Equality in a triangle

Prove that in any triangle ABC it holds that:

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

Planes and sphere

Consider two parallel planes that have a distance d \leq 2r and intersect a sphere of radius r. (both planes intersect the sphere.) Prove that the area of the surface of the sphere enclosed by both planes is only dependant by r, d and not by the position of the two planes with respect to the sphere.

 

Solution

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Factorization

Let \mathrm{ABC} be a right triangle at \mathrm{A}. Factor a^3+b^3+c^3.

Solution

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Factorial series

Prove that

    \[\sum_{n=0}^{\infty} \frac{(4n)!}{(4n+4)!} = \frac{\ln 2}{4} - \frac{\pi}{24}\]

Solution

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The composition is a metric

Let d be a metric , f be a strictly increasing function and concave on [0, +\infty) such that f(0)=0. Prove that f \circ d is a metric.

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