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India EGMO 2022 TST, 1

Let $n\ge 3$ be an integer, and suppose $x_1,x_2,\cdots ,x_n$ are positive real numbers such that $x_1+x_2+\cdots +x_n=1.$ Prove that $$x_1^{1-x_2}+x_2^{1-x_3}\cdots+x_{n-1}^{1-x_n}+x_n^{1-x_1}<2.$$ [i] ~Sutanay Bhattacharya[/i]

2009 Singapore Senior Math Olympiad, 4

Tags: weighted am-gm , logarithm , inequalities

Given that $ a,b,c, x_1, x_2, ... , x_5 $ are real positives such that $ a+b+c=1 $ and $ x_1.x_2.x_3.x_4.x_5 = 1 $. Prove that \[ (ax_1^2+bx_1+c)(ax_2^2+bx_2+c)...(ax_5^2+bx_5+c)\ge 1\]

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