IMO Shortlist 2010 problem A2
Dodao/la:
arhiva23. lipnja 2013. Let the real numbers
![a,b,c,d](/media/m/7/6/0/7605ede133e1f767d3890e0bfffb7b7f.png)
satisfy the relations
![a+b+c+d=6](/media/m/6/a/b/6ab27e515f7da8efaa53db76c36100db.png)
and
![a^2+b^2+c^2+d^2=12.](/media/m/b/2/b/b2bc15191edc0a148b868864c529d5d2.png)
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.](/media/m/8/4/1/84155b72433570fa0797d8c380201562.png)
Proposed by Nazar Serdyuk, Ukraine
%V0
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
$$36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.$$
Proposed by Nazar Serdyuk, Ukraine
Izvor: Međunarodna matematička olimpijada, shortlist 2010