Prove that with all positive real numbers a, b, c, d, e we have:

                        \(\sum\dfrac{a+b}{2}\ge\sum\dfrac{a+b+c}{3}\)

Prove that with all real numbers a, b, c, we have:

            \(6abc+2\sqrt{\prod\left(a^2-ab+b^2\right)}\ge\prod\left(a+b\right)\)

Prove that:

  1. \(2\dfrac{5}{9}< \dfrac{5}{3}+\dfrac{8}{3^2}+\dfrac{11}{3^3}+...+\dfrac{302}{3^{100}}< 3\dfrac{1}{2}\)

  2. \(3\dfrac{7}{9}< \dfrac{7}{3}+\dfrac{13}{3^2}+\dfrac{19}{3^3}+...+\dfrac{601}{3^{100}}< 5\)

  3. \(\dfrac{11}{3}+\dfrac{17}{3^2}+\dfrac{23}{3^3}+...+\dfrac{605}{3^{100}}< 7\)

  4. \(\dfrac{4}{3}+\dfrac{13}{3^2}+\dfrac{22}{3^3}+...+\dfrac{904}{3^{101}}< \dfrac{17}{4}\)

  5. \(\dfrac{7}{3}+\dfrac{11}{3^2}+\dfrac{15}{3^3}+...+\dfrac{403}{3^{100}}< 4,5\)

Quickly, please. :v

How many ordered triples of integers (m, n, p) exist such that mn = p, np = m and mp = n?

What are the maximum number of grape vines that can be planted, not closer than nine feet apart, in a square plot containing one-sixteenth of an acre? 

(Note: each side of this square plot would be 52 feet 2 inches)

Find the value of  \(P=2016x+y^{2017}+z^{2017}\) when \(\dfrac{y+z+1}{x}=\dfrac{x+z+2}{y}=\dfrac{x+y-3}{z}=\dfrac{1}{x+y+z}\).