1this is a two liner guys ... crack it !
Suppose a_1, a_2, a_3,...,a_n are n real numbers
It is obvious that if the ai are all positive, then the numbers \sum a_i, \sum_{i<j} a_i a_j \sum_{i<j<k} a_ia_ja_k,..., \prod_{i=1}^n a_i will all be positive
Prove that the converse also holds
[Students first please. Hints after day 1]
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5 Answers
39take the polynomial
(x-a1)(x-a2)......(x-an)
this equation has these sums as its coefficients with (-1)^i
if these sums are positive, this means that the i-th coefficient is positive if n-i is even and negative if n-i is odd
now such an equation cannot be ful filled by a negative number.
hence proved
3dude i just got this one in mind,in the next time give time for beginners like me.