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Nađi (jedan) cijeli broj $a$ takav da za polinom $P(x) = x^5 + ax$ tvrdnja $$\text{ako } n \mid P(k) - P(l) \text{ onda } n \mid k - l, \text{ za sve } k, l \in \mathbb{Z}$$ vrijedi samo za konačno mnogo prirodnih brojeva $n$, među kojima je i $n = 95$.

Neka su $P(x)$ i $Q(x)$ polinomi s realnim koeficijentima takvi da je

$$P(P(x)) = (Q(x))^2$$

za svaki realni broj $x$. Postoji li nužno polinom $R(x)$, također s realnim koeficijentima, takav da je $P(x) = (R(x))^2$ za svaki realni broj $x$?

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

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

Consider the sequence $\{c_n\}$, where $$\begin{aligned} c_1 &= a_1 + a_2 + \cdots + a_8 \\ c_2 &= a_1^2 + a_2^2 + \cdots + a_8^2 \\ &\cdots \\ c_n &= a_1^n + a_2^n + \cdots + a_8^n \\ &\cdots \end{aligned}$$ in which $a_1, a_2, \ldots, a_8$ are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence $\{c_n\}$ are equal to zero. Find all natural numbers $n$ for which $c_n = 0$.

Prove that the following assertion is true for $n = 3$ and $n = 5$, and that it is false for every other natural number $n > 2$: If $a_1, a_2, \ldots, a_n$ are arbitrary real numbers, then $$(a_1 - a_2)(a_1 - a_3) \cdots (a_1 - a_n) + (a_2 - a_1)(a_2 - a_3) \cdots (a_2 - a_n)$$ $$+ \cdots + (a_n - a_1)(a_n - a_2) \cdots (a_n - a_{n-1}) \geq 0$$

Let $a$ and $b$ be real numbers for which the equation $$x^4 + ax^3 + bx^2 + ax + 1 = 0$$ has at least one real solution. For all such pairs $(a,b)$, find the minimum value of $a^2 + b^2$.

Let $P$ be a non-constant polynomial with integer coefficients. If $n(P)$ is the number of distinct integers $k$ such that $(P(k))^2 = 1$, prove that $n(P) - \deg(P) \leq 2$, where $\deg(P)$ denotes the degree of the polynomial $P$.

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 $P_1(x) = x^2 - 2$ 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.

For any polynomial $P(x) = a_0 + a_1x + \cdots + a_kx^k$ with integer coefficients, the number of coefficients which are odd is denoted by $w(P)$. For $i = 0, 1, \ldots$, let $Q_i(x) = (1+x)^i$. Prove that if $i_1, i_2, \ldots, i_n$ are integers such that $0 \leq i_1 < i_2 < \cdots < i_n$, then $$w(Q_{i_1} + Q_{i_2} + \cdots + Q_{i_n}) \geq w(Q_{i_1}).$$

Let $n$ be an integer greater than or equal to 2. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{n/3}$, then $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

Let $f(x) = x^n + 5x^{n-1} + 3$, where $n > 1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

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 $$f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c).$$

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 $Q(t) = t$.

A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n) = n^2 + n + 1$. What is the least possible value of the positive integer $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a + 1), P(a + 2), \ldots, P(a + b)\}$$ is fragrant?

The equation $$(x - 1)(x - 2) \cdots (x - 2016) = (x - 1)(x - 2) \cdots (x - 2016)$$ is written on the board, with 2016 linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these 4032 linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is 1. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots, a_n$ such that, for each $(x, y)$ in $S$, we have:

$$a_0 x^n + a_1 x^{n-1} y + a_2 x^{n-2} y^2 + \cdots + a_{n-1} x y^{n-1} + a_n y^n = 1.$$

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around a circle such that the product of any two neighbours is of the form $x^2 + x + k$ for some positive integer $x$.

For each integer $k \geqslant 2$, determine all infinite sequences of positive integers $a_1, a_2, \ldots$ for which there exists a polynomial $P$ of the form $P(x) = x^k + c_{k-1}x^{k-1} + \cdots + c_1x + c_0$, where $c_0, c_1, \ldots, c_{k-1}$ are non-negative integers, such that

$$P(a_n) = a_{n+1}a_{n+2}\cdots a_{n+k}$$

for every integer $n \geqslant 1$.

Let $P(x)$ be a polynomial of degree $n \geq 2$ with rational coefficients such that $P(x)$ has $n$ pairwise different real roots forming an arithmetic progression. Prove that among the roots of $P(x)$ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients.

Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x + \alpha) \leq (x^{20} + x^{19})P(x)$$ holds for all real numbers $x$.

Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor) = \lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$

(a) there exists an $n$-pretty polynomial;

(b) any $n$-pretty polynomial has a degree of at least $\frac{2n + 1}{3}$.

(Remark. For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)

For any positive integer $n$, let $\sigma(n)$ denote the sum of positive divisors of $n$. Determine all polynomials $P$ with integer coefficients such that $P(k)$ is divisible by $\sigma(k)$ for all positive integers $k$.

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 $P^n(x) = \underbrace{P(P(\ldots P(x)\ldots))}_{n \text{ times}}$.

Nadite realna rješenja sustava jednadžbi: $$x + y + z = 2$$ $$(x + y)(y + z) + (y + z)(z + x) + (z + x)(x + y) = 1$$ $$x^{2}(y + z) + y^{2}(z + x) + z^{2}(x + y) = -6$$

Odredi najmanji prirodni broj $a$ takav da je vrijednost izraza

$$\frac{n^8 + n^6 + n^4 + n^2 + a}{n^2 - 1}$$

za $n = 2014$ cijeli broj djeljiv sa $3$.

Za realne brojeve $a$, $b$ i $c$ vrijedi $a + b + c = 0$ i $abc = 4$. Odredi vrijednost izraza $a^3 + b^3 + c^3$.

Koji je broj veći, $$A = 2022 \cdot (2024^2 - 2024 + 1) \quad \text{ili} \quad B = 2024^3 - 3 \cdot 2024^2 + 3 \cdot 2024?$$

Izračunaj $$2 \cdot \frac{2027^{3} + 2025^{3}}{2^{2} + 2027 \cdot 2025} - 4052 \cdot \frac{2027^{3} - 2025^{3}}{4052^{2} - 2027 \cdot 2025}.$$

Ako za realne brojeve $a, b, c$ vrijedi $(a + b + c)^3 = a^3 + b^3 + c^3$, dokaži da je $$(a + b)^2 ab + (b + c)^2 bc + (c + a)^2 ca + 4abc(a + b + c) = 0.$$

Neka je $f: \mathbf{R} \to \mathbf{R}$ kvadratna funkcija $f(x) = ax^2 + bx + c$. Označimo sa $D$ diskriminantu, sa $P$ umnožak, a sa $S$ zbroj njezinih nultočaka. Pokažite da postoji samo jedna funkcija $f$ za koju su $a, D, P, S$ četiri uzastopna cijela broja (u rastućem poretku).

Za koje cijele brojeve $x$ izraz $x^2 + 1$ dijeli $x^3 - 8x^2 + 2x$?

Za koje realne brojeve $a$, $b$ su moduli svih korijena jednadžbe $z^3 + az^2 + bz - 1 = 0$ jednaki $1$?

Riješite jednadžbu $$2z^3 - (5 + 6i)z^2 + 9iz + 1 - 3i = 0$$ ako se zna da je jedno njezino rješenje realno.

U zavisnosti o parametru $a$ nađite rješenja jednadžbe $$x^4 - 2ax^2 + x + a^2 - a = 0.$$ Za koje realne brojeve $a$ su sva rješenja realna?

Nadite sva rješenja jednadžbe $$(x^2 + 3x - 4)^3 + (2x^2 - 5x + 3)^3 = (3x^2 - 2x - 1)^3.$$

Dokaži da svaki kompleksni broj $z$ za koji postoji točno jedan kompleksni broj $a$ takav da je $$z^3 + (2 - a) z^2 + (1 - 3a) z + a^2 - a = 0$$ zadovoljava jednakost $z^3 = 1$.

Neka su $a$ i $b$ realni brojevi takvi da su sve nultočke polinoma

$$P(x) = x^3 + a x^2 + b x - 8$$

realne. Dokaži da vrijedi $a^2 \geqslant 2b + 12$.

Neka je $a$ kompleksni broj takav da vrijedi

$$a^5 + a + 1 = 0.$$

Koje vrijednosti može poprimiti izraz $a^2 (a - 1)$?

Odredi sve parove realnih brojeva $(x,y)$ koji zadovoljavaju sustav

$$\begin{aligned} x + y^2 &= y^3, \\ y + x^2 &= x^3. \end{aligned}$$

Branko ispisuje niz kvadratnih polinoma s realnim koeficijentima. U svakom koraku, nakon prethodno napisanog polinoma $ax^2 + bx + c$, zapisuje polinom $cx^2 + bx + a$ ili polinom $a(x + d)^2 + b(x + d) + c$ za neki realni broj $d$.

Ako započne s polinomom $x^2 - 2x - 1$, može li Branko opisanim postupkom nakon određenog broja koraka dobiti polinom:

a) $2x^2 - 1$?

b) $2x^2 - x - 1$?

Odredi sve kompleksne brojeve $a$ za koje su svi koeficijenti polinoma

$$P(x) = (x - a)(x - a^2)(x - a^3)$$

realni.

Koeficijenti $a$, $b$ i $c$ kvadratne jednadžbe $ax^2 + bx + c = 0$ tri su uzastopna prirodna broja (u nekom od šest mogućih poredaka), a njezina su rješenja realni brojevi. Dokaži da je jedno od rješenja broj $-1$.

Mihael je na ploči zapisao kvadratnu funkciju $f(x)$ s cjelobrojnim koeficijentima. Nakon toga, u svakom je koraku promijenio (povećao ili smanjio) za 1 ili koeficijent uz $x$ ili konstantni član. U zadnjem koraku je na ploči zapisana kvadratna funkcija $g(x)$.

Je li sigurno da je u nekom trenutku na ploči bila zapisana kvadratna funkcija s cjelobrojnim nultočkama ako je

a) $f(x) = x^2 + x + 2024$ i $g(x) = x^2 + 2024x + 1$?

b) $f(x) = x^2 + 2024x + 2024$ i $g(x) = x^2 - 2024x + 2024$?

Za realne brojeve $a$ i $b$ jednadžba $x^2 + ax + b = 0$ ima dva cjelobrojna rješenja (ne nužno različita). Dokaži da jednadžba $x^2 + 5ax + (6a^2 + b) = 0$ također ima dva cjelobrojna rješenja.

Neka su kompleksni brojevi $a$, $b$ i $c$ rješenja jednadžbe $x^3 - 2x + 2 = 0$. Odredi $$\frac{a + 1}{a - 1} + \frac{b + 1}{b - 1} + \frac{c + 1}{c - 1}.$$

Jednadžba $$(x^2 + ax + b)(x^2 + ax + c) = 0$$ ima četiri različita realna rješenja i to su $a$, $b$, $c$ i $-1$. Odredi brojeve $a$, $b$ i $c$.

Odredi sve polinome $P$ s realnim koeficijentima takve da za sve realne brojeve $x$ vrijedi

$$P(x^2) + 2P(x) = (P(x))^2 + 2.$$

Neka je $S = \{0,95\}$. U svakom koraku Lucija proširuje skup $S$ tako da odabire neki polinom s koeficijentima iz $S$, različit od nulpolinoma, te skupu $S$ dodaje sve cjelobrojne nultočke tog polinoma. Postupak nastavlja odabirom drugog polinoma s koeficijentima iz tako proširenog skupa $S$ dok god na taj način može dobiti nove nultočke.

Dokaži da Lucija može konačnim nizom koraka proširiti skup $S$ do skupa koji nije moguće dalje proširiti. Koliko elemenata tada ima skup $S$?