On the coordinate plane, within the square (including boundary) with vertices $O(0,0), A(0,1), B(1,1), C(1,0)$, let $R$ be the region formed by points $P(x, y)$ satisfying the following condition: the set of all points at distance $|x - y|$ from point $P(x, y)$ is completely contained within the square $OABC$ (including boundary). The area of region $R$ is . (expressed as a fraction in lowest terms)
A spiral line is drawn as shown. This spiral pattern continues indefinitely. Which one of the following points is not on the spiral line? A $( 99,100 )$ B $( 99 , - 100 )$ C $( - 99,100 )$ D $( - 99 , - 100 )$ E $( 100,99 )$ F $( 100 , - 99 )$ G $( - 100,99 )$ H ( $- 100 , - 99$ )
Circle $C _ { 1 }$ is defined as $x ^ { 2 } + y ^ { 2 } = 25$ A second circle $C _ { 2 }$ has radius 4 and centre $( a , b )$ where $$- 2 \leq a \leq 2 \text { and } - 3 \leq b \leq 3$$ If the centre of $C _ { 2 }$ is equally likely to be located anywhere within the given range, what is the probability that $C _ { 2 }$ intersects $C _ { 1 }$ ?