Reāliem skaitliem \(x\) un \(y\) ir spēkā vienādība: \(\frac{x+y}{x-y}+\frac{x-y}{x+y}=5\). Pierādīt, ka \(\frac{x^{2}+y^{2}}{x^{2}-y^{2}}+\frac{x^{2}-y^{2}}{x^{2}+y^{2}}<3\).
Vienādojot saucējus izteiksmē \(\frac{x+y}{x-y}+\frac{x-y}{x+y}\), iegūstam, ka
\[ \frac{x+y}{x-y}+\frac{x-y}{x+y}=\frac{(x+y)^{2}+(x-y)^{2}}{(x+y)(x-y)}=\frac{x^{2}+2 x y+y^{2}+x^{2}-2 x y+y^{2}}{x^{2}-y^{2}}=2 \cdot \frac{x^{2}+y^{2}}{x^{2}-y^{2}} \]
Tātad, ja \(\frac{x+y}{x-y}+\frac{x-y}{x+y}=5\), tad \(\frac{x^{2}+y^{2}}{x^{2}-y^{2}}=\frac{5}{2}\) un\[ \frac{x^{2}+y^{2}}{x^{2}-y^{2}}+\frac{x^{2}-y^{2}}{x^{2}+y^{2}}=\frac{5}{2}+\frac{2}{5}=\frac{29}{10}<3 \]