Indians find those sequences of non-negative real numbers $x_0, x_1,...$ [i]mystical [/i]t hat satisfy $x_0 < 2021$, $x_{i+1} = \lfloor x_i \rfloor \{x_i\}$ for every $i \ge 0$, furthermore the sequence contains an integer different from $0$. How many sequences are mystical according to the Indians?

Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that \[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? \]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$. $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$

Let $p,n$ be positive integers, such that $p$ is prime and $p <n$. If $p$ divides $n + 1$ and $ \left(\left[\frac{n}{p}\right], (p-1)!\right) = 1$, then prove that $p\cdot \left[\frac{n}{p}\right]^2$ divides ${n \choose p} -\left[\frac{n}{p}\right]$ . (Here $[x]$ represents the integer part of the real number $x$.)

Which natural numbers cannot be presented in that way: $[n+\sqrt{n}+\frac{1}{2}]$, $n\in\mathbb{N}$ $[y]$ is the greatest integer function.

$f(k) = k + \left[ \frac{n}{k}\right ] $,$k \in \{1,2,..., n\}$, $k_0 =\left[ \sqrt{n} \right] + 1$. Prove that $f(k_0) < f(k)$ if $k \in \{1,2,..., n\}$

For every positive integer $n$, form the number $n/s(n)$, where $s(n)$ is the sum of digits of $n$ in base 10. Determine the minimum value of $n/s(n)$ in each of the following cases: (i) $10 \leq n \leq 99$ (ii) $100 \leq n \leq 999$ (iii) $1000 \leq n \leq 9999$ (iv) $10000 \leq n \leq 99999$

The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$? $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$

Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

Find all primes $p \geq 3$ with the following property: for any prime $q<p$, the number \[ p - \Big\lfloor \frac{p}{q} \Big\rfloor q \] is squarefree (i.e. is not divisible by the square of a prime).

Find the smallest positive integer $c$ for which the following statement holds: Let $k$ and $n$ be positive integers. Suppose there exist pairwise distinct subsets $S_1$, $S_2$, $\dots$, $S_{2k}$ of $\{1, 2, \dots, n\}$, such that $S_i \cap S_j \neq \varnothing$ and $S_i \cap S_{j+k} \neq \varnothing$ for all $1 \le i,j \le k$. Then $1000k \le c \cdot 2^n$. [i]Proposed by Yang Liu[/i]

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

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

Solve the system of equations $$\begin{cases} x^2+ [y]=10 \\ y^2+[x]=13 \end{cases}$$ ($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).

For $ s,t \in \mathbb{N}$, consider the set $ M\equal{}\{ (x,y) \in \mathbb{N} ^2 | 1 \le x \le s, 1 \le y \le t \}$. Find the number of rhombi with the vertices in $ M$ and the diagonals parallel to the coordinate axes.

Determine the sixth number after the decimal point in the number $(\sqrt{1978} +\lfloor\sqrt{1978}\rfloor)^{20}$

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.

(a) We say that a hyperplane $H$ that is given with this equation \[H=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n=b\}\] ($a=(a_1,\dots,a_n)\in \mathbb R^n$ and $b\in \mathbb R$ constant) bisects the finite set $A\subseteq \mathbb R^n$ if each of the two halfspaces $H^+=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n>b\}$ and $H^-=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n<b\}$ have at most $\lfloor \tfrac{|A|}{2}\rfloor$ points of $A$. Suppose that $A_1,\dots,A_n$ are finite subsets of $\mathbb R^n$. Prove that there exists a hyperplane $H$ in $\mathbb R^n$ that bisects all of them at the same time. (b) Suppose that the points in $B=A_1\cup \dots \cup A_n$ are in general position. Prove that there exists a hyperplane $H$ such that $H^+\cap A_i$ and $H^-\cap A_i$ contain exactly $\lfloor \tfrac{|A_i|}{2}\rfloor$ points of $A_i$. (c) With the help of part (b), show that the following theorem is true: Two robbers want to divide an open necklace that has $d$ different kinds of stones, where the number of stones of each kind is even, such that each of the robbers receive the same number of stones of each kind. Show that the two robbers can accomplish this by cutting the necklace in at most $d$ places.

The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is $ \textbf{(A)}\ \Big\{\frac{5}{2}\Big\}\qquad\textbf{(B)}\ \big\{x\ |\ 2 \le x \le 3\big\}\qquad\textbf{(C)}\ \big\{x\ |\ 2\le x < 3\big\}\qquad \\ \textbf{(D)}\ \Big\{x\ |\ 2 < x \le 3\Big\}\qquad\textbf{(E)}\ \Big\{x\ |\ 2 < x < 3\Big\} $

Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$. Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).

Suppose that f is a function $f : R_{\ge 0} \to R$ so that for all $x, y \in R_{\ge 0}$ (nonnegative reals) we have that $$f(x)+f(y) = f(x+y+xy)+f(x)f(y).$$ Given that $f\left(\frac{3}{5} \right) = \frac12$ and$ f(1) = 3$, determine $$\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.$$

There are $n$ lamps and $k$ switches in a room. Initially, each lamp is either turned on or turned off. Each lamp is connected by a wire with $2020$ switches. Switching a switch changes the state of a lamp, that is connected to it, to the opposite state. It is known that one can switch the switches so that all lamps will be turned on. Prove, that it is possible to achieve the same result by switching the switches no more than $ \left \lfloor \dfrac{k}{2} \right \rfloor$ times. [i]Proposed by T. Zimanov[/i]

Find all real solutions to the equation $4x^2 - 40 \lfloor x \rfloor + 51 = 0$.

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).

Find the positive integer $n$ that satisfies the equation $$n^2 - \lfloor \sqrt{n} \rfloor = 2018$$

Suppose that the set $M=\{1,2,\cdots,n\}$ is split into $t$ disjoint subsets $M_{1}$, $\cdots$, $M_{t}$ where the cardinality of $M_i$ is $m_{i}$, and $m_{i} \ge m_{i+1}$, for $i=1,\cdots,t-1$. Show that if $n>t!\cdot e$ then at least one class $M_z$ contains three elements $x_{i}$, $x_{j}$, $x_{k}$ with the property that $x_{i}-x_{j}=x_{k}$.