６５
For $0\leq x\leq 1,\ 0\leq y\leq 1,\ 0\leq z\leq 1$,
Prove that :
$$\frac{x+y+z}{3}+\sqrt{x(1-x)+y(1-y)+z(1-z)}\leq \frac 32$$
６６
Prove the following inequality.
$$\frac 1{2\cos 1}\int_1^{\frac {\pi}2}\frac {\sin x}x\ dx\, \leq\, \frac {\log_e \pi-\log_e 2}{\pi-2}$$.
６７
Let $a,\ b,\ c$ be distinct real numbers.
Prove that :
$$|a-b|+|b-c|+|c-a|+\left|\frac{b-c}{a-b}\right|+\left|\frac{c-a}{b-c}\right|+\left|\frac{a-b}{c-a}\right|+\left|\frac{c-a}{a-b}\right|+\left|\frac{a-b}{b-c}\right|+\left|\frac{b-c}{c-a}\right|$$
$$\geq 2\sqrt{|a-b|}+2\sqrt{|b-c|}+2\sqrt{|c-a|}.$$
６８
Let $a,\ b,\ c$ be positive real numbers such that $\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=4.$
Prove that :
$$\frac{1}{2a+b+c}+\frac{1}{a+2b+c}+\frac{1}{a+b+2c}\leq 1$$
６９
Let $a,\ b,\ c$ be positive real numbers such that $a+b+c=1$.
Prove that :
$$\sqrt[3]{\left(\frac{1}{ab}-2\right)\left(\frac{1}{bc}-2\right)\left(\frac{1}{ca}-2\right)}\geq 7$$
７０
Let $a,b,c$ be non-negative real numbers such that no two of which are 0.
Prove that :
$$\frac {ab}{(a+b)^2}+\frac {bc}{(b+c)^2}+\frac {ca}{(c+a)^2}+\frac 1{4}\ge \frac{ab+bc+ca}{a^2+b^2+c^2}$$