Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.

Prove by the method of induction that: [b]a)[/b] $ a!b! $ divides $ (a+b)! , $ for any natural numbers $ a,b. $ [b]b)[/b] $ p $ divides $ (-1)^{k+1} +\binom{p-1}{k} , $ for any odd primes $ p $ and $ k\in\{ 0,1,\ldots ,p-1\} . $

The vertices of a triangle have integer coordinates and one of its sides is of length $\sqrt{n}$, where $n$ is a square-free natural number. Prove that the ratio of the circumradius and the inradius is an irrational number.

$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$. $a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite. $b)$ Find 3 different prime numbers that do not divide any terms of this sequence.

Nils has a telephone number with eight different digits. He has made $28$ cards with statements of the type “The digit $a$ occurs earlier than the digit $b$ in my telephone number” – one for each pair of digits appearing in his number. How many cards can Nils show you without revealing his number?

The football tournament, held in one round, involved $16$ teams, each two of which scored a different number of points. ($3$ points were given for a victory, $1$ point for a draw, $0$ points for a defeat.) It turned out that the Chisel team lost to all the teams that ultimately scored fewer points. What is the best result that the Chisel team could achieve (insert location)?

Let $A$ be a set of positive integers having the following property: for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$. Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.

Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties: [list] [*] $0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$ [*] $x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$[/list]

Let $a, b, r,$ and $s$ be positive integers ($a \ge 2$), where $a$ and $b$ have no common prime factor. Prove that if $a^r + b^r$ is divisible by $a^s + b^s$, then $r$ is divisible by $s$.

Let $k, l, m, n$ be positive integers. Given that $k+l+m+n=km=ln$, find all possible values of $k+l+m+n$.

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$? b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?

Let $p$ be an arbitrary odd prime and $\sigma(n)$ for $1 \le n \le p-1$ denote the inverse of $n \pmod p$. Show that the number of pairs $(a,b) \in \{1,2,\cdots,p-1\}^2$ with $a<b$ but $\sigma(a) > \sigma(b)$ is at least $$\left \lfloor \left(\frac{p-1}{4}\right)^2 \right \rfloor$$ [i]usjl[/i] Note: Partial credits may be awarded if the $4$ in the statement is replaced with some larger constant

Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.

On the computer screen there are initially two $1$'s written. The [i] insert [/i] program causes the sum of those numbers to be inserted between each pair of numbers by pressing the $Enter$ key. In the first step a number is inserted and we obtain $1-2-1$; In the second step two numbers are inserted and we have $1-3-2-3-1$; In the third, four numbers are inserted and you have $1-4-3-5-2-5-3-4-1$; etc Find the sum of all the numbers that appear on the screen at the end of step number $25$.

Let $ p>2 $ be a prime number. For any permutation $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $ of the set $ S = \{ 1, 2, \cdots , p \} $, let $ f( \pi ) $ denote the number of multiples of $ p $ among the following $ p $ numbers: \[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) \] Determine the average value of $ f( \pi) $ taken over all permutations $ \pi $ of $ S $.

Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove: For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$ ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.

For a positive integer $n$ denote by $P_0(n)$ the product of all non-zero digits of $n$. Let $N_0$ be the set of all positive integers $n$ such that $P_0(n)|n$. Find the largest possible value of $\ell$ such that $N_0$ contains infinitely many strings of $\ell$ consecutive integers. [i]Proposed by Navid Safaei, Iran[/i]

For every integer $r > 1$ find the smallest integer $h(r) > 1$ having the following property: For any partition of the set $\{1, 2, . . ., h(r)\}$ into $r$ classes, there exist integers $a \geq 0, 1 \leq x \leq y$ such that the numbers $a + x, a + y, a + x + y$ are contained in the same class of the partition.

Find the number of pairs of positive integers $(m; n)$, with $m \le n$, such that the ‘least common multiple’ (LCM) of $m$ and $n$ equals $600$.

Let us call a set of positive integers nice if the number of its elements equals to the average of its numbers. Call a positive integer $n$ an [i]amazing[/i] number if the set $\{1, 2 , . . . , n\}$ can be partitioned into nice subsets. a) Prove that every perfect square is amazing. b) Show that there are infinitely many positive integers which are not amazing.

The number $n{}$ cointains $k{}$ units in binary system. Prove that $2^{n-k}{}$ divides $n!$.

For positive integer $x>1$, define set $S_x$ as $$S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},$$ where $v_p(n)$ is the power of prime divisor $p$ in positive integer $n.$ Let $f(x)$ be the sum of all the elements of $S_x$ when $x>1,$ and $f(1)=1.$ Let $m$ be a given positive integer, and the sequence $a_1,a_2,\cdots,a_n,\cdots$ satisfy that for any positive integer $n>m,$ $a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.$ Prove that (1)there exists constant $A,B(0<A<1),$ such that when positive integer $x$ has at least two different prime divisors, $f(x)<Ax+B$ holds; (2)there exists positive integer $N,l$, such that for any positive integer $n\geq N ,a_{n+l}=a_n$ holds.

A polynomial $f(x)$ with integer coefficients is said to be [i]tri-divisible[/i] if $3$ divides $f(k)$ for any integer $k$. Determine necessary and sufficient conditions for a polynomial to be tri-divisible.

For any positive integer $n$ , Prove that there is not exist three odd integer $x,y,z$ satisfing the equation $(x+y)^n+(y+z)^n=(x+z)^n$.