Solve in prime numbers $ 2p^q \minus{} q^p \equal{} 7$.

We have two lists of numbers: One initially containing 1,6,11,...,96, and the other initially containing 4,9,14,...,99. In every turn, we erase two numbers from one of the lists, and write $\frac{1}{3}$ of their sum (not necessarily an integer) in the other list. We continue this process until there are no possible moves. [list=a] [*] Prove that at the end of the process, there is exactly one number in each list. [*] Prove that those two numbers are [u]not[/u] equal. [/list]

Two distinct positive integers are called [i]close [/i] if their greatest common divisor equals their difference. Show that for any $n$, there exists a set $S$ of $n$ elements such that any two elements of $S$ are close.

Four natural numbers are such that the square of the sum of any two of them is divisible by the product of the other two numbers. Prove that at least three of these numbers are equal.

Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$

Show that there exists a positive integer $n$ such that the decimal representations of $3^n$ and $7^n$ both start with the digits $10$.

$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying \[(b^2+11(a-b))^2=a^3 b\]

Let $n\geq 1$ be an integer. Show that there exists an integer between $\sqrt{2n}$ and $\sqrt{5n}$, exclusive.

Any positive integer $n$ can be written in the form $n = 2^b(2c+1)$. We call $2c+1$ the[i] odd part[/i] of $n$. Given an odd integer $n > 0$, define the sequence $ a_0, a_1, a_2, ...$ as follows: $a_0 = 2^n-1, a_{k+1} $ is the [i]odd part[/i] of $3a_k+1$. Find $a_n$.

Find a function $f(n)$ on the positive integers with positive integer values such that $f( f(n) ) = 1993 n^{1945}$ for all $n$.

Let $k$ be a positive integer. A natural number $m$ is called [i]$k$-typical[/i] if each divisor of $m$ leaves the remainder $1$ when being divided by $k$. Prove: [b]a)[/b] If the number of all divisors of a positive integer $n$ (including the divisors $1$ and $n$) is $k$-typical, then $n$ is the $k$-th power of an integer. [b]b)[/b] If $k > 2$, then the converse of the assertion [b]a)[/b] is not true.

Let $p$ be a prime and $n$ be a positive integer such that $p^2$ divides $\prod_{k=1}^n (k^2+1)$. Show that $p<2n$.

$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $.

[b]p1.[/b] Find the smallest whole number $n \ge 2$ such that the product $(2^2 - 1)(3^2 - 1) ... (n^2 - 1)$ is the square of a whole number. [b]p2.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p3.[/b] Three cars are racing: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p4.[/b] There are $11$ big boxes. Each one is either empty or contains $8$ medium-sized boxes inside. Each medium box is either empty or contains $8$ small boxes inside. All small boxes are empty. Among all the boxes, there are a total of $102$ empty boxes. How many boxes are there altogether? [b]p5.[/b] Ann, Mary, Pete, and finally Vlad eat ice cream from a tub, in order, one after another. Each eats at a constant rate, each at his or her own rate. Each eats for exactly the period of time that it would take the three remaining people, eating together, to consume half of the tub. After Vlad eats his portion there is no more ice cream in the tube. How many times faster would it take them to consume the tub if they all ate together? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\{a_k\}_{k\geq 0}$ be a sequence given by $a_0 = 0$, $a_{k+1}=3\cdot a_k+1$ for $k\in \mathbb{N}$. Prove that $11 \mid a_{155}$

Find all polynomials $P$ with integer coefficients such that all the roots of $P^n(x)$ are integers. (here $P^n(x)$ means $P(P(...(P(x))...))$ where $P$ is repeated $n$ times)

Call integer number $x$ as far from squares and cubes, if for every integer $k$ it is true : $|x-k^2|>10^6,|x-k^3|>10^6$. Prove, that there are infinitely many far from squares and cubes degrees of $2$

Consider the set $M=\{1,2,...,n\},n\in\mathbb N$. Find the smallest positive integer $k$ with the following property: In every $k$-element subset $S$ of $M$ there exist two elements, one of which divides the other one.

Find all pairs $ (x; y) $ of positive integers such that $$xy | x^2 + 2y -1.$$

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.

$x_1$ is a natural constant. Prove that there does not exist any natural number $m> 2500$ such that the recursive sequence $\{x_i\} _{i=1} ^ \infty $ defined by $x_{n+1} = x_n^{s(n)} + 1$ becomes eventually periodic modulo $m$. (That is there does not exist natural numbers $N$ and $T$ such that for each $n\geq N$, $m\mid x_n - x_{n+T}$). ($s(n)$ is the sum of digits of $n$.)

Find all $3$-digit numbers $\overline {abc}$ such that $\overline {abc} = a! + b! + c! $.

Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.