Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]

Let the function $f:N^*\to N^*$ such that [b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$; [b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$ Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$. (High School Affiliated to Nanjing Normal University )

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

Let $N$ be the set of those positive integers $n$ for which $n\mid k^k-1$ implies $n\mid k-1$ for every positive integer $k$. Prove that if $n_1,n_2\in N$, then their greatest common divisor is also in $N$.

Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.

Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

Let $a>1$ and $c$ be natural numbers and let $b\neq 0$ be an integer. Prove that there exists a natural number $n$ such that the number $a^n+b$ has a divisor of the form $cx+1$, $x\in\mathbb{N}$.

Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a 90 degrees left turn after every $ \ell$ kilometer driving from start, Rob makes a 90 degrees right turn after every $ r$ kilometer driving from start, where $ \ell$ and $ r$ are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair ($ \ell$, $ r$) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?

Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying \[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]

Let $ \{m_1,m_2,\dots\}$ be a (finite or infinite) set of positive integers. Consider the system of congruences (1) $ x\equiv 2m_i^2 \pmod{2m_i\minus{}1}$ ($ i\equal{}1,2,...$ ). Give a necessary and sufficient condition for the system (1) to be solvable.

Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other. [i]Linus Hamilton.[/i]

Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.