[b]6.1[/b] Which number is greater $$\underbrace{1000. . . 001}_{1965\, zeroes} / \underbrace{1000 . . . 001}_{1966\, zeroes} \,\,\, or \,\,\, \underbrace{1000. . . 001}_{1966\, zeroes} / \underbrace{1000 . . . 001}_{1967\, zeroes} \,\,?$$ [b]6.2[/b] $30$ teams participate in the football championship. Prove that at any moment there will be two teams that have played at this point the same number of matches. [b]6.3./ 7.1 [/b] All integers from $1$ to $1966$ are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. [b]6.4 / 7.5[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. [b]6.5[/b] In a chess tournament, there are more than three chess players, and each player plays each other the same number of times. There were $26$ rounds in the tournament. After the $13$th round, one of the participants discovered that he had an odd number points, and each of the other participants has an even number of points. How many chess players participated in the tournament? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

Suppose $n$ is odd and each square of an $n \times n$ grid is arbitrarily filled with either by $1$ or by $-1$. Let $r_j$ and $c_k$ denote the product of all numbers in $j$-th row and $k$-th column respectively, $1 \le j, k \le n$. Prove that $$\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0$$

A factorization of a positive integers is a way of writing it as a product of positive integers greater than $1$. Two factorizations are considered the same if they only differ in the order of terms in the product. For instance, $18$ has $4$ different factorizations: $18, 2\cdot 9, 3\cdot 6$ and $ 2\cdot 3\cdot 3$. For a positive integer $n$ we denote by $f(n)$ the number of distinct factorizations of $n$. By convention $f(1)=1$. Prove that $f(n)\leq n$ for all positive integers $n$.

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

A prime number $p$ and a positive integer $n$ are given. Prove that one can colour every one of the numbers $1,2,\ldots,p-1$ using one of the $2n$ colours so that for any $i=2,3,\ldots,n$ the sum of any $i$ numbers of the same colour is not divisible by $p$.

Find all positive integers $n$ with the folowing property: for all triples ($a$,$b$,$c$) of positive real there is a triple of non negative integers ($l$,$j$,$k$) such that $an^k$, $bn^j$ and $cn^l$ are sides of a non degenate triangle

Find all pairs of positive integers $(m,n)$ such that \[m+n-\frac{3mn}{m+n}=\frac{2011}{3}.\]

The sequence $a_1,a_2,a_3\ldots $, consisting of natural numbers, is defined by the rule: \[a_{n+1}=a_{n}+2t(n)\] for every natural number $n$, where $t(n)$ is the number of the different divisors of $n$ (including $1$ and $n$). Is it possible that two consecutive members of the sequence are squares of natural numbers?

Prove that, for all $n\in\mathbb{N}$, on $ [n-4\sqrt{n}, n+4\sqrt{n}]$ exists natural number $k=x^3+y^3$ where $x$, $y$ are nonnegative integers.

Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).

The largest one of numbers $ p_1^{\alpha_1}, p_2^{\alpha_2}, \cdots, p_t^{\alpha_t}$ is called a $ \textbf{Good Number}$ of positive integer $ n$, if $ \displaystyle n\equal{} p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t}$, where $ p_1$, $ p_2$, $ \cdots$, $ p_t$ are pairwisely different primes and $ \alpha_1, \alpha_2, \cdots, \alpha_t$ are positive integers. Let $ n_1, n_2, \cdots, n_{10000}$ be $ 10000$ distinct positive integers such that the $ \textbf{Good Numbers}$ of $ n_1, n_2, \cdots, n_{10000}$ are all equal. Prove that there exist integers $ a_1, a_2, \cdots, a_{10000}$ such that any two of the following $ 10000$ arithmetical progressions $ \{ a_i, a_i \plus{} n_i, a_i \plus{} 2n_i, a_i \plus{} 3n_i, \cdots \}$($ i\equal{}1,2, \cdots 10000$) have no common terms.

With expressions containing the symbol $*$, the following transformations can be performed: 1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$; 2) rewrite the expression in the form $x * 1$ as $x$. Conversions can only be performed with an integer expression, but not with its parts. For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$). Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$ it is possible to lead to a form in which there is not a single asterisk. Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.

Find the largest prime factor of $45^5-1.$

On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

A positive integer is called [i]piola [/i] if the $9$ is the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9$ and $10$ and it's digits are all different and nonzero. How many [i]piolas[/i] are there between $ 1$ and $100000$?

Prove that the number $512^{3} +675^{3}+ 720^{3}$ is composite.

Let $n > 1$ and $x_i \in \mathbb{R}$ for $i = 1,\cdots, n$. Set \[S_k = x_1^k+ x^k_2+\cdots+ x^k_n\] for $k \ge 1$. If $S_1 = S_2 =\cdots= S_{n+1}$, show that $x_i \in \{0, 1\}$ for every $i = 1, 2,\cdots, n.$

Juan must pay $4$ bills. He goes to an ATM, but doesn't remember the amount of the bills. Just know that a) Each account is a multiple of $1,000$ and is at least $4,000$. b) The accounts total 2$00, 000$. What is the least number of times Juan must use the ATM to make sure he can pay the bills with exact change without any excess money? The cashier has banknotes of $2, 000$, $5, 000$, $10, 000$, and $20,000$. Juan can decide how much money he asks the cashier each time, but you cannot decide how many bills of each type to give to the cashier.

Determine all triples $(x,y, p)$ with $x$, $y$ positive integers and $p$ a prime number verifying the equation $p^x -y^p = 1$.

Consider the sequence $\{a_n\}$ such that $a_0=4$, $a_1=22$, and $a_n-6a_{n-1}+a_{n-2}=0$ for $n\ge2$. Prove that there exist sequences $\{x_n\}$ and $\{y_n\}$ of positive integers such that \[ a_n=\frac{y_n^2+7}{x_n-y_n} \] for any $n\ge0$.

Each two-digit natural number is [i]assigned [/i] a digit as follows: Its digits are multiplied. If the result is a digit, this is the assigned digit. If the result is a two-digit number, these two figures are multiplied, and if the result is a digit, this is the assigned digit. Otherwise, the operation is repeated. For example, the digit assigned to $32$ is $6$ since $3 \times = 6$; the digit assigned to $93$ is $4$ since $9 \times 3 = 27$, $2 \times 7 = 14$, $1 \times 4 = 4$. Find all the two-digit numbers that are assigned $8$.

We are given a positive integer $s \ge 2$. For each positive integer $k$, we define its [i]twist[/i] $k’$ as follows: write $k$ as $as+b$, where $a, b$ are non-negative integers and $b < s$, then $k’ = bs+a$. For the positive integer $n$, consider the infinite sequence $d_1, d_2, \dots$ where $d_1=n$ and $d_{i+1}$ is the twist of $d_i$ for each positive integer $i$. Prove that this sequence contains $1$ if and only if the remainder when $n$ is divided by $s^2-1$ is either $1$ or $s$.

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$). Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$; ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$. [i](Romania) Dan Schwarz[/i]

Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that the sum \[a+a^2+a^3+\cdots+a^{(p-2)!} \]is not divisible by $p$?