Question

For a real number r we denote by $\left[r\right]$ the largest integer less than or equal to r. If x, y, are real numbers with $x,y\ge 1$ then which of the following statements is always true?

• A. $[x+y]\le \left[x\right]+\left[y\right]$
• B. $\left[xy\right]\le \left[x\right]\left[y\right]$
• C. $\left[2^x\right]\le 2^{\infty }$
• D. $\left[\frac{x}{y}\right]\le \frac{|x{|}^2}{|y{|}^2}$

Answer 1

The correct option is C $\left[2^x\right]\le 2^{\infty }$

option D : $\left[\frac{x}{y}\right]\ge \frac{|x{|}^2}{|y{|}^2}$

for option A: say $x=7/2$ and $y=9/2$

Therefore, $[x+y]=8$

However, $\left[x\right]=3$ and $\left[y\right]=4$

Thus $[x+y]\ge \left[x\right]+\left[y\right]$

Also option B: $\left[xy\right]=63/4$

and $\left[x\right]\left[y\right]=3\times 4=12$

again $\left[xy\right]\ge \left[x\right]\left[y\right]$

$\left[2^x\right]\le 2^{\infty }$

which means that for any value of $x$,

$\left[2^x\right]\le \infty$

Thus, option C is correct.

All the other options are incorrect