Let $p, q$ be coprime integers, such that $|\frac{p} {q}| \leq 1$. For which $p, q$, there exist even integers $b_1, b_2, \ldots, b_n$, such that $$\frac{p} {q}=\frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\ldots}}}? $$

For given positive integers $n$ and $p$, find neaessary and sufficient conditions for the system of equations $$x + py = n , \\ x + y = p^2$$ to have a solution $(x, y, z)$ of positive integers. Prove also that there is at most one such solution.

A sequence of positive reals $\{ a_n \}$ is defined below. $$a_0 = 1, a_1 = 3, a_{n+2} = \frac{a_{n+1}^2+2}{a_n}$$ Show that for all nonnegative integer $n$, $a_n$ is a positive integer.

Determine all the natural numbers $n$ such that exactly one fifth of the natural numbers $1,2,...,n$ are divisors of $n$.

Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.

Show that for any $n \in N$ the polynomial $f(x) = (x^2 +x)^{2^n}+1$ is irreducible over $Z[x]$.

A set of $2003$ positive numbers is such that for any two numbers $a$ and $b$ included in it ($a > b$) at least one of the numbers $a + b$ or $a - b$ also included in the set. Prove that if these numbers are ordered by increasing, then the differences between adjacent numbers will be the same.

Determine the positive integers $n$ such that the set of all positive divisors of $30^n$ can be divided into groups of three so that the product of the three numbers in each group is the same.

We call a positive integer $n$ [i]happy [/i] if there exist integers $a,b$ such that $a^2+b^2 = n$. If $t$ is happy, show that (a) $2t$ is [i]happy[/i], (b) $3t$ is not [i]happy[/i]

A positive integer $n$ is called nice if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is nice because its largest proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of nice integers not greater than $3000$.

[b]p1.[/b] Let $U = \{-2, 0, 1\}$ and $N = \{1, 2, 3, 4, 5\}$. Let $f$ be a function that maps $U$ to $N$. For any $x \in U$, $x + f(x) + xf(x)$ is an odd number. How many $f$ satisfy the above statement? [b]p2.[/b] Around a circle are written all of the positive integers from $ 1$ to $n$, $n \ge 2$ in such a way that any two adjacent integers have at least one digit in common in their decimal expressions. Find the smallest $n$ for which this is possible. [b]p3.[/b] Michael loses things, especially his room key. If in a day of the week he has $n$ classes he loses his key with probability $n/5$. After he loses his key during the day he replaces it before he goes to sleep so the next day he will have a key. During the weekend(Saturday and Sunday) Michael studies all day and does not leave his room, therefore he does not lose his key. Given that on Monday he has 1 class, on Tuesday and Thursday he has $2$ classes and that on Wednesday and Friday he has $3$ classes, what is the probability that loses his key at least once during a week? [b]p4.[/b] Given two concentric circles one with radius $8$ and the other $5$. What is the probability that the distance between two randomly chosen points on the circles, one from each circle, is greater than $7$ ? [b]p5.[/b] We say that a positive integer $n$ is lucky if $n^2$ can be written as the sum of $n$ consecutive positive integers. Find the number of lucky numbers strictly less than $2015$. [b]p6.[/b] Let $A = \{3^x + 3^y + 3^z|x, y, z \ge 0, x, y, z \in Z, x < y < z\}$. Arrange the set $A$ in increasing order. Then what is the $50$th number? (Express the answer in the form $3^x + 3^y + 3^z$). [b]p7.[/b] Justin and Oscar found $2015$ sticks on the table. I know what you are thinking, that is very curious. They decided to play a game with them. The game is, each player in turn must remove from the table some sticks, provided that the player removes at least one stick and at most half of the sticks on the table. The player who leaves just one stick on the table loses the game. Justin goes first and he realizes he has a winning strategy. How many sticks does he have to take off to guarantee that he will win? [b]p8.[/b] Let $(x, y, z)$ with $x \ge y \ge z \ge 0$ be integers such that $\frac{x^3+y^3+z^3}{3} = xyz + 21$. Find $x$. [b]p9.[/b] Let $p < q < r < s$ be prime numbers such that $$1 - \frac{1}{p} -\frac{1}{q} -\frac{1}{r}- \frac{1}{s}= \frac{1}{pqrs}.$$ Find $p + q + r + s$. [b]p10.[/b] In ”island-land”, there are $10$ islands. Alex falls out of a plane onto one of the islands, with equal probability of landing on any island. That night, the Chocolate King visits Alex in his sleep and tells him that there is a mountain of chocolate on one of the islands, with equal probability of being on each island. However, Alex has become very fat from eating chocolate his whole life, so he can’t swim to any of the other islands. Luckily, there is a teleporter on each island. Each teleporter will teleport Alex to exactly one other teleporter (possibly itself) and each teleporter gets teleported to by exactly one teleporter. The configuration of the teleporters is chosen uniformly at random from all possible configurations of teleporters satisfying these criteria. What is the probability that Alex can get his chocolate? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Do there exist three natural numbers greater than 1, such that the square of each, minus one, is divisible by each of the others? [i]A. Golovanov[/i]

15 positive integers, all less than 1998(and no one equal to 1), are relatively prime (no pair has a common factor > 1). Show that at least one of them must be prime.

p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$. p2. If $a, b$, and $c$ satisfy the system of equations $$ \frac{ab}{a+b}=\frac12$$ $$\frac{bc}{b+c}=\frac13 $$ $$ \frac{ac}{a+c}=\frac17 $$ Determine the value of $(a- c)^b$. p3. Given triangle $ABC$. If point $M$ is located at the midpoint of $AC$, point $N$ is located at the midpoint of $BC$, and the point $P$ is any point on $AB$. Determine the area of ​​the quadrilateral $PMCN$. [img]https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png[/img] p4. Given the rule of motion of a particle on a flat plane $xy$ as following: $N: (m, n)\to (m + 1, n + 1)$ $T: (m, n)\to (m + 1, n - 1)$, where $m$ and $n$ are integers. How many different tracks are there from $(0, 3)$ to $(7, 2)$ by using the above rules ? p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing $30$ marbles. For each take, the player can take the least a minimum of $ 1$ and a maximum of $6$ marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking $3$ marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.

Determine the largest number of steps for $\gcd(k,76)$ to terminate over all choices of $0 < k < 76$, using the following algorithm for gcd. Give your answer in the form $(n,k)$ where $n$ is the maximal number of steps and $k$ is the $k$ which achieves this. If multiple $k$ work, submit the smallest one. \begin{tabular}{l} 1: \textbf{FUNCTION} $\text{gcd}(a,b)$: \\ 2: $\qquad$ \textbf{IF} $a = 0$ \textbf{RETURN} $b$ \\ 3: $\qquad$ \textbf{ELSE RETURN} $\text{gcd}(b \bmod a,a)$ \end{tabular}

For any positive integer $n>1$, let $p(n)$ be the greatest prime factor of $n$. Find all the triplets of distinct positive integers $(x,y,z)$ which satisfy the following properties: $x,y$ and $z$ form an arithmetic progression, and $p(xyz)\leq 3.$

Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.

Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.

A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? $ \textbf{(A) }39 \qquad \textbf{(B) }71 \qquad \textbf{(C) }73 \qquad \textbf{(D) }75 \qquad \textbf{(E) }135 \qquad $

For each positive integer $n$ , define $a_n = 20 + n^2$ and $d_n = gcd(a_n, a_{n+1})$. Find the set of all values that are taken by $d_n$ and show by examples that each of these values is attained.

Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.

Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequences of natural numbers such that $a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1$ for every $n\ge 1$. Show that these two sequences can have only a finite number of terms in common.

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

Determine the smallest positive integer $n\geq 2$ for which there exists a positive integer $m$ such that $mn$ divides $m^{2023} + n^{2023} + n$.

Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$. Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.