MHB Finding the Maximum Value of n for a Given Equation

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The discussion revolves around finding the largest value of n for which a multiple of 4 exists between n^2 and n^2 + 2016/n^2. It is established that n^2 is less than n^2 + 2016/n^2, leading to confusion about the existence of a number that fits the criteria. Participants question the logical consistency of having a number that is both less than and greater than defined bounds. The core issue lies in reconciling the mathematical conditions set by the equation. Ultimately, the conversation highlights the complexities of the problem and the need for clarity in defining the parameters.
Marcelo Arevalo
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What is the Largest possible value of n such that there is a multiple of 4 less than n^2 but greater than n^2 + 2016/n^2 ?
 
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I think you will agree that:

$$n^2<n^2+\frac{2016}{n^2}$$

So, how can some number be smaller than the smaller value, and at the same time greater than the greater value?
 
Meaning, there's no such thing as the smaller value, and at the same time greater than the greater value?
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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