Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: A ship travels from point A to point B along a semicircular path, centered at Island X. Then it travels along a straight path from B to C. Which of these graphs best shows the ship's distance from Island X as it moves along its course?

<image1>
Choices:
A. <image2>
B. <image3>
C. <image4>
D. <image5>
E. <image6>
Answer: B
Problem: <image>Question: In the figure, the area of square WXYZ is $25 \text{cm}^2$. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In $\Delta ABC$, $AB = AC$, and when $\Delta ABC$ is folded over side BC, point A coincides with O, the center of square WXYZ. What is the area of $\Delta ABC$, in square centimeters?

<image1>
Choices:
A. $\frac{15}4$
B. $\frac{21}4$
C. $\frac{27}4$
D. $\frac{21}2$
E. $\frac{27}2$
Answer: C
Problem: <image>Question: What is the area enclosed by the geoboard quadrilateral below?

<image1>
Choices:
A. $15$
B. $18\frac{1}{2}$
C. $22\frac{1}{2}$
D. $27$
E. $41$
Answer: C
Problem: <image>Question: Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?

<image1>
Answer: 11
Problem: <image>Question: Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?

<image1>
Choices:
A. $\frac{1}{4}$
B. $\frac{1}{3}$
C. $\frac{1}{2}$
D. $\frac{2}{3}$
E. $\frac{3}{4}$
Answer: D
Problem: <image>Question: Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home?

<image1>
Choices:
A. <image2>
B. <image3>
C. <image4>
D. <image5>
E. <image6>
Answer: D
Problem: <image>Question: In the figure, $ABCD$ is a rectangle and $EFGH$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $HE$ and $FG$?

<image1>
Answer: 7.6
Problem: <image>Question: Two $4\times 4$ squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?

<image1>
Choices:
A. $16-4\pi$
B. $16-2\pi$
C. $28-4\pi$
D. $28-2\pi$
E. $32-2\pi$
Answer: D
Problem: <image>Question: What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $ \overline{BD}$ of square $ ABCD$?
<image1>
Answer: 4
Problem: <image>Question: In quadrilateral $ ABCD$, sides $ \overline{AB}$ and $ \overline{BC}$ both have length 10, sides $ \overline{CD}$ and $ \overline{DA}$ both have length 17, and the measure of angle $ ADC$ is $ 60^\circ$. What is the length of diagonal $ \overline{AC}$?
<image1>
Answer: 17
Problem: <image>Question: The area of polygon $ ABCDEF$ is 52 with $ AB=8$, $ BC=9$ and $ FA=5$. What is $ DE+EF$?
<image1>
Answer: 9
Problem: <image>Question: The results of a cross-country team's training run are graphed below. Which student has the greatest average speed?
<image1>
Choices:
A. $\text{Angela}$
B. $\text{Briana}$
C. $\text{Carla}$
D. $\text{Debra}$
E. $\text{Evelyn}$
Answer: E
Problem: <image>Question: What is the perimeter of trapezoid $ ABCD$?

<image1>
Answer: 180
Problem: <image>Question: How many distinct triangles can be drawn using three of the dots below as vertices?

<image1>
Answer: 18
Problem: <image>Question: Isosceles right triangle $ ABC$ encloses a semicircle of area $ 2\pi$. The circle has its center $ O$ on hypotenuse $ \overline{AB}$ and is tangent to sides $ \overline{AC}$ and $ \overline{BC}$. What is the area of triangle $ ABC$?

<image1>
Choices:
A. $6$
B. $8$
C. $3\pi$
D. $10$
E. $4\pi$
Answer: B
Problem: <image>Question: A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

<image1>
Choices:
A. $\frac{2}{\sqrt{\pi}}$
B. $\frac{1+\sqrt{2}}{2}$
C. $\frac{3}{2}$
D. $\sqrt{3}$
E. $\sqrt{\pi}$
Answer: A
Problem: <image>Question: Initially, a spinner points west. Chenille moves it clockwise $ 2 \frac{1}{4}$ revolutions and then counterclockwise $ 3 \frac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?

<image1>
Choices:
A. $\text{north}$
B. $\text{east}$
C. $\text{south}$
D. $\text{west}$
E. $\text{northwest}$
Answer: B
Problem: <image>Question: Points $ A, B, C$ and $ D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?

<image1>
Answer: 30
Problem: <image>Question: The letter T is formed by placing two $ 2\times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches?

<image1>
Answer: 20
Problem: <image>Question: Jorge's teacher asks him to plot all the ordered pairs $ (w, l)$ of positive integers for which $ w$ is the width and $ l$ is the length of a rectangle with area 12. What should his graph look like?
Choices:
A. <image1>
B. <image2>
C. <image3>
D. <image4>
E. <image5>
Answer: A
Problem: <image>Question: Jeff rotates spinners $ P$, $ Q$ and $ R$ and adds the resulting numbers. What is the probability that his sum is an odd number?

<image1>
Choices:
A. $\frac{1}{4}$
B. $\frac{1}{3}$
C. $\frac{1}{2}$
D. $\frac{2}{3}$
E. $\frac{3}{4}$
Answer: B
Problem: <image>Question: Triangle $ ABC$ is an isosceles triangle with $ \overline{AB} =\overline{BC}$. Point $ D$ is the midpoint of both $ \overline{BC}$ and $ \overline{AE}$, and $ \overline{CE}$ is 11 units long. Triangle $ ABD$ is congruent to triangle $ ECD$. What is the length of $ \overline{BD}$?

<image1>
Answer: 5.5
Problem: <image>Question: Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?


<image1>
Answer: 26
Problem: <image>Question: Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?

<image1>
Answer: 14
Problem: <image>Question: Six-hundred fifty students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?

<image1>
Choices:
A. $\frac{2}{5}$
B. $\frac{1}{2}$
C. $\frac{5}{4}$
D. $\frac{5}{3}$
E. $\frac{5}{2}$
Answer: E
Problem: <image>Question: In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD=AB=3$, and $DC=6$. In addition, E is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\Delta BEC$.

<image1>
Answer: 4.5
Problem: <image>Question: Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$?

<image1>
Choices:
A. $I$
B. $II$
C. $III$
D. $IV$
E. $\text{ cannot be determined}$
Answer: D
Problem: <image>Question: A unit hexagon is composed of a regular haxagon of side length 1 and its equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

<image1>
Choices:
A. 1:1
B. 6:5
C. 3:2
D. 2:1
E. 3:1
Answer: A
Problem: <image>Question: Sets A and B, shown in the venn diagram, have the same number of elements. Thier union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.

<image1>
Choices:
A. $03$
B. $1006$
C. $504$
D. $1507$
E. $1510$
Answer: C
Problem: <image>Question: Amanda Reckonwith draws five circles with radii 1, 2, 3, 4 and 5. Then for each circle she plots the point (C; A), where C is its circumference and A is its area. Which of the following could be her graph?
Choices:
A. <image1>
B. <image2>
C. <image3>
D. <image4>
E. <image5>
Answer: A
Problem: <image>Question: What is the area of the shaded pinwheel shown in the $5\times 5$ grid?

<image1>
Choices:
A. $4$
B. $6$
C. $8$
D. $10$
E. $12$
Answer: B
Problem: <image>Question: On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has radius 3. Three radii divide each circle into the three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to to the area of the region. What two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?

<image1>
Choices:
A. $\frac{17}{36}$
B. $\frac{35}{72}$
C. $\frac{1}{2}$
D. $\frac{37}{72}$
E. $\frac{19}{36}$
Answer: B
Problem: <image>Question: In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids?
<image1>
Answer: 5
Problem: <image>Question: In the figure, what is the ratio of the area of the gray squares to the area of the white squares?
<image1>
Choices:
A. 3:10
B. 3:8
C. 3:7
D. 3:5
E. 1:1
Answer: D
Problem: <image>Question: Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?
<image1>
Answer: 80
Problem: <image>Question: Three $\text{A's}$, three $\text{B's}$, and three $\text{C's}$ are placed in the nine spaces so that each row and column contain one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible?
<image1>
Answer: 4
Problem: <image>Question: A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?

<image1>
Choices:
A. 1 : 6
B. 7 : 36
C. 1 : 5
D. 7 : 30
E. 6 : 25
Answer: D
Problem: <image>Question: Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run?
<image1>
Choices:
A. $10\pi+20$
B. $10\pi+30$
C. $10\pi+40$
D. $20\pi+20$
E. $20\pi+40$
Answer: E
Problem: <image>Question: Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?
<image1>
Choices:
A. $\frac{1}{4}$
B. $\frac{2}{7}$
C. $\frac{4}{11}$
D. $\frac{1}{2}$
E. $\frac{4}{7}$
Answer: B
Problem: <image>Question: Jerry cuts a wedge from a $6$-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
<image1>
Choices:
A. 48
B. 75
C. 151
D. 192
E. 603
Answer: C
Problem: <image>Question: In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\triangle BFD$ to the area of square $ABCE$?
<image1>
Choices:
A. $\frac{1}{6}$
B. $\frac{2}{9}$
C. $\frac{5}{18}$
D. $\frac{1}{3}$
E. $\frac{7}{20}$
Answer: C
Problem: <image>Question: Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black?

<image1>
Choices:
A. 42
B. 44
C. 45
D. 46
E. 48
Answer: A
Problem: <image>Question: The graph shows the constant rate at which Suzanna rides her bike.  If she rides a total of a half an hour at the same speed, how many miles would she have ridden?
<image1>
Answer: 6
Problem: <image>Question: The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?
<image1>

<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad.  Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles.  The width of the railroad track can be ignored.  How many square miles are in the plot of land ACD?
<image1>
Answer: 4.5
Problem: <image>Question: Construct a square on one side of an equilateral triangle. One on non-adjacent side of the square, construct a regular pentagon, as shown. One a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?

<image1>
Answer: 23
Problem: <image>Question: On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does  not   touch the outer edge of the board?
<image1>
Choices:
A. $\frac{1}{16}$
B. $\frac{7}{16}$
C. $\frac{1}{2}$
D. $\frac{9}{16}$
E. $\frac{49}{64}$
Answer: D
Problem: <image>Question: The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?
<image1>
<image2>
Choices:
A. $\frac{1}{2}$
B. $\frac{2}{3}$
C. $\frac{3}{4}$
D. $\frac{7}{9}$
E. $\frac{5}{6}$
Answer: D
Problem: <image>Question: The diagram represents a $ 7$-foot-by-$ 7$-foot floor that is tiled with $ 1$-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a $ 15$-foot-by-$ 15$-foot floor is to be tiled in the same manner, how many white tiles will be needed?
<image1>
Answer: 64
Problem: <image>Question: How many non-congruent triangles have vertices at three of the eight points in the array shown below?
<image1>
Answer: 8
Problem: <image>Question: A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cub is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
<image1>

<image2>
Choices:
A. 6$
B. $7$
C. $\frac{419}{51}$
D. $\frac{158}{17}$
E. $11$
Answer: E
Problem: <image>Question: The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?

<image1>
Answer: 70
Problem: <image>Question: The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\frac{XQ}{QY}$?

<image1>
Choices:
A. $\frac{2}{5}$
B. $\frac{1}{2}$
C. $\frac{3}{5}$
D. $\frac{2}{3}$
E. $\frac{3}{4}$
Answer: D
Problem: <image>Question: A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.

<image1>
Choices:
A. $2:3$
B. $3:2$
C. $6:\pi$
D. $9: \pi$
E. $30 : \pi$
Answer: C
Problem: <image>Question: The two circles pictured have the same center $C$. Chord $\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\overline{AD}$ has length $16$. What is the area between the two circles?

<image1>
Choices:
A. $36 \pi$
B. $49 \pi$
C. $64 \pi$
D. $81 \pi$
E. $100 \pi$
Answer: C
Problem: <image>Question: Semicircles $POQ$ and $ROS$ pass through the center of circle $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$?
<image1>
Choices:
A. $\frac{\sqrt{2}}{4}$
B. $\frac{1}{2}$
C. $\frac{2}{\pi}$
D. $\frac{2}{3}$
E. $\frac{\sqrt{2}}{2}$
Answer: B
Problem: <image>Question: Extend the square pattern of $8$ black and $17$ white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
<image1>
Choices:
A. 8:17
B. 25:49
C. 36:25
D. 32:17
E. 36:17
Answer: D
Problem: <image>Question: Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?


<image1>
Choices:
A. $12\frac{1}{2}$
B. $20$
C. $25$
D. $33 \frac{1}{3}$
E. $37\frac{1}{2}$
Answer: C
Problem: <image>Question: Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?

<image1>
Answer: 5
Problem: <image>Question: The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?

<image1>
Answer: 6
Problem: <image>Question: Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded?

<image1>
Answer: 20
Problem: <image>Question: How many rectangles are in this figure?

<image1>
Answer: 11
Problem: <image>Question: Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid?

<image1>
Answer: 750
Problem: <image>Question: A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?

<image1>
Choices:
A. $\frac{1}2$
B. $1$
C. $\frac{3}2$
D. $2$
E. $\frac{5}2$
Answer: A
Problem: <image>Question: In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is $X$, in centimeters?

<image1>
Answer: 5
Problem: <image>Question: A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?

<image1>
Choices:
A. $\frac{4-\pi}\pi$
B. $\frac{1}{\pi}$
C. $\frac{\sqrt{2}}{\pi}$
D. $\frac{\pi-1}\pi$
E. $\frac{3}{\pi}$
Answer: A
Problem: <image>Question: A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$ ?

<image1>
Choices:
A. $\frac{1}{5}$
B. $\frac{2}{5}$
C. $\frac{1}{2}$
D. $1$
E. $4$
Answer: C
Problem: <image>Question: The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row?

<image1>
Answer: 4
Problem: <image>Question: Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
<image1>
Answer: 280
Problem: <image>Question: Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?

<image1>
Answer: 3932
Problem: <image>Question: Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$?
<image1>
Answer: 7.5
Problem: <image>Question: Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ ad $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares?

<image1>
Choices:
A. $\frac{1}{4}$
B. $\frac{7}{24}$
C. $\frac{1}{3}$
D. $\frac{3}{8}$
E. $\frac{5}{12}$
Answer: C
Problem: <image>Question: A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
<image1>
Choices:
A. $238\pi$
B. $240\pi$
C. $260\pi$
D. $280\pi$
E. $500\pi$
Answer: A
Problem: <image>Question: In $\bigtriangleup ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$?

<image1>
Answer: 140
Problem: <image>Question: Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?

<image1>
Answer: 13
Problem: <image>Question: The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?

<image1>
Answer: 90
Problem: <image>Question: Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

<image1>
Answer: 4.0
Problem: <image>Question: A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch?

Note: $1$ mile= $5280$ feet

<image1>
Choices:
A. $\frac{\pi}{11}$
B. $\frac{\pi}{10}$
C. $\frac{\pi}{5}$
D. $\frac{2\pi}{5}$
E. $\frac{2\pi}{3}$
Answer: B
Problem: <image>Question: Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}$. What fraction of the area of the octagon is shaded?
<image1>
Choices:
A. $\frac{11}{32}$
B. $\frac{3}{8}$
C. $\frac{13}{32}$
D. $\frac{7}{16}$
E. $\frac{15}{32}$
Answer: D
Problem: <image>Question: How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?
<image1>
Answer: 18
Problem: <image>Question: An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
<image1>
Answer: 31
Problem: <image>Question: A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
<image1>
Choices:
A. $\frac{1}{6}$
B. $\frac{1}{5}$
C. $\frac{1}{4}$
D. $\frac{1}{3}$
E. $\frac{1}{2}$
Answer: A
Problem: <image>Question: In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
<image1>
Choices:
A. $6\sqrt{2}$
B. $9$
C. $12$
D. $9\sqrt{2}$
E. $32$
Answer: C
Problem: <image>Question: One-inch squares are cut from the corners of this 5 inch square.  What is the area in square inches of the largest square that can be fitted into the remaining space?
<image1>
Choices:
A. $9$
B. $12\frac{1}{2}$
C. $15$
D. $15\frac{1}{2}$
E. $17$
Answer: C