1982 AMC 12 Problem 15

Attempt Problem 15 of the 1982 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 1982 AMC 12 solutions, or check the answer key.

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15.

Let

\lfloor z\rfloor

denote the greatest integer not exceeding

z.

Let

x

and

y

satisfy the simultaneous equations

\begin{aligned} y&=2\lfloor x\rfloor+3,\\ y&=3\lfloor x-2\rfloor+5. \end{aligned}

x

is not an integer, then

x+y

an integer

between

4

and

5

between

-4

and

4

between

15

and

16

16.5

For nonintegral

x,

relate

\lfloor x-2\rfloor

\lfloor x\rfloor

Let

n=\lfloor x\rfloor

and equate the two formulas for

y

For nonintegral

x,

let

n=\lfloor x\rfloor;

then

\lfloor x-2\rfloor=n-2.

Thus

2n+3=3(n-2)+5,

n=4

and

y=11.

Since

4\lt x\lt5,

we have

15\lt x+y\lt16.

Therefore, the correct answer is D.

Problem 15 in other years