1. Inequality challenge questions

1. Let $a,b,c,d \ge 0$ prove that $\frac{1}{a}+\frac{1}{b} + \frac{4}{c}+\frac{16}{d} \ge \frac{64}{a+b+c+d}$

2. Let $x,y,z \ge 0$ with $xyz = 1$. Find the minimum value of $\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y}$

3. Let $a,b,c \ge 0$. Prove that $\sqrt{3(a+b+c)} \ge \sqrt{a} + \sqrt{b} + \sqrt{c}$

2. In what sense are these "challenge questions"?

3. In the sense that they are for me

4. the first and the third inequality can be easily proved by using Cauchy inequality.
By let x+y=r, y+z=t, z+x=s; then the second inequality can be easily proved by using the Mean value inequality!

5. You should try your best before asking help in the forum.
And tell us where you are stuck on!