For each real number x, let [x] be the largest integer not exceeding x (i.e., the integer n such that n≤ x< n+1 ). Which of the following statements is (are) true? I. [x+1]=[x]+1 for all x II. [x+y]=[x]+[y] for all x and y III. [xy]=[x][y] for all x and y

1977 AMC 12 Problem 11

Attempt Problem 11 of the 1977 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1977 AMC 12 solutions, or check the answer key.

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11.

For each real number

x,

let

[x]

be the largest integer not exceeding

x

(i.e., the integer

n

such that

n\le x\lt n+1

). Which of the following statements is (are) true?

\mathrm{I}.

[x+1]=[x]+1

for all

x

\mathrm{II}.

[x+y]=[x]+[y]

for all

x

and

y

\mathrm{III}.

[xy]=[x][y]

for all

x

and

y

none

\mathrm{I}

only

\mathrm{I}

and

\mathrm{II}

only

\mathrm{III}

only

all

Prove the translation statement directly from

n\le x\lt n+1

Test the other two claims using noninteger values between

1

and

2

n\le x\lt n+1,

then

n+1\le x+1\lt n+2,

\mathrm{I}

is true. Taking

x=y=\frac{3}{2}

gives

[x+y]=3\ne2=[x]+[y]

and

[xy]=2\ne1=[x][y].

Thus

\mathrm{II}

and

\mathrm{III}

are false.

Therefore, the correct answer is B.