Since the last post of this kind was so wildly popular (I was the only one to post....), I thought I would try another.
"For any positive integer, k, let Sk = {x1, x2, ... , xn} be the set of real numbers for which x1 + x2 + ... + xn = k and P = x1 x2 ... xn is maximised.
For example, when k = 10, the set {2, 3, 5} would give P = 30 and the set {2.2, 2.4, 2.5, 2.9} would give P = 38.25. In fact, S10 = {2.5, 2.5, 2.5, 2.5}, for which P = 39.0625.
Prove that P is maximised when all the elements of S are equal in value and rational."
Oooh....the British spelling of maximized....this must be a good problem....right?
And follow-up questions of my own design....
For a given k, what would n be? For this, assume you can have a real number for n. (The actual n would be either the ceiling or floor of the equation for n.)
Given your equation for n....would that make each element in the set still rational?
What is each element in the set? Are you surprised by this answer?
Question
EventHorizon
Since the last post of this kind was so wildly popular (I was the only one to post....), I thought I would try another.
"For any positive integer, k, let Sk = {x1, x2, ... , xn} be the set of real numbers for which x1 + x2 + ... + xn = k and P = x1 x2 ... xn is maximised.
For example, when k = 10, the set {2, 3, 5} would give P = 30 and the set {2.2, 2.4, 2.5, 2.9} would give P = 38.25. In fact, S10 = {2.5, 2.5, 2.5, 2.5}, for which P = 39.0625.
Prove that P is maximised when all the elements of S are equal in value and rational."
Oooh....the British spelling of maximized....this must be a good problem....right?
And follow-up questions of my own design....
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