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 $q^2 + r = 2011$.

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$

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

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].

What is the units digit of $ 13^{2003}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

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}$

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.

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$

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 $gcd(x,y)$

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 $n$.

Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 \equal{} 1$, $ a_{2k} \equal{} 1 \plus{} a_k$ and $ a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.

An $\textit{Euclidean step}$ transforms a pair $(a, b)$ of positive integers, $a > b$, to the pair $(b, r)$, where $r$ is the remainder when a is divided by $b$. Let us call the $\textit{complexity}$ of a pair $(a, b)$ the number of Euclidean steps needed to transform it to a pair of the form $(s, 0)$. Prove that if $ad - bc = 1$, then the complexities of $(a, b)$ and $(c, d)$ differ at most by $2$.

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

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.