Around a circular lagoon, whose radius measures 1 km, there is a bicycle path. Due to frequent bicycle thefts, the city council plans to allocate police officers in strategic positions to patrol this bicycle path, in order to make it fully protected. A point on the bicycle path is considered protected if there is at least one police officer at most 200 m away from that point, positioned on the bicycle path. The figure illustrates a point $P$ on the bicycle path, which will be protected if there is at least one police officer positioned on the dark gray region.
Disregard the width of the bicycle path and use 3 as an approximation for $\pi$.
Under these conditions, the minimum number of police officers to be allocated along this bicycle path to make it protected is
(A) 4.
(B) 8.
(C) 15.
(D) 30.
(E) 60.

Consider the quadratic equation $x ^ { 2 } + b x + c = 0$, where $b$ and $c$ are chosen randomly from the interval $[ 0,1 ]$ with the probability uniformly distributed over all pairs $( b , c )$. Let $p ( b ) =$ the probability that the given equation has a real solution for given (fixed) value of $b$. Answer the following questions by filling in the blanks. a) The equation $x ^ { 2 } + b x + c = 0$ has a real solution if and only if $b ^ { 2 } - 4 c$ is
Answer: $\_\_\_\_$ b) The value of $p \left( \frac { 1 } { 2 } \right)$, i.e., the probability that $x ^ { 2 } + \frac { x } { 2 } + c = 0$ has a real solution is
Answer: $\_\_\_\_$ c) As a function of $b$, is $p ( b )$ increasing, decreasing or constant?
Answer: $\_\_\_\_$ d) As $b$ and $c$ both vary, what is the probability that $x ^ { 2 } + b x + c = 0$ has a real solution?
Answer: $\_\_\_\_$

7. Two numbers $x$ and $y$ are randomly chosen from the interval $[0,1]$. Let $p_1$ be the probability of the event ``$x + y \geq \frac{1}{2}$'', $p_2$ be the probability of the event ``$|x - y| \leq \frac{1}{2}$'', and $p_3$ be the probability of the event ``$xy \leq \frac{1}{2}$''. Then
A. $p_1 < p_2 < p_3$
B. $p_2 < p_3 < p_1$
C. $p_3 < p_1 < p_2$
D. $p_3 < p_2 < p_1$

7. As shown in the figure, the angle between the oblique line segment AB and plane $\alpha$ is $60 ^ { \circ }$ , B is the foot of the oblique line, and the moving point P on plane $\alpha$ satisfies $\angle \mathrm { PAB } = 30 ^ { \circ }$ . Then the locus of point P is
A. a line
B. a parabola
C. an ellipse
D. one branch of a hyperbola

8. Two numbers $\mathrm { x } , \mathrm { y }$ are randomly selected from the interval $[ 0,1 ]$. Let $p _ { 1 }$ be the probability of the event ``$x + y \leq \frac { 1 } { 2 }$'', and $P _ { 2 }$ be the probability of the event ``$x y \leq \frac { 1 } { 2 }$'', then
A. $p _ { 1 } < p _ { 2 } < \frac { 1 } { 2 }$
B. $p _ { 2 } < \frac { 1 } { 2 } < p _ { 1 }$
C. $\frac { 1 } { 2 } < p _ { 2 } < p _ { 1 }$
D. $p _ { 1 } < \frac { 1 } { 2 } < p _ { 2 }$

11. Let the complex number $z = ( x - 1 ) + y i ( x , y \in R )$. If $| z | \leq 1$, the probability that $y \geq x$ is
A. $\frac { 3 } { 4 } + \frac { 1 } { 2 \pi }$
B. $\frac { 1 } { 4 } - \frac { 1 } { 2 \pi }$
C. $\frac { 1 } { 2 } - \frac { 1 } { \pi }$
D. $\frac { 1 } { 2 } + \frac { 1 } { \pi }$

13. In the figure, the coordinates of point $A$ are $( 1,0 )$, the coordinates of point $C$ are $( 2,4 )$, and the function $f ( x ) = x ^ { 2 }$. If a point is randomly selected inside rectangle $A B C D$, then the probability that this point is in the shaded region equals $\_\_\_\_$.

As shown in the figure, the pattern inside square $ABCD$ comes from the ancient Chinese Tai Chi diagram with central symmetry about the center. If a point is randomly selected inside the square, the probability that this point is in the shaded region is
A. $\frac { 1 } { 4 }$
B. $\frac { \pi } { 8 }$
C. $\frac { 1 } { 2 }$
D. $\frac { \pi } { 4 }$

As shown in the figure, the pattern inside square $ABCD$ comes from the ancient Chinese Tai Chi diagram. The black and white parts in the inscribed circle of the square are symmetric with respect to the center of the square. If a point is randomly selected from the square, the probability that it falls in the black part is
A. $\frac{1}{4}$
B. $\frac{\pi}{8}$
C. $\frac{1}{2}$
D. $\frac{\pi}{4}$

Let $O$ be the origin of the coordinate system. A point is randomly selected in the region $\left\{ ( x , y ) \mid 1 \leqslant x ^ { 2 } + y ^ { 2 } \leqslant 4 \right\}$. The probability that the inclination angle of line $OA$ does not exceed $\frac { \pi } { 4 }$ is
A. $\frac { 1 } { 8 }$
B. $\frac { 1 } { 6 }$
C. $\frac { 1 } { 4 }$
D. $\frac { 1 } { 2 }$

Let $X = \left\{ ( x , y ) \in \mathbb { Z } \times \mathbb { Z } : \frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 20 } < 1 \right.$ and $\left. y ^ { 2 } < 5 x \right\}$. Three distinct points $P , Q$ and $R$ are randomly chosen from $X$. Then the probability that $P , Q$ and $R$ form a triangle whose area is a positive integer, is
(A) $\frac { 71 } { 220 }$
(B) $\frac { 73 } { 220 }$
(C) $\frac { 79 } { 220 }$
(D) $\frac { 83 } { 220 }$

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $( 0,0 ) , ( 0,41 )$ and $( 41,0 )$ is
(1) 780
(2) 901
(3) 861
(4) 820

Let $S$ be the set of all triangles in the $xy$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is:
(1) 36
(2) 32
(3) 9
(4) 18

If a point $A ( x , y )$ lies in the region bounded by the $y$-axis, straight lines $2 y + x = 6$ and $5 x - 6 y = 30$, then the probability that $y < 1$ is
(1) $\frac { 1 } { 6 }$
(2) $\frac { 5 } { 6 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 6 } { 7 }$

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 \hspace{2cm}. (expressed as a fraction in lowest terms)

The Earth is a sphere. Five points $A , B , C , D , E$ on the Earth's surface have the following latitude and longitude coordinates, for example, point $A$ is located at longitude 0 degrees, north latitude 60 degrees.

Location	Longitude 0 degrees	Longitude 180 degrees
North latitude 60 degrees	$A$	$B$
North latitude 30 degrees	$C$	$D$
Latitude 0 degrees	$E$

A great circle is the circle formed by the intersection of a plane passing through the center of the sphere with the sphere's surface. The shorter arc formed by two distinct points on the sphere on a great circle is the shortest path. Based on the above, select the correct options.
(1) ``The shortest path length from the North Pole to $A$'' equals ``the shortest path length from the North Pole to $B$''
(2) ``The shortest path length from $A$ to $B$'' equals ``the shortest path length from $C$ to $D$''
(3) The shortest path from $A$ to $E$ must pass through $C$
(4) The shortest path from $C$ to $D$ must pass through the North Pole
(5) The ratio of ``the shortest path length from $E$ to the North Pole'' to ``the shortest path length from $C$ to $D$'' is $2 : 3$