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5

Part of 2006 Japan MO Finals

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Japan mathematical olympiad finals 2006, problem 5

Source:

3/3/2006
For any positive real numbers x1, x2, x3, y1, y2, y3, z1, z2, z3,x_1,\ x_2,\ x_3,\ y_1,\ y_2,\ y_3,\ z_1,\ z_2,\ z_3, find the maximum value of real number AA such that if M=(x13+x23+x33+1)(y13+y23+y33+1)(z13+z23+z33+1) M = (x_1^3+x_2^3+x_3^3+1)(y_1^3+y_2^3+y_3^3+1)(z_1^3+z_2^3+z_3^3+1) and N=A(x1+y1+z1)(x2+y2+z2)(x3+y3+z3), N = A(x_1+y_1+z_1)(x_2+y_2+z_2)(x_3+y_3+z_3), then M≥NM \geq N always holds, then find the condition that the equality holds.
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