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Let a, b, c, and d be positive integers. We know that a+b = kc, b+c = kd, c+d =ka, d+a = kb. Find the maximum of a/b + b/c + c/d + d/a.

 Feb 1, 2021
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Regarded as a set of four equations in the four unknowns a, b, c and d, (k being a parameter),

the equations will possess non-zero solutions only if the determinant of coefficients is equal to zero.

That is,

\(\displaystyle \begin{array} {| cccc |}1 & 1 & -k & 0 \\ 0 & 1 & 1 & -k \\ -k & 0 & 1 & 1\\ 1 & -k & 0 & 1 \\ \end{array}=0\)

Expanding, that gets you 

\(\displaystyle k^{4}-2k^{2} -4k = 0, \\ \text{ or, }\\ k(k-2)(k^{2}+2k+2)=0, \\ \text{ so } \\ k=0 \text{ or } k=2.\)

 

k = 0 produces the solution a = t, b = -t, c = t, d = -t , t a parameter, from which a/b + b/c + c/d + d/a = -4.

k = 2 produces a = b = c = d = t, t a parameter so a/b + b/c + c/d + d/a = 4.

 

So the maximum would appear to be 4.

 Feb 2, 2021

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