[過去ﾛｸﾞ] 不等式への招待 第６章 (995ﾚｽ)
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760(3): 2012/10/28(日)08:21 AAS
３３
For $ a>0\ b>0\ c>0$, prove that $\displaystyle \frac{a^3+b^3+c^3+6}{ab+bc+ca}\left(\frac{b^2}{a}+\frac{c^2}{b}+\frac{a^2}{c}\right)\geq 9.$
３４
Let $ a,\ b,\ c$ be postive real numbers such that $\displaystyle \frac{a^2+b^2}{(a+b)^3}+\frac{b^2+c^2}{(b+c)^3}+\frac{c^2+a^2}{(c+a)^3}=\frac{3}{2}$.
Prove that $\displaystyle a^2+b^2+c^2\geq \frac{3}{4}$.
３５
Let $a,\ b,\ c$ be positive real numbers.
Prove that :
\[\left(\frac{1}{a}+b\right)\left(\frac{1}{b}+c\right)\left(\frac{1}{c}+a\right)\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)\left(c+\frac{1}{c}\right)\geq 64.\]
３６
Let $a,\ b,\ c$ be positive real numbers such that $a^ab^bc^c=1$.
Prove that $a+b+c\leq 3$.
３７
For $-2\leq a,\ b,\ c\leq 1$ and $a+b+c=0$, prove that $a^2+b^2+c^2\leq 6$.
３８
Given positive real numbers $x,\ y,\ z$ with $x+y+z=1$.
Prove that :
\[ x\sqrt[3]{1+y-z}+y\sqrt[3]{1+z-x}+z\sqrt[3]{1+x-y}\leq 1. \]
３９
For $\displaystyle 0<x<\frac{\pi}{2}$, prove that $\displaystyle \left (\frac{\sin x}{x} \right )^2 > \cos x >\left( \frac{x}{\tan x}\right)^2.$
４０
Let $ a,\ b$ be real numbers with $ 0<a<b$ and $ x,\ y,\ z$ satisfy :
$$a\leq x\leq b,\ a\leq y\leq b,\ a\leq z\leq b$$
If $ x+y+z=a+2b$, then prove that $ xyz\geq ab^2$.

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