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Za prirodni broj $d$ definiran je niz $$a_0 = 1, \qquad a_{n+1} = \begin{cases} \dfrac{a_n}{2}, & \text{ako je } a_n \text{ paran}, \\ a_n + d, & \text{inače}. \end{cases}$$

Za koje vrijednosti broja $d$ postoji prirodni broj $n$ za koji je $a_n = 1$?

Odredi sve nizove $a: \mathbb{N} \to \mathbb{N}$ takve da za sve $n \in \mathbb{N}$ vrijedi: $$a_n + a_{n+1} = a_{n+2}a_{n+3} - 200.$$

Zadan je niz realnih brojeva: $$x_0 = 1,$$ $$x_1 = 1,$$ $$x_n = \sqrt{\frac{n}{2} + x_{n-1}x_{n-2}}, \quad \text{za } n \geqslant 2.$$

Postoji li realni broj $A$ takav da je $An < x_n < An + 1$ za svaki $n \in \mathbb{N}$?

Odredi sve periodične nizove $(x_n)_{n \in \mathbb{N}}$ pozitivnih realnih brojeva sa svojstvom da za sve $n \in \mathbb{N}$ vrijedi $$x_{n+2} = \frac{1}{2}\left(\frac{1}{x_{n+1}} + x_n\right).$$

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.

A sequence $\{u_n\}$ is defined by $$u_0 = 2, \quad u_1 = 5/2, \quad u_{n+1} = u_n(u_{n-1}^2 - 2) - u_1 \text{ for } n = 1, 2, \cdots$$

Prove that for positive integers $n$, $$[u_n] = 2^{[2^n - (-1)^n]/3}$$ where $[x]$ denotes the greatest integer $\leq x$.

Let $A$ and $E$ be opposite vertices of a regular octagon. A frog starts jumping at vertex $A$. From any vertex of the octagon except $E,$ it may jump to either of the two adjacent vertices. When it reaches vertex $E,$ the frog stops and stays there. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E.$ Prove that $a_{2n-1}=0,$

$$a_{2n}=\frac{1}{\sqrt{2}}(x^{n-1}-y^{n-1}),\quad n=1,2,3,\cdots,$$

where $x=2+\sqrt{2}$ and $y=2-\sqrt{2}.$

Note. A path of $n$ jumps is a sequence of vertices $(P_0,\ldots,P_n)$ such that

(i) $P_0=A, P_n=E;$

(ii) for every $i, 0\leq i\leq n-1, P_i$ is distinct from $E;$

(iii) for every $i, 0\leq i\leq n-1, P_i$ and $P_{i+1}$ are adjacent.

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 $f(4, 1981)$.

For every real number $x_1$, construct the sequence $x_1, x_2, \ldots$ by setting $$x_{n+1} = x_n\left(x_n + \frac{1}{n}\right) \text{ for each } n \geq 1.$$ Prove that there exists exactly one value of $x_1$ for which $$0 < x_n < x_{n+1} < 1$$ for every $n$.

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 $f(n) = n$.

There are $n$ lamps $L_0, \ldots, L_{n-1}$ in a circle ($n > 1$), where we denote $L_{n+k} = L_k$. (A lamp at all times is either on or off.) Perform steps $s_0, s_1, \ldots$ as follows: at step $s_i$, if $L_{i-1}$ is lit, switch $L_i$ from on to off or vice versa, otherwise do nothing. Initially all lamps are on. Show that:

(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all the lamps are on again;

(b) If $n = 2^k$, we can take $M(n) = n^2 - 1$;

(c) If $n = 2^k + 1$, we can take $M(n) = n^2 - n + 1$.

Find the maximum value of $x_0$ for which there exists a sequence $x_0, x_1, \ldots, x_{1995}$ of positive reals with $x_0 = x_{1995}$, such that for $i = 1, \ldots, 1995$, $$x_{i-1} + \frac{2}{x_{i-1}} = 2x_i + \frac{1}{x_i}.$$

Suppose that $s_1, s_2, s_3, \ldots$ is a strictly increasing sequence of positive integers such that the subsequences

$$s_{s_1}, s_{s_2}, s_{s_3}, \ldots \quad \text{and} \quad s_{s_1 + 1}, s_{s_2 + 1}, s_{s_3 + 1}, \ldots$$

are both arithmetic progressions. Prove that the sequence $s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ by:

$$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise,} \end{cases} \quad \text{for each } n \geqslant 0.$$

Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \ldots, a_{n+2}$, such that $a_{n+1} = a_1$ and $a_{n+2} = a_2$, and $$a_i a_{i+1} + 1 = a_{i+2}$$ for $i = 1, 2, \ldots, n$.

Let $a_1, a_2, a_3, \ldots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n > N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1, a_2, \ldots, a_{n-1}$.

Prove that at least one of the sequences $a_1, a_3, a_5, \ldots$ and $a_2, a_4, a_6, \ldots$ is eventually periodic.

(An infinite sequence $b_1, b_2, b_3, \ldots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p} = b_m$ for all $m \geq M$.)

Determine all real numbers $A$ such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying

$$x_{n+1} = A - \frac{1}{x_n}$$

for every integer $n \geqslant 1$, has only finitely many negative terms.

Given a pair $(a_0, b_0)$ of real numbers, we define two sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ of real numbers by $$a_{n+1} = a_n + b_n \quad \text{and} \quad b_{n+1} = a_n \cdot b_n$$ for all $n = 0, 1, 2, \ldots$. Find all pairs $(a_0, b_0)$ of real numbers such that $a_{2022} = a_0$ and $b_{2022} = b_0$.

Let $A$ and $B$ be positive integers. Consider a sequence of positive integers $(x_{n})_{n\geq 1}$ such that

$$x_{n+1} = A \cdot \gcd(x_{n}, x_{n-1}) + B \quad \text{for every } n \geq 2.$$

Prove that the sequence attains only finitely many different values.

Remark. We denote by $\gcd(a, b)$ the greatest common divisor of positive integers $a$ and $b$.

Consider the two infinite sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ of real numbers such that $a_0 = 0$, $b_0 = 0$ and $$a_{k+1} = b_k, \qquad b_{k+1} = \frac{a_k b_k + a_k + 1}{b_k + 1}$$ for each integer $k \geq 0$. Prove that $a_{2024} + b_{2024} \geq 88$.

Niz znamenaka $1, 2, 3, 4, 0, 9, 6, 9, 4, 8, 7, \ldots$ konstruira se tako da je svaki broj, počevši od petog, jednak znamenki jedinica zbroja prethodne četiri znamenke.

a) Da li se u tom nizu redom pojavljuju znamenke $2, 0, 0, 4$, tim redom?

b) Da li se u tom nizu ikad ponavljaju početne znamenke $1, 2, 3, 4$, tim redom?

Zadan je niz $\{a_n\}$ rekurzivnom formulom $$a_{n+1} = \frac{a_n^2 + b}{2}, \quad 0 < b \leq 1, \; a_0 = 0.$$ Pokažite da je niz konvergentan i izračunajte mu limes.

Zadan je niz $x_1 = 1$, $x_2 = 2$, $x_3 = 4$, $x_{n+3} = x_{n+2} + x_{n+1} + x_n$, za svako $n \in \mathbb{N}$. Dokažite da se svaki prirodni broj može prikazati kao zbroj različitih elemenata tog niza.

Dana je funkcija $f$ definirana na pozitivnim cijelim brojevima, koja ima ova svojstva $$f(1) = 1, \quad f(2) = 2,$$ $$f(n + 2) = f(n + 2 - f(n + 1)) + f(n + 1 - f(n)), \quad (n \geq 1).$$

(a) Pokažite da je $f(n + 1) - f(n) \in \{0, 1\}$ za svaki $n \geq 1$.

(b) Ako je $f(n)$ neparan, pokažite da je $f(n + 1) = f(n) + 1$.

(c) Za dani prirodan broj $k$ odredite sve vrijednosti $n$ za koje je $$f(n) = 2^{k-1} + 1.$$

Izračunajte sumu $$\frac{a_1}{2} + \frac{a_2}{2^2} + \frac{a_3}{2^3} + \ldots + \frac{a_k}{2^k} + \ldots$$ gdje je $(a_n)$ niz brojeva definiran na ovaj način: $$a_1 = 1, \quad a_2 = 1, \quad a_n = a_{n-1} + a_{n-2}, \quad \text{za } n > 2.$$

Niz realnih brojeva $(a_n)_{n \geq 0}$ ima svojstvo da za sve $m \geq n \geq 0$ vrijedi $$a_{m+n} + a_{m-n} = \frac{1}{2}(a_{2m} + a_{2n}).$$ Odredite $a_{2003}$ ako je $a_1 = 1$.

Nizovi realnih brojeva $(x_n)$, $(y_n)$, $(z_n)$, $n \in \mathbb{N}$, definirani su formulama $$x_{n+1} = \frac{2x_n}{x_n^2 - 1}, \quad y_{n+1} = \frac{2y_n}{y_n^2 - 1}, \quad z_{n+1} = \frac{2z_n}{z_n^2 - 1},$$

a početni članovi su $x_1 = 2$, $y_1 = 4$ i $z_1$ takav da vrijedi $x_1 y_1 z_1 = x_1 + y_1 + z_1$.

a) Provjerite da su za svaki $n \in \mathbb{N}$ zadovoljeni uvjeti: $x_n^2 \neq 1$, $y_n^2 \neq 1$, $z_n^2 \neq 1$.

b) Da li postoji $k \in \mathbb{N}$ takav da je $x_k + y_k + z_k = 0$?

Niz $(a_n)_{n \in \mathbb{N}}$ je zadan rekurzivno s $a_1 = 1$, $$a_n = a_1 \cdots a_{n-1} + 1, \quad \text{za } n \geq 2.$$ Odredite najmanji realni broj $M$ takav da je $$\sum_{n=1}^{m} \frac{1}{a_n} < M \quad \text{za svaki } m \in \mathbb{N}.$$

Niz $(a_n)$ zadan je rekurzivno: $$\begin{aligned} a_0 &= 3 \\ a_n &= 2 + a_0 \cdot a_1 \cdot \dots \cdot a_{n-1}, \quad n \geq 1. \end{aligned}$$

a) Dokažite da su svi članovi tog niza u parovima relativno prosti prirodni brojevi.

b) Odredite $a_{2007}$.

Neka su $p_1$ i $q_1$ cijeli brojevi takvi da jednadžba $x^2 + p_1x + q_1 = 0$ ima dva cjelobrojna rješenja. Za svaki $n \in \mathbb{N}$ definiramo brojeve $p_{n+1}$ i $q_{n+1}$ formulama $$p_{n+1} = p_n + 1, \quad q_{n+1} = q_n + \frac{1}{2} p_n.$$

Dokaži da postoji beskonačno mnogo prirodnih brojeva $n$ za koje jednadžba $x^2 + p_nx + q_n = 0$ ima dva cjelobrojna rješenja.

Niz $(a_n)$ zadan je rekurzivno: $a_1 = 2$, $a_n = 2(n + a_{n-1})$ za $n \geqslant 2$.

Dokaži da je $a_n < 2^{n+2}$ za sve $n \in \mathbb{N}$.

Neka je $(x_n)$ niz takav da je $x_0 = 1$, $x_1 = 2$, sa svojstvom da je niz $(y_n)$ zadan relacijom $$y_n = \binom{n}{0}x_0 + \binom{n}{1}x_1 + \ldots + \binom{n}{n}x_n, \quad \text{za } n \in \mathbb{N}_0$$ geometrijski niz. Odredi $x_{2020}$.

Koristeći niz $(a_n)_{n \in \mathbb{N}}$ definirana su dva nova niza, $(b_n)_{n \in \mathbb{N}}$ i $(c_n)_{n \in \mathbb{N}}$ tako da za svaki prirodan broj $n$ vrijedi $$b_n = a_{n+1} - \sum_{i=1}^{n} a_i, \quad c_n = a_{n+2} - a_{n+1}.$$

Ako je niz $(b_n)_{n \in \mathbb{N}}$ aritmetički, dokaži da je $(c_n)_{n \in \mathbb{N}}$ geometrijski niz.

Zadan je niz $(a_n)$ takav da je $a_0 = 1$, $a_1 = 4$ i

$$a_n = 3a_{n-1} + 4a_{n-2},$$

za svaki prirodni broj $n \geqslant 2$.

Dokaži da su svi članovi niza $(a_n)$ kvadrati prirodnih brojeva.

Rekurzivno je zadan niz: $$a_1 = 1, \quad a_2 = 3,$$ $$a_n = (n + 1)a_{n-1} - na_{n-2} \quad \text{za } n \geqslant 3.$$ Odredi sve prirodne brojeve $n$ za koje je $a_n$ djeljivo s $9$.

Neka je $(a_n)$ niz definiran sa $a_1 = 1$, $a_2 = 2$ i $$a_n = a_{n-1} + (n-1)a_{n-2} \quad \text{za } n \geqslant 3.$$ Dokaži da vrijedi $a_{2024} \geqslant \sqrt{2024!}$.

Zadan je niz $(a_n)_{n\in \mathbb{N}_0}$ takav da je $a_0 = a$, $a_1 = b$, gdje su $a$, $b\in \mathbb{R}$, i

$$a_n = a_{n-1} + a_{n-2}, \quad n \geq 2.$$

Odredite $a_n^2 - a_{n-1}a_{n+1}$.

Dokaži da je za svaki prirodan broj $n$ broj $$\frac{(2 + \sqrt{2})^n - (2 - \sqrt{2})^n}{\sqrt{2}}$$ također prirodan.

Neka je $(a_n)$ niz pozitivnih realnih brojeva takav da je $a_1 = 1$ i $a_{n+1}^2 + a_{n+1} = a_n$ za sve $n \in \mathbb{N}$. Dokaži da je $a_n \geqslant \frac{1}{n}$ za sve $n \in \mathbb{N}$.

Dani su aritmetički niz $(a_n)_{n\in\mathbb{N}}$ i geometrijski niz $(b_n)_{n\in\mathbb{N}}$ takvi da su im svi članovi pozitivni realni brojevi i da vrijedi

$$a_1 = b_1, \quad a_2 = b_2, \quad \text{i} \quad a_{10} = b_3.$$

Dokaži da se svaki član niza $(b_n)_{n\in\mathbb{N}}$ pojavljuje u nizu $(a_n)_{n\in\mathbb{N}}$.