Home Competitions CGMO 2010 Problem 4
Let x 1 , x 2 , ⋯ , x n x_1,x_2,\cdots,x_n x 1 , x 2 , ⋯ , x n x 1 2 + x 2 2 + ⋯ + x n 2 = 1 x_1^2+x_2^2+\cdots+x_n^2=1 x 1 2 + x 2 2 + ⋯ + x n 2 = 1 ∑ k = 1 n ( 1 − k ∑ i = 1 n i x i 2 ) 2 ⋅ x k 2 k ≤ ( n − 1 n + 1 ) 2 ∑ k = 1 n x k 2 k \sum_{k=1}^{n}\left(1-\dfrac{k}{{ \sum_{i=1}^{n} ix_i^2}}\right)^2 \cdot \dfrac{x_k^2}{k} \leq \left(\dfrac{n-1}{n+1}\right)^2 \sum_{k=1}^{n} \dfrac{x_k^2}{k} k = 1 ∑ n ( 1 − ∑ i = 1 n i x i 2 k ) 2 ⋅ k x k 2 ≤ ( n + 1 n − 1 ) 2 k = 1 ∑ n k x k 2
Dataset text AI-MO AoPS . Quoted with source.
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