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]

You are given three lists $A$, $B$, and $C$. List $A$ contains the numbers of the form $10^k$ in base $10$, with $k$ any integer greater than or equal to $1$. Lists $B$ and $C$ contain the same numbers translated into base $2$ and $5$ respectively: $$\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}$$ Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.

Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation \[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\] that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$

Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)

A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where [x] denotes the smallest integer $\leq$ x)$.$

Let the sequence $ a(n), n \equal{} 1,2,3, \ldots$ be generated as follows with $ a(1) \equal{} 0,$ and for $ n > 1:$ \[ a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}.\] 1.) Determine the maximum and minimum value of $ a(n)$ over $ n \leq 1996$ and find all $ n \leq 1996$ for which these extreme values are attained. 2.) How many terms $ a(n), n \leq 1996,$ are equal to 0?

A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?

Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$

How many non-negative integral values of $x$ satisfy the equation $ \lfloor \frac{x}{5}\rfloor = \lfloor \frac{x}{7}\rfloor $

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

Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$. Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\]

Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$. [i]Evan O' Dorney.[/i]

Find the number of integer solutions of $\left[\frac{x}{100} \left[\frac{x}{100}\right]\right]= 5$

Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an \plus{} b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.

Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$ (Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)

Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define \[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \] where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.

An infinite sequence of natural number $\{x_n\}_{n\ge 1}$ is such that $x_{n+1}$ is obtained by adding one of the non-zero digits of $x_n$ to itself. Show this sequence contains an even number.

Compute the sum of the real solutions to $\lfloor x \rfloor \{x\} = 2020x$. Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$, and$ \{x\} = x -\lfloor x \rfloor$.

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

We say that a natural number $n$ is interesting if it can be written in the form \[ n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor, \] where $a,b,c$ are positive real numbers such that $a + b + c = 1.$ Determine all interesting numbers. ( $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.)

The symbolism $ \lfloor x\rfloor$ denotes the largest integer not exceeding $ x$. For example. $ \lfloor3\rfloor\equal{}3$, and $ \lfloor 9/2\rfloor\equal{}4$. Compute \[ \lfloor\sqrt1\rfloor\plus{}\lfloor\sqrt2\rfloor\plus{}\lfloor\sqrt3\rfloor\plus{}\cdots\plus{}\lfloor\sqrt{16}\rfloor. \]$ \textbf{(A)}\ 35 \qquad \textbf{(B)}\ 38 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 136$

Denote by $\lfloor x\rfloor$ the greatest positive integer less than or equal to $x$. Let $m\ge2$ be an integer, and let $s$ be a real number between $0$ and $1$. Define an infinite sequence of real numbers $a_1, a_2, a_3,\ldots$ by setting $a_1 = s$ and $ak = ma_{k-1}-(m-1)\lfloor a_{k-1}\rfloor$ for all $k\ge2$. For example, if $m = 3$ and $s = \tfrac58$, then we get $a_1 = \tfrac58$, $a_2 = \tfrac{15}8$, $a_3 = \tfrac{29}8$, $a_4 = \tfrac{39}8$, and so on. Call the sequence $a_1, a_2, a_3,\ldots$ $\textbf{orderly}$ if we can find rational numbers $b, c$ such that $\lfloor a_n\rfloor = \lfloor bn + c\rfloor$ for all $n\ge1$. With the example above where $m = 3$ and $s = \tfrac58$, we get an orderly sequence since $\lfloor a_n\rfloor = \left\lfloor\tfrac{3n}2-\tfrac32\right\rfloor$ for all $n$. Show that if $s$ is an irrational number and $m\ge2$ is any integer, then the sequence $a_1, a_2, a_3,\ldots$ is $\textbf{not}$ an orderly sequence.

Let $n\geq 2$ be an integer and consider an array composed of $n$ rows and $2n$ columns. Half of the elements in the array are colored in red. Prove that for each integer $k$, $1<k\leq \dsp \left\lfloor \frac n2\right\rfloor+1$, there exist $k$ rows such that the array of size $k\times 2n$ formed with these $k$ rows has at least \[ \frac { k! (n-2k+2) } {(n-k+1)(n-k+2)\cdots (n-1)} \] columns which contain only red cells.

Find the smallest positive integer $n$ such that \[0< \sqrt[4]{n}-\lfloor \sqrt[4]{n}\rfloor < 0.00001.\]

A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?