# 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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