HMMT 二月 2018 · ALGNT 赛 · 第 1 题

HMMT February 2018 — ALGNT Round — Problem 1

For some real number c , the graphs of the equation y = | x − 20 | + | x + 18 | and the line y = x + c intersect at exactly one point. What is c ?

解析

For some real number c , the graphs of the equation y = | x − 20 | + | x + 18 | and the line y = x + c intersect at exactly one point. What is c ? Proposed by: Henrik Boecken Answer: 18 We want to know the value of c so that the graph | x − 20 | + | x + 18 | − x = c has one solution. The graph of the function | x − 20 | + | x + 18 | − x consists of an infinite section of slope − 3 for x ∈ ( −∞ , − 18], then a finite section of slope − 1 for x ∈ [ − 18 , 20], then an infinite section of slope 1 for x ∈ [20 , ∞ ). Notice that this graph is strictly decreasing on ( −∞ , 20] and strictly increasing on [20 , ∞ ). Therefore any horizontal line will intersect this graph 0 or 2 times, except the one that passes through the “vertex” (20 , | 20 − 20 | + | 20 + 18 | − 20) = (20 , 18), giving a value of c = 18.