# Calculate the floor-function of the expression

Let n be a positive integer and x a real number with x $$\ge$$ $$\frac {3n^2+1}{3}$$. Calculate $$\lfloor \sqrt{x^2-nx}+\sqrt{x^2-n^2}+\sqrt{x^2+n^2}-3x \rfloor$$, where $$\lfloor t \rfloor$$ is the usual notation for the integer part of t.

I have proved that the expression is negative using the mean inequality and I have seen that the result depends on n and x, so it isn't something constant. I tried to fit every radical between 2 expressions, but I couldn't do it for the last one. Can you help me?

• I suggest letting $u = \frac nx$ and rewriting the expression inside the brackets as $$nu\left(\sqrt{1-u} + \sqrt{1 - u^2} + \sqrt{1 + u^2} - 3\right)$$ for $$u \le \frac{3n}{3n^2+1} < \frac 1n$$ You can use Taylor polynomials to estimate the expression and decide what the integer part will be. – Paul Sinclair Mar 13 at 23:47
• Thank you! But do you think I can estimate the expression without using Taylor polynomials? – Michael Goldberg Mar 14 at 5:53
• You can if you want to do a lot more work to accomplish the same thing, but why? – Paul Sinclair Mar 14 at 15:38
• Yes, you are right. I was just thinking about a more elementary solution when it comes to the estimation. – Michael Goldberg Mar 14 at 16:21