What is the Greatest Integer When Evaluating a Complex Fraction?

[anemone]

Evaluate \(\displaystyle \left\lfloor{\frac{2014^3}{(2015)(2016)}+\frac{2016^3}{2014(2015)}}\right\rfloor\).

 

[kaliprasad]

Evaluate \(\displaystyle \left\lfloor{\frac{2014^3}{(2015)(2016)}+\frac{2016^3}{2014(2015)}}\right\rfloor\).

Spoiler

let x = 2015
so we get
$\left\lfloor\frac{(x-1)^3}{x(x+1)} + \frac{(x+1)^3}{(x-1)x}\right\rfloor$
$=\left\lfloor\frac{(x-1)^4+ (x+1)^4}{x(x+1)(x-1)}\right\rfloor$
$=\left\lfloor\frac{2(x^4+6x^2+ 2)}{x(x+1)(x-1)}\right\rfloor$
$=\left\lfloor\frac{2((x^2-1)(x^2+7)+9)}{x(x^2-1)}\right\rfloor$
$=\left\lfloor(\frac{2(x^2+7)}{x}+ \frac{18}{x(x^2-1)})\right\rfloor$
$=\left\lfloor(2x + \frac{14}{x} + \frac{18}{x(x^2-1)})\right\rfloor$
now as x = 2015 and so $\frac{14}{x} < \frac{1}{2}$ and $\frac{18}{x(x^2-1)} < \frac{1}{2}$ so
ans is 2x or 4030

 

[anemone]

Thanks for participating, kaliprasad! Just so you know that my approach is exactly the same as yours. (Smile)