sqing wrote: Let $f(x)=a(3a+2c)x^2-2b(2a+c)x+b^2+(c+a)^2$ $(a,b,c\in R, a(3a+2c)\neq 0).$ If $$f(x)\leq 1$$for any real $x$, find the maximum of $|ab|.$

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sqing wrote: sqing wrote: Let $f(x)=a(3a+2c)x^2-2b(2a+c)x+b^2+(c+a)^2$ $(a,b,c\in R, a(3a+2c)\neq 0).$ If $$f(x)\leq 1$$for any real $x$, find the maximum of $|ab|.$ What book is?

From the water mark, it's from a chinese tutoring institution website that posts these contests and their solutions

https://artofproblemsolving.com/community/c1252089_2020_south_east_mathematical_olympiad

you can try $x=a+c,y=2a+c$ and max is $\frac{3\sqrt{3}}{8}$