Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$, $$\{ x,f(x),\cdots f^{p-1}(x) \} $$ is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$. [i]Proposed by IndoMathXdZ[/i]

Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA

For a positive integer $n$ let $\sigma (n)$ be the sum of all its positive divisors. Find all positive integers $n$ such that the number $\frac{\sigma (n)}{n + 1}$ is an integer.

Determine all integers $n \ge 2$ such that $n \mid s_n-t_n$ where $s_n$ is the sum of all the integers in the interval $[1,n]$ that are mutually prime to $n$, and $t_n$ is the sum of the remaining integers in the same interval.

Find all positive integers $n$ such that $7^n + 147$ is a perfect square.

Let $n?$ denote the product of all primes smaller than $n$. Prove that $n? > n$ holds for any natural number $n > 3$.

Find all pairs of positive integers $(x,n) $ that are solutions of the equation $3 \cdot 2^x +4 =n^2$.

Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.

Prove that any irreducible fraction $p/q$, where $p$ and $q$ are positive integers and $q$ is odd, is equal to a fraction $\frac{n}{2^k-1}$ for some positive integers $n$ and $k$.

Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.

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 $N$.

Solve in positive integers \[ x^y + y = y^x + x \]

Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$

Find all pairs of positive integers $(n, k)$ such that all sufficiently large odd positive integers $m$ are representable as $$m=a_1^{n^2}+a_2^{(n+1)^2}+\ldots+a_k^{(n+k-1)^2}+a_{k+1}^{(n+k)^2}$$ for some non-negative integers $a_1, a_2, \ldots, a_{k+1}$.

Call an $n$-digit integer with distinct digits [i]mountainous [/i]if, for some integer $1 \le k \le n$, the first $k$ digits are in strictly ascending order and the following $n - k$ digits are in strictly descending order. How many $5$-digit mountainous integers with distinct digits are there?

Find all $4$ consecutive even numbers, such that the sum of their squares divides the square of their product.

With expressions containing the symbol $*$, the following transformations can be performed: 1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$; 2) rewrite the expression in the form $x * 1$ as $x$. Conversions can only be performed with an integer expression, but not with its parts. For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$). Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$ it is possible to lead to a form in which there is not a single asterisk. Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.

Consider the sequence $a_1, a_2, a_3, \ldots$ with $$ a_n = \frac{1}{n(n+1)}.$$ In how many ways can the number $\frac{1}{1980}$ be represented as the sum of finitely many consecutive terms of this sequence?

Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

Consider the sequence $a_1 = 3$ and $a_{n + 1} =\frac{3a_n^2+1}{2}-a_n$ for $n = 1 ,2 ,...$. Prove that if $n$ is a power of $3$ then $n$ divides $a_n$ .

Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent: 1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$ 2) For any natural $d \leq \frac{p-1}{2}$, $$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$ where indices are taken $\pmod p$

Let $p\geq 3$ a prime number, $a$ and $b$ integers such that $\gcd(a, b)=1$. Let $n$ a natural number such that $p$ divides $a^{2^n}+b^{2^n}$, prove that $2^{n+1}$ divides $p-1$.

Let $x_1$ and $x_2$ be the roots of the equation $x^2-px+1=0$ where $p>2$ is a prime number. Prove that $x_1^p+x_2^p$ is an integer divisible by $p^2$. [i]From the folklore[/i]