1986 AMC 12 Problem 7

Attempt Problem 7 of the 1986 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 1986 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The sum of the greatest integer less than or equal to

x

and the least integer greater than or equal to

x

5.

The solution set for

x

\frac52

\{x\mid 2\le x\le3\}

\{x\mid 2\le x\lt3\}

\{x\mid 2\lt x\le3\}

\{x\mid 2\lt x\lt3\}

Treat integer and noninteger values of

x

separately

For noninteger

x,

the ceiling is one more than the floor

x

is an integer, then

\lfloor x\rfloor+\lceil x\rceil=2x,

which cannot equal

5.

x

is not an integer, then

\lfloor x\rfloor+\lceil x\rceil=2\lfloor x\rfloor+1.

This equals

5

exactly when

\lfloor x\rfloor=2,

2\lt x\lt3.

Thus the correct answer is E.

Problem 7 in other years