Find all positive integers $n$ such that the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \leq y$.

Find all the integer pairs $ (x,y)$ such that: \[ x^2\minus{}y^4\equal{}2009\]

How many $ 3$-digit positive integers have digits whose product equals $ 24$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

Let $n$ a positive integer, $n > 1$. The number $n$ is wonderful if the number is divisible by sum of the your prime factors. For example; $90$ is wondeful, because $90 = 2 \times 3^2\times 5$ and $2 + 3 + 5 = 10, 10$ divides $90$. Show that, exist a number "wonderful" with at least $10^{2002}$ distinct prime numbers.

Given a positive integer $n \geq 2$, whose canonical prime factorization is $n = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$, we define the following functions: $$\varphi(n) = n\bigg(1 -\frac{1}{p_1}\bigg) \bigg(1 -\frac{1}{p_2}\bigg) \ldots \bigg(1 -\frac {1}{p_k}\bigg) ; \overline{\varphi}(n) = n\bigg(1 +\frac{1}{p_1}\bigg) \bigg(1 +\frac{1}{p_2}\bigg) \ldots \bigg(1 + \frac{1}{p_k}\bigg)$$ Consider all positive integers $n$ such that $\overline{\varphi}(n)$ is a multiple of $n + \varphi(n) $. (a) Prove that $n$ is even. (b) Determine all positive integers $n$ that satisfy this property.

The ${\textit{distinct prime factors}}$ of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$. Let $A=2^k - 2$ and $B= 2^k \cdot A$, where $k$ is an integer ($k \ge 2$). Show that, for every integer $k$ greater than or equal to $2$, [list=i] [*] $A$ and $B$ have the same set of distinct prime factors. [*] $A+1$ and $B+1$ have the same set of distinct prime factors. [/list]

Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $? [b](A)[/b] the set of positive integers [b](B)[/b] the set of composite positive integers [b](C)[/b] the set of even positive integers [b](D)[/b] the set of integers greater than 3 [b](E)[/b] the set of integers greater than 4

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

Is it possible to place $1995$ different natural numbers along a circle so that for any two of these numbers, the ratio of the greatest to the least is a prime? I feel that my solution's wording and notation is awkward (and perhaps unnecessarily complicated), so please feel free to critique it: [hide] Suppose that we do have such a configuration $a_{1},a_{2},...a_{1995}$. WLOG, $a_{2}=p_{1}a_{1}$. Then \[\frac{a_{2}}{a_{3}}= p_{2}, \frac{1}{p_{2}}\] \[\frac{a_{3}}{a_{4}}= p_{3}, \frac{1}{p_{3}}\] \[... \] \[\frac{a_{1995}}{a_{1}}= p_{1995}, \frac{1}{p_{1995}}\] Multiplying these all together, \[\frac{a_{2}}{a_{1}}= \frac{\prod p_{k}}{\prod p_{j}}= p_{1}\] Where $\prod p_{k}$ is some product of the elements in a subset of $\{ p_{2},p_{3}, ...p_{1995}\}$. We clear denominators to get \[p_{1}\prod p_{j}= \prod p_{k}\] Now, by unique prime factorization, the set $\{ p_{j}\}\cup \{ p_{1}\}$ is equal to the set $\{ p_{k}\}$. However, since there are a total of $1995$ primes, this is impossible. We conclude that no such configuration exists. [/hide]

Let $n > 6$ be a perfect number. Let $p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}$ be the prime factorisation of $n$, where we assume that $p_1 < p_2 <...< p_k$ and $a_i > 0$ for all $ i = 1,...,k$. Prove that $a_1$ is even. Remark: An integer $n \ge 2$ is called a perfect number if the sum of its positive divisors, excluding $ n$ itself, is equal to $n$. For example, $6$ is perfect, as its positive divisors are $\{1, 2, 3, 6\}$ and $1+2+3=6$.

For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\] $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Let $n>6$ be a perfect number, and let $n=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k}$ be its prime factorisation with $1<p_1<\dots <p_k$. Prove that $e_1$ is an even number. A number $n$ is [i]perfect[/i] if $s(n)=2n$, where $s(n)$ is the sum of the divisors of $n$. (Proposed by Javier Rodrigo, Universidad Pontificia Comillas)

Let $ A$, $ M$, and $ C$ be digits with \[ (100A \plus{} 10M \plus{} C )(A \plus{} M \plus{} C ) \equal{} 2005. \]What is $ A$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 46 \times 47$ in order to make it a perfect square?

A positive integer $n$ is called [i]mythical[/i] if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors. [i]Proposed by Evan Chen[/i]

Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

The number of digits in $4^{16} 5^{25}$ (when written in the usual base 10 form) is A. 31 B. 30 C. 29 D. 28 E. 27

Prove that if the odd prime $p$ divides $a^{b}-1$, where $a$ and $b$ are positive integers, then $p$ appears to the same power in the prime factorization of $b(a^{d}-1)$, where $d=\gcd(b,p-1)$.

It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$. a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$). b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

Let $ \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0$ be the function from every positive integer $ n$ to the sum of divisors $ \sum_{d \mid n}{d}$ (i.e. $ \sigma(6) \equal{} 6 \plus{} 3 \plus{} 2 \plus{} 1$ and $ \sigma(8) \equal{} 8 \plus{} 4 \plus{} 2 \plus{} 1$). Find all primes $ p$ such that $ p \mid \sigma(p \minus{} 1)$. [i](Salvatore Tringali)[/i]