MathDB

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Part of 2015 China Second Round Olympiad

Problems(2)

China Second Round Olympiad 2015 Test 2 Q1

Source: China

9/13/2015
Let a1,a2,…,ana_1, a_2, \ldots, a_n be real numbers.Prove that you can select ε1,ε2,…,εn∈{−1,1}\varepsilon _1, \varepsilon _2, \ldots, \varepsilon _n\in\{-1,1\} such that(∑i=1nai)2+(∑i=1nεiai)2≤(n+1)(∑i=1nai2).\left( \sum_{i=1}^{n}a_{i}\right)^2 +\left( \sum_{i=1}^{n}\varepsilon _ia_{i}\right)^2 \leq(n+1)\left( \sum_{i=1}^{n}a^2_{i}\right).
inequalitiesProbabilistic Method
China Second Round Olympiad 2015 （B) Test 2 ，Q1

Source: Sep 13, 2015

9/15/2015
Let a,b,ca,b,c be nonnegative real numbers.Prove that(a−bc)2+(b−ca)2+(c−ab)2(a−b)2+(b−c)2+(c−a)2≥12.\frac{(a-bc)^2+(b-ca)^2+(c-ab)^2}{(a-b)^2+(b-c)^2+(c-a)^2}\geq\frac{1}{2}.
inequalities