For positive integers $a$ and $b$, $gcd (a, b)$ denote their greatest common divisor and $lcm (a, b)$ their least common multiple. Determine the number of ordered pairs (a,b) of positive integers satisfying the equation $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$

[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other. [b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?

Find the smallest positive integer $n$ such that [list][*] $n$ has exactly $144$ distinct positive divisors, [*] there are ten consecutive integers among the positive divisors of $n$. [/list]

For any positive integers $n>m$ prove the following inequality: $$[m,n]+[m+1,n+1]\geq 2m\sqrt{n}$$ As usual, [x,y] denotes the least common multiply of $x,y$ [I]Proposed by A. Golovanov[/i]

Suppose that $p$ is a prime number. Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have \[ n|a^{\frac{\varphi(n)}{p}}-1 \]

Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,$ $32,$ $33,$ $34,$ $35,$ $36,$ $37,$ $38,$ $39,$ and $40.$ What is the value of $\frac{N}{M}?$ $(\textbf{A})\: 1\qquad(\textbf{B}) \: 2\qquad(\textbf{C}) \: 37\qquad(\textbf{D}) \: 74\qquad(\textbf{E}) \: 2886$

Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that $$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$

When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is $\text{(A)}\ 120 \qquad \text{(B)}\ 210 \qquad \text{(C)}\ 420 \qquad \text{(D)}\ 840 \qquad \text{(E)}\ 5040$

For every natural number $a$ and $b$, define the notation $[a,b]$ as the least common multiple of $a $ and $b$ and the notation $(a,b)$ as the greatest common divisor of $a$ and $b$. Find all $n \in \mathbb{N}$ that satisfies \[ 4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)} \]

Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $.

The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ so that $T_n=T_0$.

Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$

Two distinct positive integers $a,b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. Prove that eventually the two numbers on the board will be equal.

Let $(G,\cdot)$ be a finite group with the identity element, $e$. The smallest positive integer $n$ with the property that $x^{n}= e$, for all $x \in G$, is called the [i]exponent[/i] of $G$. (a) For all primes $p \geq 3$, prove that the multiplicative group $\mathcal G_{p}$ of the matrices of the form $\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}$, with $\hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z$, is not commutative and has [i]exponent[/i] $p$. (b) Prove that if $\left( G, \circ \right)$ and $\left( H, \bullet \right)$ are finite groups of [i]exponents[/i] $m$ and $n$, respectively, then the group $\left( G \times H, \odot \right)$ with the operation given by $(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)$, for all $\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H$, has the [i]exponent[/i] equal to $\textrm{lcm}(m,n)$. (c) Prove that any $n \geq 3$ is the [i]exponent[/i] of a finite, non-commutative group. [i]Ion Savu[/i]

The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the students are freshmen?

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also \[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\] (b) Show that there are no two positive integers $a$ and $b$ such that \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]

Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n \minus{} 1.$

Find the smallest positive integer $k$ such that \[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\] for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.) [i]Alex Zhu.[/i] [hide="Clarifications"][list=1][*]${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$. In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$.[/list][/hide]

Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that $\bullet$ the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear, $\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint, $\bullet$ all of the angles at $P_1, P_2,\ldots, P_n$ are congruent, $\bullet$ all of the $n$ line segments $P_2P_3,\ldots, P_nP_1$ are congruent, and $\bullet$ the path $P_1P_2, P_2P_3,\ldots, P_nP_1$ turns counterclockwise at an angle of less than 180 degrees at each vertex. There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?

Let $n$ be an integer larger than $1$ and let $S$ be the set of $n$-element subsets of the set $\{1,2,\ldots,2n\}$. Determine \[\max_{A\in S}\left (\min_{x,y\in A, x \neq y} [x,y]\right )\] where $[x,y]$ is the least common multiple of the integers $x$, $y$.

A farmer divides his herd of $n$ cows among his four sons so that one son gets one-half the herd, a second son, one-fourth, a third son, one-fifth, and the fourth son, 7 cows. Then $n$ is: $ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 180\qquad\textbf{(E)}\ 240 $

Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n \minus{} 1.$

Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $(n, d)$ which player has a winning strategy?

Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Eugene Chen[/i]

Let $m,n\in\mathbb{Z_{+}}$ be such numbers that set $\{1,2,\ldots,n\}$ contains exactly $m$ different prime numbers. Prove that if we choose any $m+1$ different numbers from $\{1,2,\ldots,n\}$ then we can find number from $m+1$ choosen numbers, which divide product of other $m$ numbers.