In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on. If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won. Prove that Kain can force a win in a finite number of moves.

Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.

Denote by $S(k)$ the sum of the digits in the decimal representation of $k$. Prove that there are infinitely many $n\in \mathbb{Z_{+}}$ such that: ${S(2^{n}+n})<S(2^{n})$.

For each natural number $a$ we denote $\tau (a)$ and $\phi (a)$ the number of natural numbers dividing $a$ and the number of natural numbers less than $a$ that are relatively prime to $a$. Find all natural numbers $n$ for which $n$ has exactly two different prime divisors and $n$ satisfies $\tau (\phi (n))=\phi (\tau (n))$.

Guy has 17 cards. Each of them has an integer written on it (the numbers are not necessarily positive, and not necessarily different from each other). Guy noticed that for each card, the square of the number written on it equals the sum of the numbers on the 16 other cards. What are the numbers on Guy's cards? Find all of the options.

Show that there are infinitely many positive integers $a$ that cannot be written as $a = a_{1}^{6}+ a_{2}^{6} + \ldots + a_{7}^{6},$ where the $a_i$ are positive integers. State and prove a generalization.

Let $p(n)$ be the number of partitions of the natural number $n$ into natural summands. The diversity of a partition is by definition the number of different summands in it. Denote by $q(n)$ the sum of the diversities of all the $p(n) $ partitions of $n$. (For example, $p(4) = 5$, the five distinct partitions of $4$ being $4, 3 + 1, 2+2, 2 + 1 + 1, 1 + 1 + 1 + 1,$ and $g(4) =1 + 2+1+ 2+1 = 7$.) Prove that, for all natural numbers $n$, (a) $q(n)= 1 + P(1) + P(2) + p(3) + ...+ p(n -1)$, (b) $q(n) < \sqrt{2n} p(n)$. (AV Zelevinskiy, Moscow)

Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1$ (mod $2017$). ($2017$ is prime.)

Find the number of solutions to the equation: $6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $ With {x} we denote the fractional part of the number x.

Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\in \mathbb N$, number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\le j$, then $a_i\le a_j$.)

A factorian is defined to be a number such that it is equal to the sum of it's digits' factorials. What is the smallest three digit factorian?

Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.

[b]1A.[/b] The iTest, by virtue of being the first national internet-based high school math competition, saves a lot of paper every year. The quantity of trees saved (“$a$”) is determined by the following formula: $a = x^2 + 3x + 9$, where $x$ is the number of participating students in the competition. If $x$ is the correct answer from short answer [hide=problem 22]x=20[/hide], then find $a$. [i](1 point)[/i] [b]1B.[/b] Let $q$ be the sum of the digits of $a$. If $q = b! - (b-1)! + (b-2)! - (b-3)!$, find $b$. [i](2 points)[/i] [b]1C.[/b] Find the number of the following statements that are false: [i] (4 points)[/i] 1. $q$ is the first prime number resulting from the sum of cubes of distinct fractions, where both the numerator and denominator are primes. 2. $q$ is composite. 3. $q$ is composite and is the sum of the first four prime numbers and $1$. 4. $q$ is the smallest prime equal to the difference of cubes of two consecutive primes. 5. $q$ is not the smallest prime equal to the product of twin primes plus their arithmetic mean. 6. The sum of $q$ consecutive Fibonacci numbers, starting from the $q^{th}$ Fibonacci number, is prime. 7. $q$ is the largest prime factor of $1bbb$. 8. $q$ is the $8^{th}$ largest prime number. 9. $a$ is composite. 10. $a + q + b = q^2$. 11. The decimal expansion of $q^q$ begins with $q$. 12. $q$ is the smallest prime equal to the sum of three distinct primes. 13. $q^5 + q^2 + q^1 + q^3 + q^5 + q^6 + q^4 + q^0 = 52135640$. 14. $q$ is not the smallest prime such that $q$ and $q^2$ have the same sum of their digits. 15. $q$ is the smallest prime such that $q$ = (the product of its digits + the sum of its digits). [hide=ANSWER KEY]1A. 469 1B. 4 1C. 6[/hide]

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that \[n \mid a^{f(n)}-1.\] Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.

Prove that for each positive integer n,the equation $x^{2}+15y^{2}=4^{n}$ has at least $n$ integer solution $(x,y)$

Determine all integers $n$ that can be written in the form \[ n = \frac{a^2 - b^2}{b}, \] where $a$ and $b$ are positive integers. [i](Walther Janous)[/i]

There´s a ping pong tournament with $n\geq 3$ participants that we´ll call $1, 2, \dots n$. The tournament rules are the following ones: at the start, all the players form a line, ordered from $1$ to $n$. Players $1$ and $2$ play the first match. The winner is at the beginning of the line and the loser is placed behind the last person in the line.In the next play, the two who at that moment are the first two in line face each other, the winner is first in line and the loser goes to the end of the line, just behind the last loser. And so on. After $N$ matches, the tournament ends.Player number $1$ won $a_1$ matches, player number $2$ won $a_2$, and so on till player $n$, that has won $a_n$ matches (it is trivial that $a_1+a_2+\dots+a_n=N)$.Determine how many games each player has lost, based on $a_1, a_2, \dots , a_n$

For which complex numbers $\alpha$ does there exist a completely multiplicative, complex-valued arithmetic function $f$ such that \[ \sum_{n<x}f(n)=\alpha x+O(1)\,\,? \]

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

Let $a$ be an odd digit and $b$ an even digit. Prove that for every positive integer $n$ there exists a positive integer, divisible by $2^n$, whose decimal representation contains no digits other than $a$ and $b$.

Show that $ x^3 + x+ a^2 = y^2 $ has at least one pair of positive integer solution $ (x,y) $ for each positive integer $ a $.

Let $k,m,n$ be positive integers such that $n^m$ is divisible by $m^n$, and $m^k$ is divisible by $k^m$. [list=a] [*] Prove that $n^k$ is divisible by $k^n$. [*] Find an example of $k,m,n$ satisfying the above conditions, where all three numbers are distinct and bigger than 1. [/list]

Find a pair of relatively prime four digit natural numbers $A$ and $B$ such that for all natural numbers $m$ and $n$, $\vert A^m -B^n \vert \ge 400$.