Home Competitions ILL 1971 Problem 18
Let a 1 , a 2 , … , a n a_1, a_2, \ldots, a_n a 1 , a 2 , … , a n m g = ( a 1 a 2 ⋯ a n ) n m_g = \sqrt[n]{(a_1a_2 \cdots a_n)} m g = n ( a 1 a 2 ⋯ a n ) m a = ( a 1 + a 2 + ⋯ + a n ) n m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n} m a = n ( a 1 + a 2 + ⋯ + a n ) ( 1 + m g ) n ≤ ( 1 + a 1 ) ⋯ ( 1 + a n ) ≤ ( 1 + m a ) n . (1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n. ( 1 + m g ) n ≤ ( 1 + a 1 ) ⋯ ( 1 + a n ) ≤ ( 1 + m a ) n .
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