Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.

The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \begin{eqnarray*} f(x,x) &=& x, \\ f(x,y) &=& f(y,x), \quad \text{and} \\ (x + y) f(x,y) &=& yf(x,x + y). \end{eqnarray*} Calculate $f(14,52)$.

Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties: [list] [*] $f(1,1)=0$. [*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1; [*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$. [/list] Prove that $$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$

Let $m,n$ be natural numbers and let $d = gcd(m,n)$. Let $x = 2^{m} -1$ and $y= 2^n +1$ (a) If $\frac{m}{d}$ is odd, prove that $gcd(x,y) = 1$ (b) If $\frac{m}{d}$ is even, Find $gcd(x,y)$

Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$, [*]$x+y=0$, [*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?

Determine the largest number of steps for $\gcd(k,76)$ to terminate over all choices of $0 < k < 76$, using the following algorithm for gcd. Give your answer in the form $(n,k)$ where $n$ is the maximal number of steps and $k$ is the $k$ which achieves this. If multiple $k$ work, submit the smallest one. \begin{tabular}{l} 1: \textbf{FUNCTION} $\text{gcd}(a,b)$: \\ 2: $\qquad$ \textbf{IF} $a = 0$ \textbf{RETURN} $b$ \\ 3: $\qquad$ \textbf{ELSE RETURN} $\text{gcd}(b \bmod a,a)$ \end{tabular}

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.

By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder. As the division of the remainder with $n$ continues, the new quotient is $0.4$ and the new remainder is $0.2$. Find $m$ and $n$.

Let $ P_1$ be a regular $ r$-gon and $ P_2$ be a regular $ s$-gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$. What's the largest possible value of $ s$?

What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?

Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$. For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows. (i) $a_1=<a>$ (ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$, if $a_n=0$, then $a_{n+1}=0$. (1) For $a=\sqrt{2}$, find $a_n$. (2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$. (3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$. [i]2011 Tokyo University entrance exam/Science, Problem 2[/i]

Find all solutions $(x, y) \in \mathbb Z^2$ of the equation \[x^3 - y^3 = 2xy + 8.\]

Find the largest positive integer $k$ such that $\phi ( \sigma ( 2^k)) = 2^k$. ($\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$). As a hint, you are given that $641|2^{32}+1$.

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 $x \in \mathbb{Q^+}.$

Look at these fractions. At firs step we have $ \frac{0}{1}$ and $ \frac{1}{0}$, and at each step we write $ \frac{a\plus{}b}{c\plus{}d}$ between $ \frac{a}{b}$ and $ \frac{c}{d}$, and we do this forever \[ \begin{array}{ccccccccccccccccccccccccc}\frac{0}{1}&&&&&&&&\frac{1}{0}\\ \frac{0}{1}&&&&\frac{1}{1}&&&&\frac{1}{0}\\ \frac{0}{1}&&\frac{1}{2}&&\frac{1}{1}&&\frac{2}{1}&&\frac{1}{0}\\ \frac{0}{1}&\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{1}{1}&\frac{3}{2}&\frac{2}{1}&\frac{3}{1}&\frac{1}{0}\\ &&&&\dots\end{array}\] a) Prove that each of these fractions is irreducible. b) In the plane we have put infinitely many circles of diameter 1, over each integer on the real line, one circle. The inductively we put circles that each circle is tangent to two adjacent circles and real line, and we do this forever. Prove that points of tangency of these circles are exactly all the numbers in part a(except $ \frac{1}{0}$). [img]http://i2.tinypic.com/4m8tmbq.png[/img] c) Prove that in these two parts all of positive rational numbers appear. If you don't understand the numbers, look at [url=http://upload.wikimedia.org/wikipedia/commons/2/21/Arabic_numerals-en.svg]here[/url].

Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.

Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$. Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$. (a) Suppose that $c = 0$. Show that $d_n = 1,\ \forall n \geq 1$. (b) Suppose that $c = 1$. Show that $d_n \in \{1,5\},\ \forall n \geq 1$. (c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$.

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

2. Find all natural $n>1$ for which value of the sum $2^2+3^2+...+n^2$ equals $p^k$ where p is prime and k is natural

Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.

Find the least positive integer $ n$ for which $ \frac{n\minus{}13}{5n\plus{}6}$ is non-zero reducible fraction. $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 68 \qquad \textbf{(C)}\ 155 \qquad \textbf{(D)}\ 226 \qquad \textbf{(E)}\ \text{none of these}$

A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible? $ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$

What is the greatest common factor of $12345678987654321$ and $12345654321$? [i]Proposed by Evan Chen[/i]

If the Highest Common Divisor of $ 6432$ and $ 132$ is diminished by $ 8$, it will equal: $ \textbf{(A)}\ \minus{}6 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ \minus{}2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$