exo fesable..3

[img]http://latex.codecogs.com/gif.latex?a,b,c%3E0...MQ:\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{6\sqrt[3]{abc}}{a+b+c}%3E=5[/img]

si on pose x=a/b y=b/c z=c/a l'inegalite devient plus facile

aymas a écrit:

si on pose x=a/b y=b/c z=c/a l'inegalite devient plus facile

mais le terme 5racicubique(abc)\(a+b+c) devient plus compliqué

[img]http://latex.codecogs.com/gif.latex?d%27apres%20%5C%20la%20%5C%20remarque%20%5C%20precedent%20%5C%20l%27inegalite%20%5C%20devient%20%5C%20%5Cfrac%7B1%7D%7Bx%7D%20+%5Cfrac%7B1%7D%7By%7D%20+%5Cfrac%7B1%7D%7Bz%7D%20+6%20%5Cfrac%7B1%7D%7B%5Csum%20%5Csqrt[3]%7B%5Cfrac%7Bx%7D%7By%7D%7D%7D%20%5Cgeq%205%20%5C%20equivalent%20%5C%20%C3%A0%20%5C%20%5Csum%20xy%20+%5Cfrac%7B6%7D%7B%5Csum%20%5Csqrt[3]%7Bxy%7D%7D%20%5Cgeq%205%20%5C%5C%20posons%20%5C%20on%20%5C%20suite%20%5C%20%5Csqrt[3]%7Bxy%7D=r%20%5C%20et%20%5C%20%5Csqrt[3]%7Bxz%7D=p%20%5C%20et%20%5C%20%5Csqrt[3]%7Byz%7D=q%20%5C%20il%20%5C%20s%27ensuit%20%5C%20que%20%5C%20pqr=1%20%5C%20d%27apres%20%5C%20l%27hypothese%20.%5C%5C%20l%27inegalite%20%5C%20devient%20%5C%20%5Csum%20p%5E%7B3%7D%20+%20%5Cfrac%7B6%7D%7B%5Csum%20p%7D%5Cgeq%205%20%5C%5C%20D%27apres%20%5C%20l%27inegalite%20%5C%20de%20%5C%20holder%20%5C%20%5Csum%20p%5E%7B3%7D%5Cgeq%201/9%20%28%5Csum%20p%29%5E%7B3%7D%20%5C%5C%20alors%20%5C%20il%20%5C%20suffit%20%5C%20de%20%5C%20prouver%20%5C%20que%20%5C%201/9%20%28%5Csum%20p%29%5E%7B3%7D%20+%20%5Cfrac%7B6%7D%7B%5Csum%20p%7D%20%5Cgeq%205%20%5C%5C%20posons%20%5C%20%5Csum%20p%20=t%20%5C%20le%20%5C%20polyn%C3%B4me%20%5C%20%5Cfrac%7B1%7D%7B9%7Dt%5E%7B3%7D%20+%20%5Cfrac%7B6%7D%7Bt%7D%20-%205%20%5C%20et%20%5C%20toujours%20%5C%20positive%20%5C%20si%20%5C%20t%20%5Cgeq%203%20%5C%20ce%20%5C%20qui%20%5C%20est%20%5C%20facile%20%5C%20a%20%5C%20prouver%20%5C%20puisque%20%5C%20pqr%20=1%20%5CRightarrow%20%5Csum%20p%20%5Cgeq%203%20%5C%5C[/img][/img]

sauf ereur

je propose une generalisation

[img]http://latex.codecogs.com/gif.latex?soit%20%5C%20n%20%5C%20un%20%5C%20entier%20%5C%20positive%20%5C%20prouver%20%5C%20que%5C%5C%20a;b;c%3E%20O%20%5C%20%5Csum%20%5Cfrac%7Ba%7D%7Bb%7D%20+%20n%5Cfrac%7B%5Csqrt[3]%7Babc%7D%7D%7Ba+b+c%7D%20%5Cgeq%20%5Cfrac%7Bn%7D%7B3%7D%20+%203%5C%5C[/img]