Let $p$ be an odd prime number. Show that the smallest positive quadratic nonresidue of $p$ is smaller than $\sqrt{p}+1$.

Find the highest power of $2$ that divides exactly into $1996!=1\times2\times\cdots\times1996$.

Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ $\emph{balanced}$ if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.

Let $f(n)=n+\lfloor \sqrt{n}\rfloor$. Prove that, for every positive integer $m$, the sequence \[m, f(m), f(f(m)), f(f(f(m))), \cdots\] contains at least one square of an integer.

Determine all real numbers $x$ satisfying $$x \lfloor x \lfloor x \rfloor \rfloor =\sqrt2.$$

A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that $$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$ holds for all $m \in \mathbb{Z}$.

For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.

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 $a+b$.

Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$. (i) Find $r_5$. (ii) Find $r_7$. (iii) Find $r_k$ for $k \in \mathbb{N}$.

Solve the equation $x^3 - [x] = 3$.

In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game. If after ending of tournament participant have at least $ 75 % $ of maximum possible points he called $winner$ $of$ $tournament$. Find maximum possible numbers of $winners$ $of$ $tournament$.

Let $x =\sqrt{a}+\sqrt{b}$, where $a$ and $b$ are natural numbers, $x$ is not an integer, and $x < 1976$. Prove that the fractional part of $x$ exceeds $10^{-19.76}$.

$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.

Determine the integral part of $A$, where $A =\frac{1}{672}+\frac{1}{673}+... +\frac{1}{2014}$

What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers?

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 $n$ be a positive integer and $x$ positive real number such that none of numbers $x,2x,\dots,nx$ and none of $\frac{1}{x},\frac{2}{x},\dots,\frac{\left\lfloor nx\right\rfloor }{x}$ is an integer. Prove that \[ \left\lfloor x\right\rfloor +\left\lfloor 2x\right\rfloor +\dots+\left\lfloor nx\right\rfloor +\left\lfloor \frac{1}{x}\right\rfloor +\left\lfloor \frac{2}{x}\right\rfloor +\dots+\left\lfloor \frac{\left\lfloor nx\right\rfloor }{x}\right\rfloor =n\left\lfloor nx\right\rfloor \]

Let $n$ be a positive integer such that $\lfloor\sqrt n\rfloor-2$ divides $n-4$ and $\lfloor\sqrt n\rfloor+2$ divides $n+4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x\rfloor$ refers to the greatest integer less than or equal to $x$.)

Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years. (a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time. Consider a period of $rb$ years in which the column is initially empty. Determine, in terms of $r$ and $b$, the number of blocks in the column at the end. (b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd. At time $t=0$, the column is initially empty. Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years. Prove that at time $t=rb$, the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$, where \[ S = \left\lfloor \frac{2r}{r+b} \right\rfloor + \left\lfloor \frac{4r}{r+b} \right\rfloor + ... + \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor . \] [i]Proposed by Sammy Luo[/i]

Call a subset $ S$ of $ \{1,2,\dots,n\}$ [i]mediocre[/i] if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)\equal{}7.$] Find all positive integers $ n$ such that $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions : 1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$ 2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.

The National Foundation of Happiness (NFoH) wants to estimate the happiness of people of country. NFoH selected $n$ random persons, and on every morning asked from each of them whether she is happy or not. On any two distinct days, exactly half of the persons gave the same answer. Show that after $k$ days, there were at most $n-\frac{n}{k}$ persons whose “yes” answers equals their “no” answers.

For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$. Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.

Let $x_1,x_2,\dots,x_{31}$ be real numbers. Then find the maximum value can $$\sum_{i,j=1,2,\dots,31, \; i\neq j}{\lceil x_ix_j \rceil }-30\left(\sum_{i=1,2,\dots,31}{\lfloor x_i^2 \rfloor } \right)$$ achieve. P.S.: For a real number $x$ we denote the smallest integer that does not subseed $x$ by $\lceil x \rceil$ and the biggest integer that does not exceed $x$ by $\lfloor x \rfloor$. For example $\lceil 2.7 \rceil=3$, $\lfloor 2.7 \rfloor=2$ and $\lfloor 4 \rfloor=\lceil 4 \rceil=4$

Let $a, b, c$, and $d$ be real numbers. Suppose that $\lfloor na\rfloor +\lfloor nb\rfloor =\lfloor nc\rfloor +\lfloor nd\rfloor $ for all positive integers $n$. Show that at least one of $a+b$, $a-c$, $a-d$ is an integer.