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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?

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  • $\begingroup$ 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. $\endgroup$ – Paul Sinclair Mar 13 at 23:47
  • $\begingroup$ Thank you! But do you think I can estimate the expression without using Taylor polynomials? $\endgroup$ – Michael Goldberg Mar 14 at 5:53
  • $\begingroup$ You can if you want to do a lot more work to accomplish the same thing, but why? $\endgroup$ – Paul Sinclair Mar 14 at 15:38
  • $\begingroup$ Yes, you are right. I was just thinking about a more elementary solution when it comes to the estimation. $\endgroup$ – Michael Goldberg Mar 14 at 16:21

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