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Quadratic Function

Source: 2002 National High School Mathematics League, Exam One, Problem 15

March 16, 2020
function

Problem Statement

Quadratic function f(x)=ax2+bx+cf(x)=ax^2+bx+c satisfies that: (1)∀x∈R,f(x−4)=f(2−x),f(x)≥x(1)\forall x\in\mathbb{R},f(x-4)=f(2-x),f(x)\geq x; (2)∀x∈(0,2),f(x)≤(x+12)2(2)\forall x\in(0,2),f(x)\leq\left(\frac{x+1}{2}\right)^2; (3)min⁡x∈Rf(x)=0(3)\min\limits_{x\in\mathbb{R}}f(x)=0 Find the maximum of m(m>1)m(m>1), satisfying: There exists t∈Rt\in\mathbb{R}, as long as x∈[1,m]x\in[1,m], then f(x+t)≤xf(x+t)\leq x.
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