Iniciante
. Chame
então temos
onde claramente
é inteiro se variarmos
, logo para todo
vale que
é inteiro e positivo, logo basta achar o número de
. Vejamos que o menor
é
pois
0 \Rightarrow k>0" /> e o maior
é
pois
2014" />.
Intermediário
.
Avançado
Façamos
,
,
. Temos então que
. Logo, transformando os 1’s dos numeradores e denomindores em abc, nossa desigualdade fica assim:
![x=a^3](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_e157574f080f35f6a6346fa9cc1dfd8f.gif?w=640&ssl=1)
![y=b^3](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_bed9a8b8c0a657ddb35dd8304672f01c.gif?w=640&ssl=1)
![z=c^3](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_0db7f82b8869d0be8e7f07e92a69e8b2.gif?w=640&ssl=1)
![abc=1](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_eb2edd33b6c2b81ff9ee629c4956bbe9.gif?w=640&ssl=1)
![\dfrac{x^3}{(1+y)(1+z)}+\dfrac{y^3}{(1+x)(1+z)}+\dfrac{z^3}{(1+x)(1+y)} \ge \dfrac{3}{4}](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_498247dde2765669ea6a03e31aa5fe2d.gif?w=640&ssl=1)
![\dfrac{a^9}{(abc+b^3)(abc+c^3)}+\dfrac{b^9}{(abc+a^3)(abc+c^3)}+\dfrac{c^9}{(abc+a^3)(abc+b^3)} \ge \dfrac{3abc}{4}](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_7bf9575a2dc49bb35cf9eba68d48cb24.gif?w=640&ssl=1)
Abrindo e multiplicando por
, temos:
![2](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_c81e728d9d4c2f636f067f89cc14862c.gif?w=640&ssl=1)
![\displaystyle\sum_{sym} a^9(abc+a^3)\ge \dfrac{3abc}{4} \displaystyle\sum_{sym} (abc+a^3)(abc+b^3)(abc+c^3) \Rightarrow 4\displaystyle\sum_{sym}a^{12}+a^{10}bc \ge 3a^2b^2c^2 \displaystyle\sum_{sym} (a^2+bc)(b^2+ac)(c^2+ab) \Rightarrow \displaystyle\sum_{sym} 4a^{12} + 4a^{10}bc \ge \displaystyle\sum_{sym} 6a^4b^4c^4 + a^5b^5c^2 + a^6b^3c^3.](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_8767a9773a4fc91ca3f7b10248f4b72e.gif?w=640&ssl=1)
Veja que, por Muirhead:
![\displaystyle\sum_{sym} 4a^{12}bc \ge \displaystyle\sum_{sym} 4a^4b^4c^4](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_1c661071a19596a3217f4caa75b3ce13.gif?w=640&ssl=1)
![\displaystyle\sum_{sym} 2a^{10}bc \ge \displaystyle\sum_{sym} 2a^4b^4c^4](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_d4e9eaee44dbf2f1e94e162f2b040ed0.gif?w=640&ssl=1)
![\displaystyle\sum_{sym} a^{10}bc \ge \displaystyle\sum_{sym} a^5b^5c^2](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_78434fd74f321ce352bf01686ebb015c.gif?w=640&ssl=1)
![\displaystyle\sum_{sym} a^{10}bc \ge \displaystyle\sum_{sym} a^6b^3c^3](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_2d0e837d78613cd86d4c3cccb79d2124.gif?w=640&ssl=1)
Logo, o problema acabou.