Olympiad Corner #1

$Q1$ If $S_{ n }$ is the sum

$S_{ n }=1+\frac { 1 }{ 2 } +\frac { 1 }{ 3 } +.........+\frac { 1 }{ n } (n>2),\quad prove\quad that\\ n{ (n+1) }^{ \frac { 1 }{ n } }-n<{ S }_{ n }<n-(n-1){ n }^{ \frac { -1 }{ (n-1) } }$

$Q2$ If $a_{1}$ , $a_{2}$ , $a_{3}$ ,........, $a_{n}$ are positive real numbers, prove that

$\sum _{ i<j }^{ }{ \sqrt { { a }_{ i }{ a }_{ j } } } \le \frac { n-1 }{ 2 } ({ a }_{ 1 }+{ a }_{ 2 }+...+{ a }_{ n })$

#Algebra #RMO #Proofs #INMO

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