Track / Stage 4 / 51 of 340 #311 of 2000

Problem 311

AMC 12 late, AIME early

Number theory Difficulty 4.6 Find the answer

33 -21 Let $[x]$ denote the greatest integer not exceeding $x$ . For any real number $x$ , which of the following statements is correct?
(A) $[x]=|x|$ .
(B) $[x] \geqslant \sqrt{x^{2}}$ .
(C) $[x]>-x$
(D) $[x]>x-1$ .
(China Beijing Junior High School Mathematics Competition, 1985)

Your answer

Official solution

[Solution] The range of $[x]$ takes all integers. $[-2]=-2$ , while $|-2|=2$ , $\sqrt{(-2)^{2}}=2$ . We can eliminate $(A),(B)$ ,
also $-(-2)=2$ , so $[-2]<-(-2)$ , we can eliminate $(C)$ .
Therefore, the answer is $(D)$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.