1973 AMC 12 Problem 30

Question

Let

[t]

denote the greatest integer not exceeding

t,

where

t\ge0,

and

\begin{aligned} S=\{(x,y):{}&(x-T)^2+y^2\\ &\le T^2\},\\ T&=t-[t]. \end{aligned}

Then we have

the point

(0,0)

does not belong to

S

for any

t

0\le\operatorname{Area} S\le\pi

for all

t

S

is contained in the first quadrant for all

t\ge5

the center of

S

for any

t

is on the line

y=x

none of the other statements is true

Recognize

T=t-[t]

as the fractional part of

t

Interpret the equation for

S

as a disk and identify its center and radius

The fractional part satisfies

0\le T\lt1.

The set

S

is the closed disk centered at

(T,0)

with radius

T.

Its area is

\pi T^2,

so

0\le\operatorname{Area}S\lt\pi,

which in particular gives

0\le\operatorname{Area}S\le\pi.

The origin lies on every such disk, the disk extends below the

x

-axis when

T\gt0,

and its center is generally not on

y=x.

Therefore, the correct answer is B.

Problem 30 in other years