Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

Determine all $n \in \mathbb{N}$ for which [list][*] $n$ is not the square of any integer, [*] $\lfloor \sqrt{n}\rfloor ^3$ divides $n^2$. [/list]

For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and \[ \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots\] Prove that \[ \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1\] holds for every positive integer $ n$.

Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows: $$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$ Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.

For a group $\left( G, \star \right)$ and $A, B$ two non-void subsets of $G$, we define $A \star B = \left\{ a \star b : a \in A \ \text{and}\ b \in B \right\}$. (a) Prove that if $n \in \mathbb N, \, n \geq 3$, then the group $\left( \mathbb Z \slash n \mathbb Z,+\right)$ can be writen as $\mathbb Z \slash n \mathbb Z = A+B$, where $A, B$ are two non-void subsets of $\mathbb Z \slash n \mathbb Z$ and $A \neq \mathbb Z \slash n \mathbb Z, \, B \neq \mathbb Z \slash n \mathbb Z, \, \left| A \cap B \right| = 1$. (b) If $\left( G, \star \right)$ is a finite group, $A, B$ are two subsets of $G$ and $a \in G \setminus \left( A \star B \right)$, then prove that function $f : A \to G \setminus B$ given by $f(x) = x^{-1}\star a$ is well-defined and injective. Deduce that if $|A|+|B| > |G|$, then $G = A \star B$. [hide="Question."]Does the last result have a name?[/hide]

Prove that for any positive integer $m$, the equation \[ \frac{n}{m}\equal{}\lfloor\sqrt[3]{n^2}\rfloor\plus{}\lfloor\sqrt{n}\rfloor\plus{}1\] has (at least) a positive integer solution $n_{m}$. [i]Cezar Lupu & Dan Schwarz[/i]

Determine the number of positive divisors of $2008^8$ that are less than $2008^4$.

Compute \[\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Find the sum of the decimal digits of \[ \left\lfloor \frac{51525354555657\dots979899}{50} \right\rfloor. \] Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$. [i]Proposed by Evan Chen[/i]

Determine all real numbers $a$ such that \[4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.\]

The positive integers from 1 to $n^2$ are placed arbitrarily on the $n^2$ squares of a $n\times n$ chessboard. Two squares are called [i]adjacent[/i] if they have a common side. Show that two opposite corner squares can be joined by a path of $2n-1$ adjacent squares so that the sum of the numbers placed on them is at least $\left\lfloor \frac{n^3} 2 \right\rfloor + n^2 - n + 1$. [i]Radu Gologan[/i]

Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N} $ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ . (Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)

Let $ p$ be a prime number with $ p>5$. Consider the set $ X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$. Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$. [i]Albania[/i]

Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that \[a<S_n-[S_n]<b\] where $[x]$ represents the largest integer not exceeding $x$.

Let $c$ be the probability that the cards are neither from the same suit or the same rank. Compute $\lfloor 1000c\rfloor$.

How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?

Let $ \alpha$ be the positive root of the equation $ x^{2} \equal{} 1991x \plus{} 1$. For natural numbers $ m$ and $ n$ define \[ m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor. \] Prove that for all natural numbers $ p$, $ q$, and $ r$, \[ (p*q)*r \equal{} p*(q*r). \]

Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.

Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]

Find the maximum possible number of kings on a $12\times 12$ chess table so that each king attacks exactly one of the other kings (a king attacks only the squares that have a common point with the square he sits on).

Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by \[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\] for all $n \in \mathbb{N}^{*}$. Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$. (Here $[x]$ denotes the integral part of real number $x$).

Find the number of solutions in positive integers of the equation $\lfloor \frac{x}{2} \rfloor = \lfloor \frac{x}{11} \rfloor +1$ where $\lfloor A\rfloor$ denotes the integer part of the number $A$, e.g. $\lfloor 2.031\rfloor = 2$, $\lfloor 2\rfloor = 2$, etc.

Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies \[ a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1 \] for all $i,j \geq 1$ with $i + j \leq 1997$. Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$) for all $1 \leq n \leq 1997$.

Let $ p$ be a prime number with $ p>5$. Consider the set $ X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$. Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$. [i]Albania[/i]

Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.