2

Note: we have $a>0,b>0,c>0$.

Solve $(1-a)^2+(1-b)^2+(1-c)^2 \ge \frac{c^2(1-a^2)(1-b^2)}{(ab+c)^2}+\frac{b^2(1-a^2)(1-c^2)}{(ac+b)^2}+\frac{a^2(1-c^2)(1-b^2)}{(bc+a)^2}$ where $a>0,b>0,c>0$. Elementary (high-school) methods are preferred.

I'm not an expert in inequalities. I only know GM-AM and it does not appear to be working. It is not clear to me how to take advantage of the symmetry here.

Daniel Li
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  • @dezdichado this is proved in the answer below for $a,b,c\ge 0$ – Daniel Li Aug 18 '19 at 20:45
  • @dezdichado how come? I see it may not hold if the assumption $a,b,c\ge 0$ is not given. Otherwise, it looks fine. – Daniel Li Aug 18 '19 at 20:56
  • @dezdichado I double checked. This inequality should be true and we have additional condition that $a,b,c>0$. – Daniel Li Aug 18 '19 at 23:04
  • Some years ago, I saw this problem. In a blog entry posted in 2013, it was said that this problem was proposed by Dongyi Wei (full marks both at 50th IMO 2009 and 49th IMO 2008) and was solved by Zipei Nie (full marks at 51st IMO 2010). Nie's proof is ugly. Some years ago, I also gave a ugly (long) proof. – River Li Aug 19 '19 at 03:15
  • @RiverLi so the proof in the linked question is legit? – dezdichado Aug 19 '19 at 18:48
  • @dezdichado I don't know. I have not yet looked at that proof. – River Li Aug 19 '19 at 23:39

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