09/03/28 00:59:42
>>847
Solution 1.
f(x) = (1-x)log(x), とおくと
f "(x) = -(1+x)/(x^2) < 0,
∴ y=f(x) は上に凸。
∴ log(左辺) = f(a) + f(b) + f(c) ≦ 3f((a+b+c)/3) = 3f(1/3) = 2log(1/3) = log(1/9) = log(右辺),
Solution 312.
相加相乗平均２回
(a+1)^2 = (a-1)^2 + 4a ≧ 4a, etc.
(左辺) ≧ 16{(a/b)^2 + (b/c)^2 + (c/a)^2}
= 16{(aac)^2 + (bba)^2 + (ccb)^2}/(abc)^2
= 48 + {(aac)^2 + (bba)^2 + (ccb)^2 -3(abc)^2}/(abc)^2
= 48 + (X^3 + Y^3 + Z^3 -3XYZ)/(abc)^2
≧ 48 = (右辺).
ここに X=(AAC)^(2/3), Y=(BBA)^(2/3), Z=(CCB)^(2/3),
X^3 +Y^3 +Z^3 -3XYZ = (1/2)(X+Y+Z){(X-Y)^2 +(Y-Z)^2 +(Z-X)^2} ≧ 0,