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2

Part of 1987 Spain Mathematical Olympiad

Problems(1)

Σκ \sqrt{{n \choose k}} <\sqrt{2^{n-1}n^3}, from k=1 to n

Source: Spanish Mathematical Olympiad 1987 P2

8/2/2018
Show that for each natural number n>1n > 1 1⋅(n1)+2⋅(n2)+...+n⋅(nn)<2n−1n31 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}
binomial coefficientsinequalitiespositive integers