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Am I Missing Something?

This is supposed to be one of the easiest problems from BdMO 2014. But I can't seem to find a rigorous way of solving this problem. Am I missing something?

\(x\), \(a\), \(b\) are positive numbers such that if \(a>b\), \(f(a)>f(b)\) and \(f(f(x))=x^2+2\). What is the value of \(f(3)\)?

This also happens to be my first note!

Thanks!

Note by Siam Habib
3 years, 3 months ago

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\(f(f(1))=3\) => \(f(1)=\) \(1\) or \(2\) or \(3\). Easy investigation gives

\(f(1) =2\).

\(f(f(1))= f(2)= 3\).

\(f(f(2))=f(3)= 2^2 +2 =6\). Fatin Farhan · 3 years, 3 months ago

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@Fatin Farhan I don't understand with your solving.. f(1)=f(2)=1/2/3=3/2=3.. am i wrong to understand?? Hafizh Ahsan Permana · 3 years, 3 months ago

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@Hafizh Ahsan Permana Sorry. I was saying \(f(1)=\) 1 or 2 or 3. Fatin Farhan · 3 years, 3 months ago

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@Fatin Farhan And how you conclude that f(1)=1or2or3?? and than continue to f(1)=2? sorry i just learned this.. Hafizh Ahsan Permana · 3 years, 3 months ago

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@Fatin Farhan I think you ought to prove \(f(n)>n\) for this Alessio Di Lorenzo · 3 years, 3 months ago

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@Alessio Di Lorenzo If \(f(1)= 1\) then \(f(f(1))= f(1) = 1\). But \(f(f(1))=3\) If \(f(1)=3\) then \(f(f(1))=f(3) = 3\) So, \(f(f(3)) = f(3)= 3\) But \(f(f(3))=11\)

So, \(f(1)=2\). Fatin Farhan · 3 years, 3 months ago

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Congrats on your first note Siam! It was a good one. Peter Taylor Staff · 3 years, 3 months ago

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