４９
Given real numbers $a,b,c,d$ with $a+b+c+d=6,\ a^2+b^2+c^2+d^2=12.$
Prove that :
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
５０
Let $a,\ b$ be positive real numbers such that $a+b=1.$
Prove that : $a^{a}b^{b}+a^bb^a\leq 1.$
５１
Let $a,\ b$ be positive real numbers.
Prove that :
$$ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1).$$
５２
For $0\leq a\leq 1,\ 0\leq b\leq 1,\ 0\leq c\leq 1$,
Prove that :
$$\frac{a}{bc+1}+\frac{b}{ca+1}+\frac{c}{ab+1}\leq 2.$$
５３
For real numbers $a,\ b,\ c$, prove that :
$$a(a+b)^3+b(b+c)^3+c(c+a)^3\ge \frac{8}{27}(a+b+c)^4.$$
５４
For any real numbers $a,b,c$, prove that
$5a^2 + 5b^2 + 5c^2 \ge 4ab + 4ac + 4bc $
and determine when equality occurs.
５５
For positive real numbers $a,\ b,\ c,\ d$ with $abcd=1.$
Prove that :
$$\displaystyle \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geq \frac{{25}}{4}$$.
５６
Let $x,\ y,\ z$ be positive real numbers such that $x+y+z=1$.
Prove that :
$$\displaystyle \sqrt{xy(1-z)}+\sqrt{yz(1-x)}+\sqrt{zx(1-y)}\leq \sqrt{\frac {2}{3}}$$