Functions, RMO

It is given that ${\color{#3D99F6}f(x+y)} = {\color{#D61F06}f(xy)}$ and that $\boxed{f(8)}=\boxed 9$ . Then

$\begin{array} {rrc} {\color{#3D99F6}4+4=8}, & \color{#D61F06}4\times 4=16 & \implies {\color{#3D99F6}\boxed{f(8)}} = {\color{#D61F06}f(16)} = \boxed 9 \\ {\color{#3D99F6}8+8=16}, & \color{#D61F06}8\times 8=64 & \implies {\color{#3D99F6}\boxed{f(16)}} = {\color{#D61F06}f(64)} = \boxed 9 \\ {\color{#3D99F6}4+16=20}, & \color{#D61F06}4\times 16=64 & \implies {\color{#3D99F6}f(20)} = {\color{#D61F06}\boxed{f(64)}} = \boxed 9 \\ {\color{#3D99F6}4+5=9}, & \color{#D61F06}4\times 5=20 & \implies {\color{#3D99F6}f(9)} = {\color{#D61F06}\boxed{f(20)}} = \boxed 9 \end{array}$

Therefore, $f(9) = \boxed 9$ .

Here, f(x+y)=f(xy) For, x=y=4, f(4+4)=f(4.4) =>f(8)=f(16)=9 now, f(9)=f(4+5) =f(4.5) =f(20) =f(16 + 4) =f(16.4) =f(8.8) =f(8+8) =f(16)=9 therefore, f(9)=9. For more or for sending problems contact me at: sambit.mishra.2003@gmail.com