A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

Show that the numbers $18^n$ and $2^n + 18^n$ are having the same number of digits (as written in base 10), for every natural number $n$.

Find all integers $a$ such that there are infinitely many positive integers $n$ such that $n$ divides $\phi(n)!+a$.

Prove that for $N>1$ that $(N^{2})^{2014} - (N^{11})^{106}$ is divisible by $N^6 + N^3 +1$ Is this just a proof by induction or is there a more elegant method? I don't think calculating $N = 2$ was expected.

Find all pairs $(x, y)$ of positive integers such that $x^2 + y^2 + 33^2 =2010\sqrt{x-y}$.

Let \( A \) and \( B \) be positive integers, and let \( S \) be a set of positive integers with the following properties: (1) For every non-negative integer $k$, $\text{ } A^k \in S$; (2) If a positive integer $ n \in S$, then every positive divisor of $ n$ is in $S$; (3) If $m ,n \in S$ and $m,n$ are coprime, then $mn \in S$; (4) If $n \in S$, then $An + B \in S$. Prove that all positive integers coprime to \( B \) are in \( S \).

For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$. Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.

Find the number of positive integers less than or equal to $2019$ that are no more than $10$ away from a perfect square.

Initially, a positive integer $N$ is written on a blackboard. We repeatedly replace the number according to the following rules: 1) replace the number by a positive multiple of itself 2) replace the number by a number with the same digits in a different order. (The new number is allowed to have leading digits, which are then deleted.) [i]A possible sequence of moves is given by $5 \to 20 \to 140 \to 041=41$.[/i] Determine for which values of $N$ it is possible to obtain $1$ after a finite number of such moves.

$n$ is a positive integer. Let $a(n)$ be the smallest number for which $n\mid a(n)!$ Find all solutions of:$$\frac{a(n)}{n}=\frac{2}{3}$$

Solve the equation $28^x = 19^y +87^z$, where $x, y, z$ are integers.

Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.

Set $A$ consists of $7$ consecutive positive integers less than $2011$, while set $B$ consists of $11$ consecutive positive integers. If the sum of the numbers in $A$ is equal to the sum of the numbers in $B$ , what is the maximum possible element that $A$ could contain?

$p(x)$ is an irreducible polynomial in $\mathbb Q[x]$ that $\mbox{deg}\ p$ is odd. $q(x),r(x)$ are polynomials with rational coefficients that $p(x)|q(x)^2+q(x).r(x)+r(x)^2$. Prove that \[p(x)^2|q(x)^2+q(x).r(x)+r(x)^2\]

Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

Suppose $p > 3$ is a prime number and \[S = \sum_{2 \le i < j < k \le p-1} ijk\] Prove that $S+1$ is divisible by $p$.

Find all four-digit numbers $\overline{abcd}$ such that they are multiples of $3$ and that $\overline{ab}-\overline{cd}=11$. ($\overline{abcd}$ is a four-digit number; $\overline{ab}$ is a two digit-number as $\overline{cd}$ is).

Determine all sets of natural numbers $ A $ that have at least two elements, and satisfying the following proposition: $$ \forall x,y\in A\quad x>y\implies \frac{x-y}{\text{gcd} (x,y)} \in A. $$ [i]Marius Perianu[/i]

Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Find all integer pair $(m,n)$ such that $7^m=5^n+24$.

Show that there exist strictly increasing infinite arithmetic sequence of integers which has no numbers in common with the Fibonacci sequence.

Let $p\ge5$ be a prime number. Prove that there exists $a\in\{1,2,\ldots,p-2\}$ satisfying $p^2\nmid a^{p-1}-1$ and $p^2\nmid(a+1)^{p-1}-1$.

Let $a_1,a_2,\ldots, a_{2013}$ be a permutation of the numbers from $1$ to $2013$. Let $A_n = \frac{a_1 + a_2 + \cdots + a_n} {n}$ for $n = 1,2,\ldots, 2013$. If the smallest possible difference between the largest and smallest values of $A_1,A_2,\ldots, A_{2013}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Ray Li[/i]

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.