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Neka je $n > 1$ prirodni broj. Dokaži da jednadžba

$$(x + 1)^n - x^n = ny$$

nema rješenja u skupu prirodnih brojeva.

Dokaži da ne postoje prosti broj $p$ i prirodni brojevi $a$ i $n$ ($n \geqslant 2$) takvi da vrijedi

$$2^p + 3^p = a^n.$$

Odredi sve parove $(x, y)$ cijelih brojeva za koje vrijedi $$x^3 + x^2 + x = y^2 + y.$$

Postoje li cijeli brojevi $a$ i $b$ takvi da su oba broja $a + b$ i $a \cdot b - 1$ potpuni kvadrati?

Za dani prirodni broj $k$ neka je $S(k)$ zbroj svih brojeva iz skupa $\{1,2,\ldots,k\}$ koji su relativno prosti s $k$. Neka je $m$ prirodni i $n$ neparni prirodni broj. Dokaži da postoje prirodni brojevi $x$ i $y$, pri čemu $m$ dijeli $x$, takvi da vrijedi $2S(x) = y^n$.

Nađi sve prirodne brojeve $a$ i $b$ takve da

$$(a^2 + b) \mid (a^2b + a) \quad \text{i} \quad (b^2 - a) \mid (ab^2 + b).$$

Neka su $a$, $b$, $c$ različiti prirodni brojevi i neka su $r$, $s$, $t$ prirodni brojevi takvi da vrijedi: $$ab + 1 = r^2, \quad ac + 1 = s^2, \quad bc + 1 = t^2.$$

Dokaži da ne mogu sva tri razlomka $\dfrac{rs}{t}$, $\dfrac{rt}{s}$, $\dfrac{st}{r}$ biti prirodni brojevi.

Odredi sve prirodne brojeve $x$ i $y$ takve da vrijedi $$x(x^2 + 19) = y(y^2 - 10).$$

Odredi sve parove $(p, q)$ prostih prirodnih brojeva takve da vrijedi

$$p(p^2 - p - 1) = q(2q + 3).$$

Odredi sve parove prirodnih brojeva $(m,n)$ takve da vrijedi

$$3 \cdot 5 ^ {m} - 2 \cdot 6 ^ {n} = 3.$$

Dokaži da ne postoje prirodni brojevi $m$ i $n$ takvi da je

$$5m^3 = 27n^4 - 2n^2 + n.$$

Odredi sve prirodne brojeve $n$ za koje postoje prirodni brojevi $a$ i $b$ takvi da vrijedi

$$(n^2 + 2)^a = (2n - 1)^b.$$

Odredi sve parove prirodnih brojeva $(m,n)$ za koje vrijedi

$$2^m = 7n^2 + 1.$$

Odredi sve parove $(a,b)$ prirodnih brojeva za koje vrijedi $$a^5 + a^4 = 7^b - 1.$$

Odredi sve trojke prirodnih brojeva $(p, m, n)$ takve da je $p$ prost, $m < n$, sa svojstvom da postoje prirodni brojevi $a$, $b$, $c$, $d$ koji nisu djeljivi s $p$ takvi da vrijedi

$$\begin{aligned} ab + cd &= p^m, \\ ac + bd &= p^n. \end{aligned}$$

Odredi sve parove različitih prirodnih brojeva $(a, b)$ za koje vrijedi

$$ab(b - a) \mid a^4 - b^2.$$

Odredi sve parove prirodnih brojeva $(m,n)$ za koje postoje prirodni brojevi $a$, $b$, $c$ i $d$ takvi da je

$$\frac{m^a}{n^b} + \frac{n^c}{m^d}$$

prirodan broj, ali broj $\dfrac{m^a}{n^b}$ nije prirodan.

Odredi sve trojke prirodnih brojeva $(m,n,p)$ takve da je $p$ prost i da vrijedi

$$n(n + 1)(n + p)(n + p + 1)p^6m^2 = (4n + 3)^6(p + 1)^2(p - 4)^2.$$

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$.

Let $a$ and $b$ be positive integers. When $a^2 + b^2$ is divided by $a + b$, the quotient is $q$ and the remainder is $r$. Find all pairs $(a, b)$ such that $q^2 + r = 1977$.

Determine the maximum value of $m^3 + n^3$, where $m$ and $n$ are integers satisfying $m, n \in \{1, 2, \ldots, 1981\}$ and $(n^2 - mn - m^2)^2 = 1$.

Prove that if $n$ is a positive integer such that the equation

$$x^3 - 3xy^2 + y^3 = n$$

has a solution in integers $(x, y)$, then it has at least three such solutions.

Show that the equation has no solutions in integers when $n = 2891$.

Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Show that

$$\frac{a^2 + b^2}{ab + 1}$$

is the square of an integer.

Find all integers $a, b, c$ with $1 < a < b < c$ such that

$(a - 1)(b - 1)(c - 1)$ is a divisor of $abc - 1$.

Determine all ordered pairs $(m, n)$ of positive integers such that

$$\frac{n^3 + 1}{mn - 1}$$

is an integer.

The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Determine all pairs $(a,b)$ of positive integers such that $ab^2 + b + 7$ divides $a^2b + a + b$.

Find all pairs of integers $m>2,n>2$ such that there are infinitely many positive integers $k$ for which $k^{n}+k^{2}-1$ divides $k^{m}+k-1$.

Find all pairs $(m,n)$ of positive integers such that $\frac{m^2}{2mn^2 - n^3 + 1}$ is a positive integer.

Determine all pairs $(x, y)$ of integers such that $$1 + 2^x + 2^{2x+1} = y^2.$$

Let $a$ and $b$ be positive integers. Show that if $4ab - 1$ divides $(4a^2 - 1)^2$, then $a = b$.

Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that $$\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.$$

Determine all triples $(a, b, c)$ of positive integers such that each of the numbers $$ab - c, \quad bc - a, \quad ca - b$$ is a power of 2.

(A power of 2 is an integer of the form $2^n$, where $n$ is a non-negative integer.)

Find all triples $(a, b, p)$ of positive integers with $p$ prime and

$$a^p = b! + p.$$

Determine all pairs $(a, b)$ of positive integers for which there exist positive integers $g$ and $N$ such that $$\gcd(a^n + b, b^n + a) = g$$ holds for all integers $n \geq N$. (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y$.)

Find all pairs $(m, n)$ of integers which satisfy the equation $$(m + n)^4 = m^2n^2 + m^2 + n^2 + 6mn.$$

Find all non-negative integer solutions of the equation $$2^x + 2009 = 3^y5^z.$$

Let $a$ and $b$ be positive integers. Prove that there exist positive integers $x$ and $y$ such that $$\binom{x + y}{2} = ax + by.$$

Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that $$\frac{a^m + b^m}{a^n + b^n}$$ is an integer.

Find all pairs of positive integers $(a,b)$ such that $$a! + b! = a^b + b^a.$$

We consider the equation $a^2 + b^2 + c^2 + n = abc$, where $a, b, c$ are positive integers.

Prove:

(a) There are no solutions $(a,b,c)$ for $n = 2017$.

(b) For $n = 2016$, $a$ must be divisible by $3$ for every solution $(a, b, c)$.

(c) The equation has infinitely many solutions $(a, b, c)$ for $n = 2016$.

An integer $n$ is called Silesian if there exist positive integers $a$, $b$ and $c$ such that $$n = \frac{a^2 + b^2 + c^2}{ab + bc + ca}.$$

(a) Prove that there are infinitely many Silesian integers.

(b) Prove that not every positive integer is Silesian.

Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \ldots, x_n$ such that

$$\dfrac{1}{x_1^2} + \dfrac{2}{x_2^2} + \dfrac{4}{x_3^2} + \cdots + \dfrac{2^{n-1}}{x_n^2} = 1.$$

Find all positive integers $n$ for which there exist positive integers $a > b$ satisfying

$$n = \frac{4ab}{a - b}.$$

Da li jednadžba $y^2 = x^2 + 1990$ ima cjelobrojno rješenje?

Nađite sva cjelobrojna rješenja jednadžbe $$\frac{1}{m} + \frac{1}{n} - \frac{1}{mn^2} = \frac{3}{4}.$$

Nađite sve prirodne brojeve $m$ i $n$ koji zadovoljavaju jednadžbu $$10(m + n) = mn.$$

Riješite jednadžbu $$\frac {1}{x} + \frac {2}{y} - \frac {3}{z} = 1$$ u skupu prirodnih brojeva.

Nadite sve trojke $(x, y, z)$ prirodnih brojeva koji zadovoljavaju jednadžbu $$2x^2 y^2 + 2y^2 z^2 + 2z^2 x^2 - x^4 - y^4 - z^4 = 576.$$ Naputak: Izraz s lijeve strane jednadžbe rastavite na faktore.

Odredi sve parove $(x, y)$ cijelih brojeva za koje vrijedi $$6x^2y^2 - 4y^2 = 2012 - 3x^2.$$