#### TheSqueezeMethod

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

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

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 $$\sqrt{y-\frac{1}{5}} + \sqrt{x-\frac{1}{5}} = \sqrt{5}$$

How many ordered triples of integers $(a,b,c)$, with $a \ge 2$, $b\ge 1$, and $c \ge 0$, satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$?

Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$. What is $x + y$?

Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$. What is $a$?

For any given positive integer $n$, prove $(n^2 +n +1)$ cannot be a perfect square.

Solve in integers $y^2=x^4 + x^3 + x^2 +x +1$.

Solve in integers $x^3 + (x+1)^3 + \cdots + (x+7)^3 = y ^3$

Solve in positive integers $y^2 = x^2 + x + 1$

Solve in integers $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} = \frac{3}{5}$

Solve the following diophantine equation in natural numbers: $$x^2 = 1 + y + y^2 + y^3 + y^4$$

Find all integers $a$, $b$, $c$ with $1 < a < b < c$ such that the number $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.

Solve in positive integers $\big(1+\frac{1}{x}\big)\big(1+\frac{1}{y}\big)\big(1+\frac{1}{z}\big)=2$

Find all positive integers $n$ and $k_i$ $(1\le i \le n)$ such that $$k_1 + k_2 + \cdots + k_n = 5n-4$$ and $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$

Solve in positive integers the equation $$3(xy+yz+zx)=4xyz$$

Solve this equation in positive integers $$x^3 - y^3 = xy + 61$$

Solve in integers the equation $$(x+y)^2 = x^3 + y^3$$

Solve the following question in integers $$x^6 + 3x^3 +1 = y^4$$

Solve in positive integers the equation $x^2y + y^2z +z^2x = 3xyz$

Find all positive integer $n$ such that $n^2 + 2^n$ is a perfect square.

Find all the ordered integers $(a, b, c)$ which satisfy $a+b+c=450$ and $\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}=2c$.