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

Answer: B

Concepts:floor and ceiling functions circle area coordinate geometry

Difficulty rating: 1720

Small Hint:

Recognize $T=t-[t]$ as the fractional part of $t$

Big Hint:

Interpret the equation for $S$ as a disk and identify its center and radius

Solution:

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.

1973 AMC 12 Problem 30

Answer: B

Difficulty rating: 1720

Small Hint:

Big Hint:

Solution:

Problem 30 in Other Years