Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

Show that the number $n^{2} - 2^{2014}\times 2014n + 4^{2013} (2014^{2}-1)$ is not prime, where $n$ is a positive integer.

Let $ p>3$ be a prime. If $ p$ divides $ x$, prove that the equation $ x^2-1=y^p$ does not have positive integer solutions.

Find the number of positive integer n, which follows the following $ \bigstar $ $ n=[\frac{m^3}{2024}] $ $n$ has a positive integer $m$ that follows this equation ($ m \le 1000$)

Prove that there are infinitely many integers $k\geqslant 2024$ for which there exists a set $\{a_1,\ldots,a_k\}$ with the following properties:[list] [*]$a_1{}$ is a positive integer and $a_{i+1}=a_i+1$ for all $1\leqslant i<k,$ and [*]$2(a_1\cdots a_{k-2}-1)^2$ is divisible by $2(a_1+\cdots+a_k)+a_1-a_1^2.$ [/list][i](Proposed by Prajit Adhikari, Nepal)[/i]

Find the least positive integer with the property that if its digits are reversed and then $450$ is added to this reversal, the sum is the original number. For example, $621$ is not the answer because it is not true that $621 = 126 + 450$.

Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

Let $n \geq 2$ be a natural. Define $$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$. For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define $$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$ $$s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\})$$ Find the largest possible size of a proper subset $A$ of $X$ such that for any $s,t \in A$, one has $s \vee t \in A, s \wedge t \in A$.

Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, \[[r^m] \equiv -1 \pmod k .\] [i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients. [i]Proposed by Yugoslavia.[/i]

Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.

Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$

For every positive integer $n$ let $\tau (n)$ denote the number of its positive factors. Determine all $n \in N$ that satisfy the equality $\tau (n) = \frac{n}{3}$

Let $a\%b$ denote the remainder when $a$ is divided by $b$. Find $\Sigma_{i=1}^{100}(100\%i)$.

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

Show that the equation $x^3 +y^3 = 4(x^2y+xy^2 +1)$ has no integer solutions.

Let $n$ be an odd positive integer. Prove that if the equation $\frac1x+\frac1y=\frac4n$ has a solution in positive integers $x,y$, then $n$ has at least one divisor of the form $4k-1$, $k\in\mathbb N$.

For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that: a.) $N_a$ is odd; b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

Find the maximum number of colors used in coloring integers $n$ from $49$ to $94$ such that if $a, b$ (not necessarily different) have the same color but $c$ has a different color, then $c$ does not divide $a+b$.

Let $p,q$ be prime numbers$.$ Prove that if $p+q^2$ is a perfect square$,$ then $p^2+q^n$ is not a perfect square for any positive integer $n.$

Ana and Bojan are playing a game: Ana chooses positive integers $a$ and $b$ and each one gets $2016$ pieces of paper, visible to both - Ana gets the pieces with the numbers $a+1$, $a+2$, $\ldots$, $a+2016$ and Bojan gets the pieces with the numbers $b+1$, $b+2$, $\ldots$, $b+2016$ on them. Afterwards, one of them writes the number $a+b$ on the board. In every move, Ana chooses one of her pieces of paper and hands it to Bojan who chooses one of his own, writes their sum on the board and removes them both from the game. When they run out of pieces, they multiply the numbers on the board together. If the result has the same remainder than $a+b$ when divided by $2017$, Bojan wins, otherwise, Ana wins. Who has the winning strategy?

Prove that the equation $4^x +6^x =9^x$ has no rational solutions.

Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.

Let $p_n$ be the probability that $c+d$ is a perfect square when the integers $c$ and $d$ are selected independently at random from the set $\{1,2,\ldots,n\}$. Show that $\lim_{n\to\infty}p_n\sqrt n$ exists and express this limit in the form $r(\sqrt s-t)$, where $s$ and $t$ are integers and $r$ is a rational number.

Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.