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Find seven positive integers $x_1, x_2, x_3, ...x_7$ such that

$\left(x_1x_2x_3...x_7\right)^2=2\left(x_1^2+x_2^2+x_3^2 + ...+x_7^2\right)$

Type your answer as the sum of these seven positive integers. If you think that there is …

Evaluate the following sum: $\sum_{n=1}^\infty \frac{\phi(n)}{2^n-1}$

$$

Notation: $\phi(\cdot)$ denotes the Euler's totient function.

Consider the following diophantine equation:

$\displaystyle x^2 + xy + y^2 = n$

For a particular positive integer $n$, the number of solutions $(x, y)$ such that $x$ and $y$ …

Let $N=2^n\ (n \in \mathbb N^+, n \geq 2)$, $N$ distinct numbers denoted as $x_1,x_2,\cdots,x_N$ are put into $N$ positions labeled $1,2,\cdots,N$, then we will get the permutation ...

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