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2013年新疆维吾尔自治区高中数学预赛的最后一题是:设a、b、c为正实数,求证:(a~2+2)(b~2+2)(c~2+2)≥3(a+b+c)~2。①此题清爽、简洁,思路开阔,背景深刻,值得进行深入地探究。1试题的证明命题组给出的参考答案如下:先证明:(a~2+2)(b~2+2)≥3/2[(a+b)~2+2]。②事实上,由②式(?)2(a~2b~2+2a~2+2b~2+4)≥3(a~2+b~2+2ab+2)(?)2a~2b~2+a~2+b~2-6ab+2≥0(?)2(ab-1)~2+(a-b)~2≥0。这显然成立。再由②式及柯西不等式,得
In the 2013 Xinjiang Uygur Autonomous Region high school mathematics preliminaries, the final question is: Let a, b and c be positive real numbers, and verify: (a ~ 2 + 2) (b ~ 2 + 2) (c ~ 2 + 2) a + b + c) ~ 2. ① This topic is fresh, simple, open-minded, profound background, it is worth further exploration. 1 Proof of test questions Proposition group given reference answer is as follows: First prove: (a ~ 2 +2) (b ~ 2 +2) ≥3 / 2 [(a + b) ~ 2 +2]. (2) In fact, it is easy to find out that there are two kinds of arsenic in the surface, which are composed of 2a ~ 2b ~ 2 + 2b ~ 2 + 2 + a ~ 2 + b ~ 2-6 ab + 2 ≧ 0 (?) 2 (ab-1) ~ 2 + (ab) ~ 2 ≧ 0. This is obviously true. Then by ② and Cauchy inequality, too