Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.

How many $0$s are there at the end of the decimal representation of $2000!$? $ \textbf{(A)}\ 222 \qquad\textbf{(B)}\ 499 \qquad\textbf{(C)}\ 625 \qquad\textbf{(D)}\ 999 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Find all numbers $\alpha$ for which the equation \[x^2 - 2x[x] + x -\alpha = 0\] has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)

A flea moves in the positive direction on the real Ox axis, starting from the origin. He can only jump over distances equal with $\sqrt 2$ or $\sqrt{2005}$. Prove that there exists $n_0$ such that the flea can reach any interval $[n,n+1]$ with $n\geq n_0$.

For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that \[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \] \[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]

For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{\left\lfloor \frac{n - 1}{2}\right\rfloor}_{i=0} \binom{n}{2 \cdot i + 1} 3^i$. Find all $n$ such that $f(n) = 1996.$ [hide="old version"]For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{n + 1/2}_{i=1} \binom{2 \cdot i + 1}{n}$. Find all $n$ such that $f(n) = 1996.$[/hide]

$ n$ points (no three being collinear) are given in a plane. Some points are connected and they form $ k$ segments. If no three of these segments form triangle ( equiv. there are no three points, such that each two of them are connected) prove that $ k \leq \left \lfloor \frac {n^{2}}{4}\right\rfloor$

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

Determine all positive integers $n$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor +2$ divides $ n + 4$.

Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race. Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.

Find all possible values of $$ \lfloor \frac{x - p}{p} \rfloor + \lfloor \frac{-x-1}{p} \rfloor $$ where $x$ is a real number and $p$ is a nonzero integer. Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.

Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is defined as follows: we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer. Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.

Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that \[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\] where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$