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BALTIC-WAY 1996 Problem 15

  • Answer

For which positive real numbers a,ba, b does the inequality

x1⋅x2+x2⋅x3+⋯+xn−1⋅xn+xn⋅x1≥x1a⋅x2b⋅x3a+x2a⋅x3b⋅x4a+⋯+xna⋅x1b⋅x2ax_{1} \cdot x_{2}+x_{2} \cdot x_{3}+\cdots+x_{n-1} \cdot x_{n}+x_{n} \cdot x_{1} \geq x_{1}^{a} \cdot x_{2}^{b} \cdot x_{3}^{a}+x_{2}^{a} \cdot x_{3}^{b} \cdot x_{4}^{a}+\cdots+x_{n}^{a} \cdot x_{1}^{b} \cdot x_{2}^{a}

hold for all integers n>2n>2 and positive real numbers x1,x2,…,xnx_{1}, x_{2}, \ldots, x_{n} ?

Dataset text NuminaMath olympiads. Quoted with source.

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b=1, a=\frac{1}{2}

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