IMO Shortlist 2010 problem A8
Dodao/la:
arhiva23. lipnja 2013. Given six positive numbers
![a,b,c,d,e,f](/media/m/b/8/0/b801033b9f6c95b49242dd5b44afc876.png)
such that
![a < b < c < d < e < f.](/media/m/b/9/3/b9307e3ab0780941913e42dda3b62000.png)
Let
![a+c+e=S](/media/m/8/b/b/8bb9170cf754153aadfb95e2291e6ad9.png)
and
![b+d+f=T.](/media/m/0/7/8/078297f51ed6fe413a49dcb33aee1072.png)
Prove that
![2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.](/media/m/8/8/1/88130e229dbedcab0a72ce41818a8a60.png)
Proposed by Sung Yun Kim, South Korea
%V0
Given six positive numbers $a,b,c,d,e,f$ such that $a < b < c < d < e < f.$ Let $a+c+e=S$ and $b+d+f=T.$ Prove that
$$2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.$$
Proposed by Sung Yun Kim, South Korea
Izvor: Međunarodna matematička olimpijada, shortlist 2010