A factorization

Let ABC be a right triangle at A. Factor a^3 + b^3 + c^3.

Solution

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Equality

Prove that in any triangle ABC it holds that

    \[16\sum m_a^2 m_b^2 = 9 \left ( s^2 + r^2 + 4Rr \right )^2 - 144 s^2 Rr\]

Solution

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On a Lambert W integral

Prove that

    \[\int_{0}^{\infty} W_0 \left ( x e^{-x} \right )\, \mathrm{d}x = \int_{0}^{\infty} \frac{W_0 \left ( xe^{-x} \right )}{x} \, \mathrm{d}x = \frac{\pi^2}{12}\]

A Fibonacci series

Let \mathcal{F}_n denote the n – th Fibonacci number. Prove that

    \[\sum_{n=2}^{\infty}\arctan \left(\frac{\mathcal{F}_{n-1}}{\mathcal{F}_n \mathcal{F}_{n+1}+1}\right)\arctan \left(\frac{\mathcal{F}_{n+2}}{\mathcal{F}_n \mathcal{F}_{n+1}-1}\right)=\frac{\pi^2}{16}\]

Solution

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A floor series

Let x \in [0, 1). Prove that

  1. \displaystyle x = \sum_{n=1}^{\infty} \frac{\left \lfloor 2^n x \right \rfloor - 2 \left \lfloor 2^{n-1} x \right \rfloor}{2^n}.
  2. \displaystyle x = \sum_{n \geq 1 , 2 \nmid \left \lfloor 2^n x \right \rfloor} \frac{1}{2^n}.
  3. \displaystyle \sum_{n=1}^{\infty} \frac{1 - \mathrm{sgn} \left ( \tan 2^n \right )}{2^{n+2}} = \frac{1}{\pi} where \mathrm{sgn} denotes the sign function.

Solution

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