More problems in 2016

use weighted am-gm: $\dfrac{200\dfrac{a}{200}+500\dfrac{b}{500}+16\dfrac{c}{16}+600\dfrac{d}{600}+700\dfrac{e}{700}}{200+500+16+600+700}≥\sqrt[200+500+16+600+700]{\left(\dfrac{a}{200}\right)^{200}\left(\dfrac{b}{500}\right)^{500}\left(\dfrac{c}{16}\right)^{16}\left(\dfrac{d}{600}\right)^{600}\left(\dfrac{e}{700}\right)^{700}}$ the LHS is 1.raise 2016 to both sides. multiply both sides with the huge numbers. we have $\mathfrak{S}≤200^{200}\times500^{500}\times16^{16}\times600^{600}\times700^{700}$

Only one of these numbers is divisible by 3, that is $600^{600} = 3^{600} \times 200^{200}$ . The highest power of 3 that divides $M$ is $\boxed{600}$ .

I took the same approach with using weighted the AM/GM inequality as Aareyan except I weighted it over 50, 125, 4, 150, 175 terms for a, b, c, d, e respectively. This works because 4 is the gcd of all of the exponents in the expression of interest.

I still raised to the equation to the 2016th power but this end result is slightly easier numbers to work with because I had an additional $4^{2016}$ factored out (ie $M = 4^{2016} 50^{200} 125^{500} 4^{16} 150^{600} 175^{700}$ ).

Equality holds when $\frac{a}{50} = \frac{b}{125} = \frac{c}{4} = \frac{d}{150} = \frac{e}{175} = \frac{2016}{504} = 4$