Prove that the number $x^8+\frac{1}{x^8}$ is an integer if $x+\frac{1}{x }$ is an integer.

Let $a_0 ,a_1 , \ldots, a_n$ be real numbers and let $f(x) =a_n x^n +\ldots +a_1 x +a_0.$ Suppose that $f(i)$ is an integer for all $i.$ Prove that $n! \cdot a_k$ is an integer for each $k.$

Show that there exist infinitely many mutually non- congruent triangles $T$, satisfying (i) The side lengths of $T $ are consecutive integers. (ii) The area of $T$ is an integer.

Show that there are infinitely many positive integers $a$ that cannot be written as $a = a_{1}^{6}+ a_{2}^{6} + \ldots + a_{7}^{6},$ where the $a_i$ are positive integers. State and prove a generalization.

(a) Let $a_0, a_1,a_2$ be real numbers and consider the polynomial $P(x) = a_0 + a_1x + a_2x^2$ . Assume that $P(-1), P(0)$ and $P(1)$ are integers. Prove that $P(n)$ is an integer for all integers $n$. (b) Let $a_0,a_1, a_2, a_3$ be real numbers and consider the polynomial $Q(x) = a0 + a_1x + a_2x^2 + a_3x^3 $. Assume that there exists an integer $i$ such that $Q(i),Q(i+1),Q(i+2)$ and $Q(i+3)$ are integers. Prove that $Q(n)$ is an integer for all integers $n$.

Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.

Given rational numbers $a_1,...,a_n$ such that $\sum_{i=1}^n \{ka_i\}<\frac{n}{2}$ for any positive integer $k$. a) Prove that at least one of $a_1,...,a_n$ is integer. b) Is the previous statement true, if the number $\frac{n}{2}$ is replaced by the greater number? (Here $\{x\}$ means a fractional part of $x$.) (N. Selinger)

Find all integers that can be written in the form $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{9}{a_9}$ where $a_1,a_2, ...,a_9$ are nonzero digits, not necessarily different.

Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$?

Find all triples $(a, b, c)$ of positive integers for which $\frac{32a + 3b + 48c}{4abc}$ is also an integer.

In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on. If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won. Prove that Kain can force a win in a finite number of moves.

Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $ Let $s,t$ be two different positive integers with the following property: If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$. Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer. (FYROM)

The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$

Let $n$ be a positive integer. Prove that the interval $I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)$ does not contain any integer.