代數與最小值 (1)

a＋b＋c＝x ⋯⋯⋯⋯ (i)
ab＋bc＋ca＝y ⋯⋯ (ii)
abc＝z ⋯⋯⋯⋯⋯⋯ (iii)
x＋y＋z＝6967 ⋯⋯ (iv)

From (i), (ii), (iii), (iv), we get
(a＋1)(b＋1)(c＋1)
＝abc＋ab＋bc＋ca＋a＋b＋c＋1
＝z＋y＋x＋1
＝6967＋1
＝2*2*2*13*67

Assume 0＜a ≦ b ≦ c, so, 1＜(a+1) ≦ (b+1) ≦ (c+1)
Therefore, (a, b, c) can be
(1, 1, 1741), (1, 3, 870), (1, 25, 133), (3, 25, 66), (3, 12, 133), (7, 12, 66)
So, x can be 1742, 874, 159, 94, 148, 85.
ie. the minimum value of x is 85.

answer is 85