Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.

Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

Let $(a_n)_{n=1}^{\infty}$ be an integer sequence such that $a_{n+48} \equiv a_n \pmod {35}$ for every $n \geq 1$. Let $i$ and $j$ be the least numbers satisfying the conditions $a_{n+i} \equiv a_n \pmod {5}$ and $a_{n+j} \equiv a_n \pmod {7}$ for every $n\geq 1$. Which one below cannot be an $(i,j)$ pair? $ \textbf{(A)}\ (16,4) \qquad\textbf{(B)}\ (3,16) \qquad\textbf{(C)}\ (8,6) \qquad\textbf{(D)}\ (1,48) \qquad\textbf{(E)}\ (16,18) $

Find all natural numbers $n$ for which each natural number written with $~$ $n-1$ $~$ 'ones' and one 'seven' is prime.

Nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ are going to be inputted to the empty cells in the following table: $\begin{array} {|c|c|c|} \cline{1-3} 10 & & \\ \cline{1-3} & & 9 \\ \cline{1-3} & 3 & \\ \cline{1-3} 11 & & 17 \\ \cline{1-3} & 20 & \\ \cline{1-3} \end{array}$ such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.

Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}$ such that $$f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}$$ for all integers $x\ge 1$.

For a number $n$ in base $10$, let $f(n)$ be the sum of all numbers possible by removing some digits of $n$ (including none and all). For example, if $n = 1234$, $f(n) = 1234 + 123 + 124 + 134 + 234 + 12 + 13 + 14 + 23 + 24 + 34 + 1 + 2 + 3 + 4 = 1979$; this is formed by taking the sums of all numbers obtained when removing no digit from $n$ (1234), removing one digit from $n$ (123, 124, 134, 234), removing two digits from $n$ (12, 13, 14, 23, 24, 34), removing three digits from $n$ (1, 2, 3, 4), and removing all digits from $n$ (0). If $p$ is a 2011-digit integer, prove that $f(p)-p$ is divisible by $9$. Remark: If a number appears twice or more, it is counted as many times as it appears. For example, with the number $101$, $1$ appears three times (by removing the first digit, giving $01$ which is equal to $1$, removing the first two digits, or removing the last two digits), so it is counted three times.

Let $n$ be a positive integer. Prove that $n$ is prime if and only if \[{{n-1}\choose k}\equiv (-1)^{k}\pmod{n}\] for all $k \in \{ 0, 1, \cdots, n-1 \}$.

Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$. [i]Valentin Vornicu[/i]

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.

Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.

An odd integer $ n \ge 3$ is said to be nice if and only if there is at least one permutation $ a_{1}, \cdots, a_{n}$ of $ 1, \cdots, n$ such that the $ n$ sums $ a_{1} \minus{} a_{2} \plus{} a_{3} \minus{} \cdots \minus{} a_{n \minus{} 1} \plus{} a_{n}$, $ a_{2} \minus{} a_{3} \plus{} a_{3} \minus{} \cdots \minus{} a_{n} \plus{} a_{1}$, $ a_{3} \minus{} a_{4} \plus{} a_{5} \minus{} \cdots \minus{} a_{1} \plus{} a_{2}$, $ \cdots$, $ a_{n} \minus{} a_{1} \plus{} a_{2} \minus{} \cdots \minus{} a_{n \minus{} 2} \plus{} a_{n \minus{} 1}$ are all positive. Determine the set of all `nice' integers.

In conference there $n>2$ mathematicians. Every two mathematicians communicate in one of the $n$ offical languages of the conference. For any three different offical languages the exists three mathematicians who communicate with each other in these three languages. Find all $n$ such that this is possible.

Two countries ($A$ and $B$) organize a conference, and they send an equal number of participants. Some of them have known each other from a previous conference. Prove that one can choose a nonempty subset $C$ of the participants from $A$ such that one of the following holds: [list][*]the participants from $B$ each know an even number of people in $C$, [*]the participants from $B$ each know an odd number of participants in $C$.[/list]

Find the least positive multiple of 999 that does not have a 9 as a digit.

\begin{quote} Ted quite likes haikus, \\ poems with five-seven-five, \\ but Ted knows few words. He knows $2n$ words \\ that contain $n$ syllables \\ for every int $n$. Ted can only write \\ $N$ distinct haikus. Find $N$. \\ Take mod one hundred. \end{quote} Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. In particular, for integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Furthermore, words cannot cross between lines, but may be repeated. If Ted can make $N$ distinct haikus, compute the remainder when $N$ is divided by $100$. [i]Proposed by Lewis Chen[/i]

Let $p_1, p_2, \ldots$ be a sequence of primes such that $p_1 =2$ and for $n\geq 1, p_{n+1}$ is the largest prime factor of $p_1 p_2 \ldots p_n +1$ . Prove that $p_n \not= 5$ for any $n$.

Let $n$ be a natural number. Solve in integers the equation \[x^n + y^n = (x - y)^{n+1}.\]

Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]

For a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Show that there are infinitely many natural numbers $n$ such that $s(2^n) > s(2^{n+1})$.

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have different remainders when divided by $n + 1$?

Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$