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Given that a<b<c<d<e and p=abcde, a, b, c, d and e are positive integers. When just one of these integers is increased by 1 and multiplied by the other four, the new product is n. The difference (n - p) is greatest when which of the five integers is increased by 1?
How can I prove that it is a?
Suppose that the numbers are r,s,t,u,v, where one of these is a second is b and so on, p = rstuv.
Increase r by 1 and you have (r+1)stuv = rstuv + stuv, so n - p = (r+1)stuv - rstuv = stuv.
n - p will take its greatest value when stuv takes its greatest value and that will be when s,t,u and v are the four greatest numbers (and r is the smallest).
So how did we use the given inequality? I still don't see it how to prove it (I followed everything except the very end)