Pierādīt, ka eksistē tādi pozitīvi skaitļi \(x\) un \(y\), ka
\[x^{y}+y^{x}+x+y<1+\frac{1}{2011}\]
Apskatām skaitļus \(N=9000,\ x=\frac{1}{N}\) un \(y=\frac{1}{N^{N}}\). Tad
\[x^{y}+y^{x}+x+y=\left(\frac{1}{N}\right)^{y}+\left(\frac{1}{N^{N}}\right)^{\frac{1}{N}}+\frac{1}{N}+\frac{1}{N^{N}}=\left(\frac{1}{N}\right)^{y}+\frac{1}{N}+\frac{1}{N}+\frac{1}{N^{N}}\]
Tā kā \(\frac{1}{N} \in(0; 1)\), tad arī \(\left(\frac{1}{N}\right)^{y} \in(0; 1)\). Tāpēc\[x^{y}+y^{x}+x+y<1+\frac{2}{N}+\frac{1}{N^{N}}<1+\frac{2}{9000}+\frac{1}{9000}=1+\frac{1}{3000}<1+\frac{1}{2011}.\]