[不等式] kuing 来玩几个？ :D

problem 1:
For $a,b,c>0$ show that:
\[ \frac{(a+b+c)^{2}}{2(ab+bc+ca)}\geq \frac{a^{2}}{a^{2}+bc}+\frac{b^{2}}{b^{2}+ca}+\frac{c^{2}}{c^{2}+ab} \]

Problem 2:
For $a,b,c>0$ such that:$ ab+bc+ca=3 $ prove that:
\[ (a+2b)(b+2c)(c+2a)\geq 8\]

Problem 3:
Let $a,b,c\geq 0$ with$ ab+bc+ca+abc=4 $ Show that:
\[ \sqrt{a+3}+\sqrt{b+3}+\sqrt{c+3}\geq 6 \]

Have fun!

1# pxchg1200


人呢？！

没空啊这几天都