljedećim svojstvom:

za sve $a, b \in \mathbb{N}$ za koje je $a + b > C$, vrijedi $a + f(b) \mid a^2 + bf(a)$.

Funkcija $f: \mathbb{N}_0 \to \mathbb{N}_0$ je pseudopolinom ako za svaka dva različita broja $a, b \in \mathbb{N}_0$ vrijedi $$a - b \mid f(a) - f(b).$$

Odredi sve pseudopolinome takve da za svaki $n \in \mathbb{N}_0$ vrijedi $f(n) \leq n\sqrt{n}$.

Neka je $f: \mathbb{R} \to \mathbb{R}$ funkcija sa svojstvima:

(a) Postoji realan broj $M$ takav da je $|f(x)| \leq M$, za sve $x \in \mathbb{R}$.

(b) Za svaki realan broj $x$ vrijedi $$f\left(x + \frac{1}{2}\right) + f\left(x + \frac{1}{3}\right) = f(x) + f\left(x + \frac{5}{6}\right).$$

Pokaži da je funkcija $f$ periodična, odnosno da postoji pozitivan realan broj $T$ takav da je $f(x + T) = f(x)$ za sve $x \in \mathbb{R}$.

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

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

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

$$m^{f(n)} - n^{f(m)} \leq mn.$$

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 $a_1, a_2, \cdots, a_n$ be real constants, $x$ a real variable, and

$$f(x) = \cos(a_1 + x) + \frac{1}{2}\cos(a_2 + x) + \frac{1}{4}\cos(a_3 + x) + \cdots + \frac{1}{2^{n-1}}\cos(a_n + x).$$

Given that $f(x_1) = f(x_2) = 0$, prove that $x_2 - x_1 = m\pi$ for some integer $m$.

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 $G$.

Let $P_1(x) = x^2 - 2$ and $P_j(x) = P_1(P_{j-1}(x))$ for $j = 2, 3, \cdots$. Show that, for any positive integer $n$, the roots of the equation $P_n(x) = x$ are real and distinct.

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 set of all positive integers is the union of two disjoint subsets $\{f(1), f(2), \ldots, f(n), \ldots\}$, $\{g(1), g(2), \ldots, g(n), \ldots\}$, where

$$f(1) < f(2) < \cdots < f(n) < \cdots,$$ $$g(1) < g(2) < \cdots < g(n) < \cdots,$$

and

$$g(n) = f(f(n)) + 1 \text{ for all } n \geq 1.$$

Determine $f(240)$.

A function $f$ is defined on the positive integers by

$$\begin{aligned} f(1) &= 1, \quad f(3) = 3, \\ f(2n) &= f(n), \\ f(4n + 1) &= 2f(2n + 1) - f(n), \\ f(4n + 3) &= 3f(2n + 1) - 2f(n), \end{aligned}$$

for all positive integers $n$.

Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.

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 $0 < x$.

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 $x,y,u,v$.

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

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 $wx = yz$.

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m - n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfying the equation $$f \big (x + f (x + y) \big) + f (xy) = x + f (x + y) + yf (x)$$ for all real numbers $x$ and $y$.

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, we have $$f(x + y) + f(x)f(y) = f(xy) + (y + 1)f(x) + (x + 1)f(y).$$

Find all surjective functions $f: \mathbb{N} \to \mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true: $$f(a) = f(b),$$ $$f(a + b) = \min\{f(a), f(b)\}.$$

Remarks: $\mathbb{N}$ denotes the set of all positive integers. A function $f: X \to Y$ is said to be surjective if for every $y \in Y$ there exists $x \in X$ such that $f(x) = y$.

Determine all functions $f: \mathbb{R}\backslash\{0\} \to \mathbb{R}\backslash\{0\}$ such that $$f(x^2 y f(x)) + f(1) = x^2 f(x) + f(y)$$ holds for all nonzero real numbers $x$ and $y$.

Let $\mathbb{R}$ denote the set of real numbers. Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that $$f(x)f(y) = x f(f(y - x)) + x f(2x) + f(x^2)$$ holds for all real numbers $x$ and $y$.

Let $\mathbb{N}$ be the set of positive integers. Determine all positive integers $k$ for which there exist functions $f\colon \mathbb{N}\to \mathbb{N}$ and $g\colon \mathbb{N}\to \mathbb{N}$ such that $g$ assumes infinitely many values and such that

$$f^{g(n)}(n) = f(n) + k$$

holds for every positive integer $n$.

(Remark. Here, $f^i$ denotes the function $f$ applied $i$ times, i.e., $f^i(j) = \underbrace{f(f(\ldots f(f(j)) \ldots))}_{i \text{ times}}$.)

Let $\mathbb{N}$ denote the set of positive integers. Determine all functions $f\colon \mathbb{N}\to \mathbb{N}$ such that $f(1)\leq f(2)\leq f(3)\leq \ldots$ and the numbers $f(n) + n + 1$ and $f(f(n)) - f(n)$ are both perfect squares for every positive integer $n$.

Let $\mathbb{Z}$ denote the set of all integers and $\mathbb{Z}_{>0}$ denote the set of all positive integers.

(a) A function $f\colon \mathbb{Z}\to \mathbb{Z}$ is called $\mathbb{Z}$-good if it satisfies $f(a^{2} + b) = f(b^{2} + a)$ for all $a,b\in \mathbb{Z}$. Determine the largest possible number of distinct values that can occur among $f(1),f(2),\ldots ,f(2023)$, where $f$ is a $\mathbb{Z}$-good function.

(b) A function $f\colon \mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ is called $\mathbb{Z}_{>0}$-good if it satisfies $f(a^{2} + b) = f(b^{2} + a)$ for all $a,b\in \mathbb{Z}_{>0}$. Determine the largest possible number of distinct values that can occur among $f(1),f(2),\ldots ,f(2023)$, where $f$ is a $\mathbb{Z}_{>0}$-good function.

Determine all $k \in \mathbb{N}_0$ for which there exists a function $f \colon \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and

$$f(f(n)) \leq f(n + 1) - f(n)$$

for all $n \in \mathbb{N}_0$.

Remark. Here $\mathbb{N}_0$ denotes the set of nonnegative integers.

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

Let $\mathbb{R}^+$ be the set of positive real numbers. Let $f\colon \mathbb{R}^{+}\to \mathbb{R}^{+}$ be a function such that for all $x,y\in \mathbb{R}^{+}$ it holds that

$$y f^{2025}(x) \geq x f(y).$$

Show that there exists a positive integer $n_0$ such that for all positive integers $n \geq n_0$ and for all $x \in \mathbb{R}^+$ it holds that

$$f^n(x) \geq x.$$

Remark. Here $f^n$ denotes the function $f$ applied $n$ times, this means $f^n(x) = \underbrace{f(f(\ldots f(x)\ldots))}_{n \text{ times}}$.

Let $\mathbb{R}^+$ be the set of positive real numbers. Determine all functions $f\colon \mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all numbers $x,y\in \mathbb{R}^{+}$, we have $$f(xy) + f(x) = f(y)f(xf(y)) + f(x)f(y),$$

and there exists at most one number $a \in \mathbb{R}^+$ such that $f(a) = 1$.

Determine whether the following statement is true for every polynomial $P$ of degree at least 2 with nonnegative integer coefficients:

There exists a positive integer $m$ such that for infinitely many positive integers $n$ the number $P^n(m)$ has more than $n$ distinct positive divisors.

Remark. Here $P^n$ denotes $P$ applied $n$ times, this means $P^n(x) = \underbrace{P(P(\ldots P(x)\ldots))}_{n \text{ times}}$.

Odredi sve uređene trojke $(x, y, z)$ realnih brojeva za koje vrijedi $$\frac{2x^2}{1 + x^2} = y, \quad \frac{2y^2}{1 + y^2} = z, \quad \frac{2z^2}{1 + z^2} = x.$$

Ako funkcija $f$ zadovoljava uvjete

(a) $f(1) = 1$,

(b) $f(x + y) = f(x) + f(y), \quad \forall x, y \in \mathbf{R}$,

(c) $f\left(\frac{1}{x}\right) = \frac{f(x)}{x^2}, \quad \forall x \in \mathbf{R}, \quad x \neq 0$,

koliko je $f(\sqrt{1996})$?

Neka je $\overline{OA}$ polumjer i $\overline{OB}$ tetiva kružnice $k$ polumjera $R$, $C$ sjecište pravca $OB$ i tangente na $k$ u točki $A$, $T$ točka na dužini $\overline{OB}$ takva da je $|OT| = |BC|$ i $T'$ projekcija od $T$ na $\overline{OA}$. Izrazite $y = |T'T|$ kao funkciju od $x = |OT'|$.

Odredi broj različitih vrijednosti koje poprima izraz $$\frac{n^2 - 2}{n^2 - n + 2},$$ za $n \in \{1, 2, 3, \ldots, 2026\}$.

Odredite sve $x \in \mathbb{R}$ za koje je funkcija $f, f: \mathbb{R} \to \mathbb{R}$, $$f(x) = \sqrt{x^4 - 2x^2 + 1} + \sqrt{x^2 + 2x + 1}$$ rastuća.

Dane su dvije kvadratne funkcije $f_1(x)$ i $f_2(x)$.

Funkcija $f_1(x)$ postiže najmanju vrijednost za $x = -1$, a jedna nultočka joj je $x = 3$. Funkcija $f_2(x)$ postiže najveću vrijednost za $x = 3$, a jedna nultočka joj je $x = -1$.

Odredi sve vrijednosti $x$ za koje umnožak $f_1(x)f_2(x)$ postiže najveću vrijednost.

U ovisnosti o parametru $a \in \mathbb{R}$, odredi sliku funkcije $f(x) = \dfrac{2023}{x^2 - a^2 - x - a}$.

Dokaži da ne postoji prirodni broj $n \geqslant 2$ takav da je funkcija $$f(x) = \cos(x\sqrt{1}) + \cos(x\sqrt{2}) + \cdots + \cos(x\sqrt{n})$$ periodična.

Odredi realne brojeve $a$ i $b$ tako da skup vrijednosti funkcije $f(x) = 2022^{-2(x-2a)(x+a)} + 1$ bude interval $\langle b, 2023]$.