Determine all integers $n\ge 2$ such that for any two infinite sequences of positive integers $a_1<a_2< \cdots $ and $b_1, b_2, \cdots$, such that $a_i\mid a_j$ for all $i<j$, there always exists a real number $c$ such that $$\lfloor{ca_i}\rfloor \equiv b_i \pmod {n}$$ for all $i\ge 1$. [i]Proposed by Wong Jer Ren & Ivan Chan Kai Chin[/i]

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

Prove the following two inequalities: 1) If $n > 49$, then exist positive integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}<1$$ 2) If $n > 4$, then exist integer integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}>1$$

Let $n$ be a positive integer. Prove that there exists a natural number $m$ with exactly $n$ prime factors, such that for every positive integer $d$ the numbers in $\{1,2,3,\ldots,m\}$ of order $d$ modulo $m$ are multiples of $\phi (d)$. (15 points)

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers? [i]Proposed by F. Petrov[/i]

Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.

Let $x, y$ and $k$ be three positive integers. Prove that there exist a positive integer $N$ and a set of $k + 1$ positive integers $\{b_0,b_1, b_2, ... ,b_k\}$, such that, for every $i = 0, 1, ... , k$ , the $b_i$-ary expansion of $N$ is a $3$-digit palindrome, and the $b_0$-ary expansion is exactly $\overline{\mbox{xyx}}$. proposed by Bojan Basic, Serbia

How many positive integers $ N$ less than $ 1000$ are there such that the equation $ x^{\lfloor x\rfloor} \equal{} N$ has a solution for $ x$? (The notation $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.)

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$.

Find the number of eight-digit positive integers that are multiples of $9$ and have all distinct digits.

Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$

Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$

Determine the smallest natural number that has exactly $1994$ divisors.

Find the largest number formed by the digits 1 to 9, without repetition, that is divisible by 18.

In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.

Do there exist $2011$ positive integers $a_1 < a_2 < \ldots < a_{2011}$ such that $\gcd(a_i,a_j) = a_j - a_i$ for any $i$, $j$ such that $1 \le i < j \le 2011$?

Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

Max has $2015$ jars labeled with the numbers $1$ to $2015$ and an unlimited supply of coins. Consider the following starting configurations: (a) All jars are empty. (b) Jar $1$ contains $1$ coin, jar $2$ contains $2$ coins, and so on, up to jar $2015$ which contains $2015$ coins. (c) Jar $1$ contains $2015$ coins, jar $2$ contains $2014$ coins, and so on, up to jar $2015$ which contains $1$ coin. Now Max selects in each step a number $n$ from $1$ to $2015$ and adds $n$ to each jar [i]except to the jar $n$[/i]. Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar. (Birgit Vera Schmidt)

Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$, $n+\frac 1{pm}$, $p+\frac 1{mn}$ are integers. [i]Valentin Vornicu, Romania[/i]

Every officially published book used to have an ISBN code (International Standard Book Number) which consisted of $10$ symbols. Such code looked like this: $$a_1a_2 . . . a_9a_{10}$$ with $a_1, . . . , a_9 \in \{0, 1, . . . , 9\}$ and $a_{10} \in \{0, 1, . . . , 9, X\}$. The symbol $X$ stood for the number $10$. With a valid ISBN code was $$a_1 + 2a2 + . . . + 9a_9 + 10a_{10}$$ a multiple of $11$. Prove the following statements. (a) If one symbol is changed in a valid ISBN code, the result is no valid ISBN code. (b) When two different symbols swap places in a valid ISBN code then the result is not a valid ISBN.

Show that there exist two consecutive squares such that there are at least $1000$ primes between them.

Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the first $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?