Understand the problem

Find, with justification, all positive real numbers   $a,b,c$   satisfying the system of equations:    \[a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b.\]
Source of the problem
SMO (senior)-2014 stage 2 problem 2

Topic
Number Theory
Difficulty Level
Easy
Suggested Book
Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

Given all three relations are cyclic and symmetric . So without loss of generality it can be assumed that \( a \geq b \geq c >0 \) .     Then proceed .               [ Note \( (0, 0, 0) \) can’t be a solution since \( a , b , c \) are positive reals .]
So \( a \sqrt b = a + c \Rightarrow a(\sqrt b – 1) = c \ [and \ we \ have \ a \geq c]   \Rightarrow ( \sqrt b – 1 ) \leq 1 \Rightarrow b \leq 4 \)
Similarly \( b \sqrt c = b + a \Rightarrow b(\sqrt c – 1) = a \  [and \ we \  have \ a \geq b ] \Rightarrow \sqrt c – 1 \geq 1 \Rightarrow c \geq 4 \)

Till now we have \( b \leq 4 \ and  \  c \geq 4 \) , but we assumed that \( b \geq c \) . So it is clear that \( b = c =4 \)  \( \Rightarrow a = 4 \) also. So the only triplet \( (a , b , c)\) is \( (4,4 ,4) \) .  

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