Question

考虑满足下列条件的函数 $f$ ：对所有实数 $x$ 和 $y$ ，都有。在这些函数中，再要求 $f(300) = f(900)$ 。求的最大可能值。 $|f(x)-f(y)|\leq \dfrac{1}{2}|x-y|$ $f(f(800))-f(f(400))?$

Consider functions $f$ that satisfy $|f(x)-f(y)|\leq \dfrac{1}{2}|x-y|$ for all real numbers $x$ and $y.$ Of all such functions that also satisfy the equation $f(300) = f(900),$ what is the greatest possible value of $f(f(800))-f(f(400))?$

$25$

$50$

$100$

$150$

$200$

答案：B

知识点：函数不等式极端原理

难度评级：2390

解答：

由收缩条件可得同理， $\begin{aligned} &|f(f(400))-f(f(300))|\\ &\quad\le\frac12|f(400)-f(300)|\\ &\quad\le25, \end{aligned}$ $|f(f(800))-f(f(900))|\le25.$

因为 $M=f(f(300))=f(f(900))$ ，由三角不等式， $\begin{aligned} &|f(f(800))-f(f(400))|\\ &\quad\le|f(f(800))-M|\\ &\qquad+|M-f(f(400))|\\ &\quad\le50. \end{aligned}$

为使上界能够达到，定义 $f$ 为依次经过下列各点的分段线性函数：并规定当 $x\leq300$ 或 $x\geq900$ 时 $f(x)=600$ 。每一段斜率的绝对值都不超过 $\dfrac12$ ，因此满足收缩条件。特别地， $f(300)=f(900)=600$ 、 $f(400)=550$ 、 $f(800)=650$ 。所以 $f(f(400))=f(550)=575$ ，而 $f(f(800))=f(650)=625$ ，两者之差为 $50$ 。 $\begin{gathered}(300,600),(400,550),(550,575),\\ (650,625),(800,650),(900,600).\end{gathered}$

上述分段线性函数满足所需条件并达到该上界，因此这个上界就是最大值。所以正确答案是 B。

Applying the contraction inequality twice gives $\begin{aligned} &|f(f(400))-f(f(300))|\\ &\quad\le\frac12|f(400)-f(300)|\\ &\quad\le25, \end{aligned}$ and similarly $|f(f(800))-f(f(900))|\le25.$

Set $M=f(f(300))=f(f(900)).$ The triangle inequality now yields $\begin{aligned} &|f(f(800))-f(f(400))|\\ &\quad\le|f(f(800))-M|\\ &\qquad+|M-f(f(400))|\\ &\quad\le50. \end{aligned}$

To attain the bound, define $f$ by linear interpolation through the points $\begin{gathered}(300,600),(400,550),(550,575),\\ (650,625),(800,650),(900,600).\end{gathered}$ and set $f(x)=600$ for $x\leq300$ or $x\geq900.$ Every segment has slope with absolute value at most $\dfrac12,$ so the contraction condition holds. In particular, $f(300)=f(900)=600,$ $f(400)=550,$ and $f(800)=650.$ Hence $f(f(400))=f(550)=575,$ while $f(f(800))=f(650)=625,$ giving the difference $50.$

Thus, the answer is B .

2022 AMC 10B 第 24 题

答案：B

难度评级：2390

解答：

其他年份的第 24 题