exercise 1.108 (Greece 2008) [BOMC]
Posted: Sun Oct 30, 2011 9:28 pm
can we use anything else than "the helpful inequality"
for $x_{1},x_{2},x_{3},\cdot \cdot \cdot x_{n}$ positive integers, prove that
\[(\frac{x_{1}^{2}+x_{2}^{2}+\cdot \cdot \cdot +x_{n}^{2}}{x_{1}+x_{2}+\cdot \cdot \cdot +x_{n}})^{\frac{kn}{t}}\geq x_{1}\cdot x_{2}\cdot \cdot \cdot \cdot \cdot x_{n}\],
where $k=$ max {${{x_{1}, x_{2},\cdot \cdot \cdot ,x_{n}}}$} and $t=$ min {${{x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}}}$}.
under which conditions the equality holds?
for $x_{1},x_{2},x_{3},\cdot \cdot \cdot x_{n}$ positive integers, prove that
\[(\frac{x_{1}^{2}+x_{2}^{2}+\cdot \cdot \cdot +x_{n}^{2}}{x_{1}+x_{2}+\cdot \cdot \cdot +x_{n}})^{\frac{kn}{t}}\geq x_{1}\cdot x_{2}\cdot \cdot \cdot \cdot \cdot x_{n}\],
where $k=$ max {${{x_{1}, x_{2},\cdot \cdot \cdot ,x_{n}}}$} and $t=$ min {${{x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}}}$}.
under which conditions the equality holds?