Problem E:15756 GMB 6-7-8/2020, pag 369

 

 Proposed for the eighth grade. In translation:

          " Let $n \geqslant 2$ be a natural number and $x$ a nonzero real number so that $\frac{x^{2}+1}{x}=\sqrt{2n+1}$. Prove that $\frac{x^{4}+(n+1)x^{2}+1}{x^{2}}$ is a natural number, divisible by 3."

 

 ANSWER CiP

$\frac{x^{4}+(n+1)x^{2}+1}{x^{2}}=3\cdot n$

 

Solution CiP

            Squaring given relation $x+\frac{1}{x}=\sqrt{2n+1}$ it results

$x^{2}+2 \cdot x \cdot \frac{1}{x}+ \frac{1}{x^{2}}=2n+1$ $\Rightarrow$ $x^{2}+\frac{1}{x^{2}}=2n+1-2$ $\Rightarrow$

$\Rightarrow$ $x^{2}+(n+1)+\frac{1}{x^{2}}=2n-1+(n+1)$ $\Leftrightarrow$ $\frac{x^{4}+(n+1)x^{2}+1}{x^{2}}=3 \cdot n$.

$\blacksquare$