Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.

Find all triples of integers $(x, y, z)$ not zero and relative primes in pairs such that $\frac{(y+z-x)^2}{4x}$, $\frac{(z+x-y)^2}{4y}$ and $\frac{(x+y-z)^2}{4z}$ are all integers.

Given a square-free positive integer $n$. Show that there do not exist coprime positive integers $x,y$ such that $x^n+y^n$ is a multiple of $(x+y)^3$.

Let $n$ and $k$ be two positive integers. Prove that there exist infinitely many perfect squares of the form $n \cdot 2^k - 7$.

An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

[b]Problem 1[/b] : Find the smallest positive integer $n$, such that $\sqrt[5]{5n}$, $\sqrt[6]{6n}$ , $\sqrt[7]{7n}$ are integers.

Find all polynomials $P$ with integer coefficients which satisfy the property that, for any relatively prime integers $a$ and $b$, the sequence $\{P (an + b) \}_{n \ge 1}$ contains an infinite number of terms, any two of which are relatively prime.

Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers

Consider the set $E = \{1,2,\ldots,2n\}$. Prove that an element $c \in E$ can belong to a subset $A \subset E$, having $n$ elements, and such that any two distinct elements in $A$ do not divide one each other, if and only if \[c > n \left( \frac{2}{3}\right )^{k+1},\] where $k$ is the exponent of $2$ in the factoring of $c$.

Show that for any positive integers $k$ and $1 \leq a \leq 9$, there exists $n$ such that satisfies the below statement. When $2^n=a_0+10a_1+10^2a_2+ \cdots +10^ia_i+ \cdots $ $(0 \leq a_i \leq 9$ and $a_i$ is integer), $a_k$ is equal to $a$.

Determine all positive integer numbers $n$ satisfying the following condition: the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.

There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$. (a) The first machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (a + 13, b + 5)$ If the input pair is $(5,2)$, is it possible to obtain the pair $(20,22)$ after a series of transformations? (b) The second machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (2a, 2b)$ If the input pair is $(15,10)$, is it possible to obtain the pair $(27,23)$ after a series of transformations? (c) The third machine can perform two transformations: - $(a, b) \to (a - 2, b + 2)$ - $(a, b) \to (2a - b + 1, 2b - 1 - a)$ If the input pair is $(5,8)$, is it possible to obtain the pair $(13,17)$ after a series of transformations?

Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Rajiv Movva[/i]

Consider all pairs $(a, b)$ of natural numbers such that the product $a^ab^b$ written in decimal system ends with exactly $98$ zeros. Find the pair $(a, b)$ for which the product $ab$ is the smallest.

Let $A=2^7(7^{14}+1)+2^6\cdot 7^{11}\cdot 10^2+2^6\cdot 7^7\cdot 10^{4}+2^4\cdot 7^3\cdot 10^6$. Prove that the number $A$ ends with $14$ zeros. [i](I. Gorodnin)[/i]

Let $S \subset \{ 1, \dots, n \}$ be a nonempty set, where $n$ is a positive integer. We denote by $s$ the greatest common divisor of the elements of the set $S$. We assume that $s \not= 1$ and let $d$ be its smallest divisor greater than $1$. Let $T \subset \{ 1, \dots, n \}$ be a set such that $S \subset T$ and $|T| \ge 1 + \left[ \frac{n}{d} \right]$. Prove that the greatest common divisor of the elements in $T$ is $1$. ----------- [Second Version] Let $n(n \ge 1)$ be a positive integer and $U = \{ 1, \dots, n \}$. Let $S$ be a nonempty subset of $U$ and let $d (d \not= 1)$ be the smallest common divisor of all elements of the set $S$. Find the smallest positive integer $k$ such that for any subset $T$ of $U$, consisting of $k$ elements, with $S \subset T$, the greatest common divisor of all elements of $T$ is equal to $1$.

Are there three pairwise distinct non-zero integers whose sum is zero and whose sum of thirteenth powers is the square of some natural number?

Suppose that $p> 2$ is a prime number. For each integer $k = 1, 2,..., p-1$, denote $r_k$ as the remainder of the division $k^p$ by $p^2$. Prove that $r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}$

Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?

Let $n, m$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$, such that \[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \] where $\gcd(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.

The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \begin{eqnarray*} f(x,x) &=& x, \\ f(x,y) &=& f(y,x), \quad \text{and} \\ (x + y) f(x,y) &=& yf(x,x + y). \end{eqnarray*} Calculate $f(14,52)$.

find all pairs of non-negative integer pairs $(m,n)$,satisfies $107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}$

In a plane we choose a cartesian system of coordinates. A point $A(x,y)$ in the plane is called an integer point if and only if both $x$ and $y$ are integers. An integer point $A$ is called invisible if on the segment $(OA)$ there is at least one integer point. Prove that for each positive integer $n$ there exists a square of side $n$ in which all the interior integer points are invisible.

Consider the expansion \[(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.\] [b](a)[/b] Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$ [b](b)[/b] Prove that $10^{340 }< a_{992} < 10^{347}.$

Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]