#### NumberTheory

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Solve in integers the equation $$x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.$$

Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.

There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.

Let $m$ be the least positive integer divisible by $17$ whose digits sum is $17$. Find $m$.

The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.

Show that, if $a,b$ are positive integers satisfying $4(ab-1)\mid (4a^2-1)$, then $a=b$

Let $a > b > c$ be three positive integers. If their remainders are $2$, $7$, and $9$ respectively when being divided by $11$. Find the remainder when $(a+b+c)(a-b)(b-c)$ is divided by $11$.

Find all positive integer $n$ such that $2^n+1$ is divisible by 3.

Show that $2x^2 - 5y^2 = 7$ has no integer solution.

How many ordered pairs of positive integers $(x, y)$ can satisfy the equation $x^2 + y^2 = x^3$?

Solve in positive integers $x^2 - 4xy + 5y^2 = 169$.

Let $b$ and $c$ be two positive integers, and $a$ be a prime number. If $a^2 + b^2 = c^2$, prove $a < b$ and $b+1=c$.

How many ordered pairs of positive integers $(a, b, c)$ that can satisfy $$\left\{\begin{array}{ll}ab + bc &= 44\\ ac + bc &=23\end{array}\right.$$

Solve in integers the equation $2(x+y)=xy+7$.

Solve in integers the question $x+y=x^2 -xy + y^2$.

Find the ordered pair of positive integers $(x, y)$ with the largest possible $y$ such that $\frac{1}{x} - \frac{1}{y}=\frac{1}{12}$ holds.

How many ordered pairs of integers $(x, y)$ satisfy $0 < x < y$ and $\sqrt{1984} = \sqrt{x} + \sqrt{y}$?

Find the number of positive integers solutions to $x^2 - y^2 = 105$.

Find any positive integer solution to $x^2 - 51y^2 = 1$.

Solve in positive integers $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{4}{5}$

Find all ordered pairs of integers $(x, y)$ that satisfy the equation $$\sqrt{y-\frac{1}{5}} + \sqrt{x-\frac{1}{5}} = \sqrt{5}$$

What is the remainder when $\left(8888^{2222} + 7777^{3333}\right)$ is divided by $37$?

Solve in integers $\frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$

Let the lengths of two sides of a right triangle be $l$ and $m$, respectively, and the length of the hypotenuse be $n$. If both $m$ and $n$ are positive integers, and $l$ is a prime number, show that $2(l+m+n)$ must be a perfect square.