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The Pythagorean Theorem is one of the most famous quadratic diophantine equations, which are equations with squared variables solved over the integers. How many solutions can you find?

\[ \large \color{blue}{x}^2-\color{red}{y}^2=2011\]

How many integral solutions \((\color{blue}{x},\color{red}{y})\) are there for the equation above?

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\[ \begin{eqnarray} ab+cd&=&38 \\ ac+bd&=&34\\ ad+bc&=&43\\ \end{eqnarray} \]

Let \(a,b,c\) and \(d\) positive integers such that they satisfy the system of equations above. Find \(a+b+c+d\).

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Find the number of ordered quadruples of positive integers \((x,y,p,q)\) satisfying

\[x^3y-xy^3=pq,\]

and \(p,q\) are prime numbers.

This problem is posed by Daniel C.

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\[\large \frac{1}{x}+\frac{1}{y}=\frac{1}{156}\]

Find the number of ordered pairs of positive integers \((x, y) \) that satisfy the equation above.

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