let$\displaystyle a,b,c,d$ be distinct real numbers such that
$\displaystyle a/b+b/c+c/d+d/a=4$
and $\displaystyle ac=bd$
find the maximum value of
$\displaystyle a/c+b/d+c/a+d/b$
If the numbers were positive, then AM-GM would have killed it without the need for the second condition
Anyway my solution is tedious, but it works. Let me know if you have a problem somewhere.
We repeatedly use $\displaystyle ac = bd$.
let us first put $\displaystyle \frac{a}{b} = x = \frac{d}{c}$ -----------(1)
and then $\displaystyle \frac{b}{c} = y = \frac{a}{d}$ ----------------(2)
Data says $\displaystyle x + \frac1{x} + y + \frac1{y} = 4$
We want to know the maximum value of
$\displaystyle W = \frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$
But we can use (1) and (2) above to reduce W to
$\displaystyle W = (x + \frac1{x})(y + \frac1{y})$
Now set $\displaystyle x + \frac1{x} = X$ and $\displaystyle y + \frac1{y} = Y$
Our simplified conditions now read....$\displaystyle
\boxed{X+Y = 4, max(XY) = ??}$
This is simple.... :P
$\displaystyle XY = X(4-X) = 4 - (X^2 - 4X + 4) = 4 - (X-2)^2$
The maximum value is 4
(and it occurs for X=Y=2, further you can substitute it above and find the values for which this happens.However the question does not demand the numbers but only the maximum)
I have left a little bit for you, in particular getting $\displaystyle W = (x + \frac1{x})(y + \frac1{y})$. Try it!
I am a little bit worried about that word distinct because I got a=b=c=d for W = 4.
Do we need to worry about this at all?
I think that the maximum for distinct solutions will be when a,b,c and d are infinitely close and that the maximum for W will then be infinitely close to 4. So maybe we don't need to worry.
Very nice proof by the way Isomorphism.
Not necessarily. In other words, you got one set of solutions which you have to discard because of the data in the problem.
However note that the problem does not say they are positive real numbers.
So to satisfy
$\displaystyle x + \frac1{x} + y + \frac1{y} = 4$
Solve
$\displaystyle x + \frac1{x} = -3$ and $\displaystyle y + \frac1{y} = 7$
And get some distinct solutions
Thank youVery nice proof by the way Isomorphism.
I don't think I explained myself clearly enough the first time so I will have another go.
X=2The maximum value is 4
(and it occurs for X=Y=2,
$\displaystyle x+\frac {1}{x} = 2$
$\displaystyle x^2+1 = 2x$
$\displaystyle x^2-2x+1=0$
x=1
$\displaystyle \frac {a}{b} = 1$
a=b
So the point at which the maximum (4) occurs has a = b, so the numbers are not distinct. This is what I am worried about.
If we do this then $\displaystyle XY \not = 4$x + \frac1{x} = -3 and y + \frac1{y} = 7