Maximum value

Let $ABC$ be a triangle. Find the maximum value of:

$$R=\sqrt{2}\sin A + \sqrt{5} \sin B +\sqrt{10}\sin C$$

Solution

Closed form of a sum

Let $n$ be a non negative integer. Find a closed expression for:

$$\sum_{k=0}^{n}(-1)^k 2^k \binom{n}{k}\binom{2n-k}{n}$$

(Michael Bataille, Rouen, France)

Solution  ( José H. Nieto, Universidad del Zulia, Maracaibo, Venezuela)

Binomial sum invoking roots of unity

Evalute (in a closed form) the sum:

$$\sum_{k=0}^{n}\binom{3n}{3k}$$

Solution

From Russian Mathematical Olympiad

Evaluate:

$$  \int_0^1 \frac{\arctan \dfrac{x}{x+1}}{\arctan \dfrac{1+2x-2x^2}{2}}\,\,{\rm d}x$$

Solution: