Apologize problem for Comrade Otto B

Algebra Level 5

Suppose a quadratic function \(ax^2+bx+c\) where \(a,b,c \in \mathbb{R}\) and \(a\neq 0\) , satisfies the following conditions:

1: When \(x\in \mathbb{R}\) , \(f(x-4)=f(2-x)\) and \(f(x)\geq x\).

2: When \(x \in (0,2) \) , \(f(x) \leq \left(\dfrac{x+1}{2}\right)^2\) .

3: The minimum value of \(f(x)\) on \(\mathbb{R}\) is 0.

Find the maximum value of \(m\) (\(m>1\)) such that there exists \(t\in \mathbb{R}\) such that \(f(x+t) \leq x\) holds so long as \(x\in [1,m]\) .


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