Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

Determine all integers $n>1$ whose positive divisors add up to a power of $3.$ [i]Andrei Bâra[/i]

Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$

Solve the following diophantine equation in the set of nonnegative integers: $11^{a}5^{b}-3^{c}2^{d}=1$.

Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

[b]p1.[/b] The entrance fee the county fair is $64$ cents. Unfortunately, you only have nickels and quarters so you cannot give them exact change. Furthermore, the attendent insists that he is only allowed to change in increments of six cents. What is the least number of coins you will have to pay? [b]p2.[/b] At the county fair, there is a carnival game set up with a mouse and six cups layed out in a circle. The mouse starts at position $A$ and every ten seconds the mouse has equal probability of jumping one cup clockwise or counter-clockwise. After a minute if the mouse has returned to position $A$, you win a giant chunk of cheese. What is the probability of winning the cheese? [b]p3.[/b] A clown stops you and poses a riddle. How many ways can you distribute $21$ identical balls into $3$ different boxes, with at least $4$ balls in the first box and at least $1$ ball in the second box? [b]p4.[/b] Watch out for the pig. How many sets $S$ of positive integers are there such that the product of all the elements of the set is $125970$? [b]p5.[/b] A good word is a word consisting of two letters $A$, $B$ such that there is never a letter $B$ between any two $A$'s. Find the number of good words with length $8$. [b]p6.[/b] Evaluate $\sqrt{2 -\sqrt{2 +\sqrt{2-...}}}$ without looking. [b]p7.[/b] There is nothing wrong with being odd. Of the first $2006$ Fibonacci numbers ($F_1 = 1$, $F_2 = 1$), how many of them are even? [b]p8.[/b] Let $f$ be a function satisfying $f (x) + 2f (27- x) = x$. Find $f (11)$. [b]p9.[/b] Let $A$, $B$, $C$ denote digits in decimal representation. Given that $A$ is prime and $A -B = 4$, nd $(A,B,C)$ such that $AAABBBC$ is a prime. [b]p10.[/b] Given $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ , find $\frac{x^8+y^8}{x^8-y^8}$ in term of $k$. [b]p11.[/b] Let $a_i \in \{-1, 0, 1\}$ for each $i = 1, 2, 3, ..., 2007$. Find the least possible value for $\sum^{2006}_{i=1}\sum^{2007}_{j=i+1} a_ia_j$. [b]p12.[/b] Find all integer solutions $x$ to $x^2 + 615 = 2^n$ for any integer $n \ge 1$. [b]p13.[/b] Suppose a parabola $y = x^2 - ax - 1$ intersects the coordinate axes at three points $A$, $B$, and $C$. The circumcircle of the triangle $ABC$ intersects the $y$ - axis again at point $D = (0, t)$. Find the value of $t$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]6.1 [/b] Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway. [b]6.2.[/b] A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus [b]6.3. [/b] Prove that the difference $43^{43} - 17^{17}$ is divisible by $10$. [b]6.4. [/b] Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay? [b]6.5.[/b] The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C. [b]6.6.[/b] Is it possible to write down the numbers from $ 1$ to $1963$ in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

Find all positive integers such that they have $6$ divisors (without $1$ and the number itself) and the sum of the divisors is $14133$.

Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$

Find all pairs of integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+2$, $B=2n^2+3mn+m^2+2$, $C=3n^2+mn+2m^2+1$ have a common divisor greater than $1$.

The sequence $ (F_n)$ of Fibonacci numbers satisfies $ F_1 \equal{} 1, F_2 \equal{} 1$ and $ F_n \equal{} F_{n\minus{}1} \plus{}F_{n\minus{}2}$ for all $ n \ge 3$. Find all pairs of positive integers $ (m, n)$, such that $ F_m . F_n \equal{} mn$.

Find the number of solutions to the equation$$x_1^4+x_2^4+\ldots+x_{10}^4=2011$$in the set of positive integers.

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

Natural numbers have been divided in groups as follow: $(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots$. Let $S_n$ be the sum of the elements of the $n$th group. Prove that $\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n}$ is even.

Consider positive integers $m <n,p < q$ such that $(m, n) = 1, (p, q) = 1$ and satisfy the condition that if $\frac{m}{n}= tan\alpha$ and $\frac{p}{q} = tan\beta$, then $\alpha + \beta = 45^o$. i) Given $m, n$, find $p, q$. ii) Given $n, q$, find $m, p$. ii) Given $m, q$, find $n, p$.

Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

Find the remainder when $2^{5^9}+5^{9^2}+9^{2^5}$ is divided by $11$.

$$Problem 1$$ ;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$.

A positive integer $k$ is said to be [i]visionary[/i] if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020.$ How many visionary integers are there?

Given the rebus: $$AB \cdot AC \cdot BC = BBBCCC $$ where different letters correspond to different digits and the same letters to the same digits, find the sum $AB + AC + BC.$ [i]Proposed by Giorgi Arabidze, Georgia [/i]

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

A $5$-tuple $(1,1,1,1,2)$ has the property that the sum of any three of them is divisible by the sum of the remaining two. Is there a $5$-tuple with this property whose all terms are distinct?

Decide, whether exists positive rational number $w$, which isn't integer, such that $w^w$ is a rational number.

If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$

Let $f_n$ be $n$th Fibonacci number defined by recurrence $f_{n+1} - f_n - f_{n-1} = 0$, $n \in \mathbb{N}$ and initial conditions $f_0 = 0$, $f_1 = 1$. Prove that for any $n \in \mathbb{N}$ $$(n - 1) (n + 1) (2nf_{n+1} - (n + 6) f_n)$$is divisible by 150 for any $n \in \mathbb{N}$.