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A positive rational function (Posted on 2019-11-01) Difficulty: 3 of 5
Let Q' denote the set of all positive rational numbers. If f:Q'→Q' satisfies

f(x2f(y)2)=f(x)2f(y),

evaluate f(2019).

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Lengthy solution Comment 3 of 3 |
The square root function would also work with this equation. But whoops, it is not rational! Can that get us anywhere? Let's see...

Call the original equation "I":

f( x^2 f(y)^2 ) = f(x)^2 f(y)

With x=1, this becomes

II: f( f(y)^2 ) = c^2 f(y)

where c:=f(1). This almost looks like the square root function... Likewise, with y=1 we get

III: f(c^2 x^2) = c f(x)^2

Let's figure out the constant c. When we set x=1/c in III, we get

c = f(1) = f(c^2 (1/c)^2) =III c f(1/c)^2 => f(1/c) = 1

and setting y=1/c in II gives

f( f(1/c)^2 ) = c^2 f(1/c) = c^2

The left side is f(1^2) = c and thus c=1.

So II and III become a little handier (also replace y by x in II):

II: f( f(x)^2 ) = f(x)

III: f( x^2 ) = f(x)^2

Now plug this into I again:

f( x^2 f(y)^2 ) = f( (x f(y))^2 ) = III f( x f(y) )^2 = I f(x)^2 f(y)

With x = f(y) the last equation becomes

f( f(y)^2 )^2 = f( f(y) )^2 f(y)

Using II on the left side gives

f(y)^2 = f( f(y) )^2 f(y) <=> f(y) = f( f(y) )^2

So for any w in the image of f, i.e. w=f(y) for some y, we have

w = f(w)^2

or

IV: f(w) = sqrt(w)

What does this mean? Well, because f is rational, we can repeatedly apply f to w (i.e. take square roots) and we keep getting rational numbers. The only w you can do this with is w=1. Hence f(y) = 1 for all rational y, so f(2019) = 1.

I am afraid the solution has not quite level-3 length, but this is the shortest I could come up with.

Edited on November 27, 2019, 3:58 pm
  Posted by JLo on 2019-11-27 15:51:00

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