Home Competitions CMO 2006 Problem 5
Let { a n } \{a_n\} { a n } a 1 = 1 2 a_1 = \frac{1}{2} a 1 = 2 1 a k + 1 = − a k + 1 2 − a k a_{k+1}=-a_k+\frac{1}{2-a_k} a k + 1 = − a k + 2 − a k 1 k = 1 , 2 , … k = 1, 2,\ldots k = 1 , 2 , … ( n 2 ( a 1 + a 2 + ⋯ + a n ) − 1 ) n ≤ ( a 1 + a 2 + ⋯ + a n n ) n ( 1 a 1 − 1 ) ( 1 a 2 − 1 ) ⋯ ( 1 a n − 1 ) . \left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right). ( 2 ( a 1 + a 2 + ⋯ + a n ) n − 1 ) n ≤ ( n a 1 + a 2 + ⋯ + a n ) n ( a 1 1 − 1 ) ( a 2 1 − 1 ) ⋯ ( a n 1 − 1 ) .
Dataset text AI-MO AoPS . Quoted with source.
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