Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

1) Cells of $8 \times 8$ table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exists integer written in the same row or in the same column such that it is not relatively prime with $a$. Find maximum possible number of prime integers in the table. 2) Cells of $2n \times 2n$ table, $n \ge 2,$ contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exist integers written in the same row and in the same column such that they are not relatively prime with $a$. Find maximum possible number of prime integers in the table.

Let $\{x\} = x- \lfloor x \rfloor$ . Consider a function f from the set $\{1, 2, . . . , 2020\}$ to the half-open interval $[0, 1)$. Suppose that for all $x, y,$ there exists a $z$ so that $\{f(x) + f(y)\} = f(z)$. We say that a pair of integers $m, n$ is valid if $1 \le m, n \le 2020$ and there exists a function $f$ satisfying the above so $f(1) = \frac{m}{n}$. Determine the sum over all valid pairs $m, n$ of ${m}{n}$.

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]

Do there exist irrational numbers $a,b$ both greater than $1$, such that $\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor$ for all $m,n\in\mathbb{N}$ ?

A doubly-indexed sequence $a_{m,n}$, for $m$ and $n$ nonnegative integers, is defined as follows: [list] [*]$a_{m,0}=0$ for all $m>0$ and $a_{0,0}=1$. [*]$a_{m,1}=0$ for all $m>1$, $a_{1,1}=1$, and $a_{0,1}=0$. [*]$a_{0,n}=a_{0,n-1}+a_{0,n-2}$ for all $n\geq 2$. [*]$a_{m,n}=a_{m,n-1}+a_{m,n-2}+a_{m-1,n-1}-a_{m-1,n-2}$ for all $m>0$, $n\geq 2$. [/list] Then there exists a unique value of $x$ so $\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{a_{m,n}x^m}{3^{n-m}}=1$. Find $\lfloor 1000x^2 \rfloor$.

In the non-decreasing sequence of odd integers $\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}$ each odd positive integer $k$ appears $k$ times. It is a fact that there are integers $b$, $c$, and $d$ such that for all positive integers $n$, \[ a_n=b\lfloor \sqrt{n+c} \rfloor +d, \] where $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. The sum $b+c+d$ equals $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4$

Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$. (Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)

Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes. [i]Author: Alex Zhu[/i]

Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.

An infinite sequence $(x_{n})$ is defined by its first term $x_{1}>1$, which is a rational number, and the relation $x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor}$ for all positive integers $n$. Prove that this sequence contains an integer. [i]A. Golovanov[/i]

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$. [i]David Yang.[/i]

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=2, \; a_{n+1}=\left\lfloor \frac{3}{2}a_{n}\right\rfloor.\] Show that it has infinitely many even and infinitely many odd integers.

For every natural number $n$ we define \[f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor\] Show that for every integer $k \geq 1$ the equation \[f(f(n))-f(n)=k\] has exactly $2k-1$ solutions.

Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \] [i]Carl Schildkraut[/i]

How many positive integer multiples of 1001 can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0\leq i < j \leq 99$?

Consider the function \[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\] Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.

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?

Let $\alpha$ be an irrational number with $0<\alpha < 1$, and draw a circle in the plane whose circumference has length $1$. Given any integer $n\ge 3$, define a sequence of points $P_1, P_2, \ldots , P_n$ as follows. First select any point $P_1$ on the circle, and for $2\le k\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k-1}P_k$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k-1}$ to $P_k$. Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a+b\le n$.

Let $\left[\sqrt{(n+1)^2 + n^2} \right], n = 1,2, \ldots,$ where $[x]$ denotes the integer part of $x.$ Prove that [b]i.)[/b] there are infinitely many positive integers $m$ such that $a_{m+1} - a_m > 1;$ [b]ii.)[/b] there are infinitely many positive integers $m$ such that $a_{m+1} - a_m = 1.$

Given a real number $x,$ let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \ldots, n_{70}$ such that $k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \leq i \leq 70.$ Find the maximum value of $\frac{n_{i}}{k}$ for $1\leq i \leq 70.$

Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.

If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$. Show that the sequence $\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$ contains infinitely many odd numbers.

How many positive integers not exceeding $ 2001$ are multiples of $ 3$ or $ 4$ but not $ 5$? $ \textbf{(A)}\ 768 \qquad \textbf{(B)}\ 801 \qquad \textbf{(C)}\ 934 \qquad \textbf{(D)}\ 1067 \qquad \textbf{(E)}\ 1167$

For real number $x$, let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\}$ denote the fractional part of $x$, that is $\{x\} = x -\lfloor x\rfloor$. The sum of the solutions to the equation $2\lfloor x\rfloor^2 + 3\{x\}^2 = \frac74 x \lfloor x\rfloor$ can be written as $\frac{p}{q} $, where $p$ and $q$ are prime numbers. Find $10p + q$.