id stringlengths 1 4 | question stringlengths 31 709 | options listlengths 6 6 | image_path stringlengths 12 15 | image imagewidth (px) 56 2.52k | answer stringclasses 5
values | solution stringclasses 259
values | level int64 1 5 | subject stringclasses 16
values | Answer(Option) stringclasses 5
values |
|---|---|---|---|---|---|---|---|---|---|
1974 | Twelve congruent rectangles are placed together to make a rectangle $P Q R S$ as shown. What is the ratio $P Q: Q R$ ? <image1> | [
"$2: 3$",
"$3: 4$",
"$5: 6$",
"$7: 8$",
"$8: 9$",
"I don't know"
] | images/1974.jpg | E | Let $l$ be the length of the long side, and $w$ be the length of the short side of each rectangle. Then $P S=3 l$ and $Q R=3 w+l$ so $2 l=3 w$ (since $P S=Q R$ ). Then $Q R=3 w+l=3 w+\frac{3}{2} w=\frac{9}{2} w$. Also, $P Q=2 l+w=4 w$. Hence the ratio $P Q: Q R$ is $4: \frac{9}{2}$, which is $8: 9$. | 3 | combinatorial geometry | E | |
1975 | The diagram shows a square $P Q R S$ of side-length $1 . W$ is the centre of the square and $U$ is the midpoint of $R S$. Line segments $T W, U W$ and $V W$ split the square into three regions of equal area. What is the length of $S V$ ? <image1> | [
"$\\frac{1}{2}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"$\\frac{4}{5}$",
"$\\frac{5}{6}$",
"I don't know"
] | images/1975.jpg | E | Let the length of $S V$ be $x$. Since the three areas are equal, each must be equal to one third. We know $U W=\frac{1}{2}$ and $S U=\frac{1}{2}$, so the area of the trapezium $S V W U$ is $\frac{1}{2} \times\left(x+\frac{1}{2}\right) \times \frac{1}{2}=\frac{1}{3}$. Multiplying both sides by 4 , we get $\left(x+\frac{... | 4 | metric geometry - length | E | |
1977 | Cuthbert is going to make a cube with each face divided into four squares. Each square must have one shape drawn on it; either a cross, a triangle or a circle. Squares that share an edge must have different shapes on them. One possible cube is shown in the diagram. Which of the following combinations of crosses and tri... | [
"6 crosses, 8 triangles",
"7 crosses, 8 triangles",
"5 crosses, 8 triangles",
"7 crosses, 7 triangles",
"none of these are possible",
"I don't know"
] | images/1977.jpg | E | Each vertex of the cube consists of three squares each sharing a common edge with the other two. Hence each vertex must have one of each shape drawn on its three squares. Since there are 8 vertices, there must be 8 of each shape. Hence none of the options listed is possible. | 4 | combinatorics | E | |
1979 | The cube shown has sides of length 2 units. Holes in the shape of a hemisphere are carved into each face of the cube. The six hemispheres are identical and their centres are at the centres of the faces of the cube. The holes are just large enough to touch the hole on each neighbouring face. What is the diameter of each... | [
"1",
"$\\sqrt{2}$",
"$2-\\sqrt{2}$",
"$3-\\sqrt{2}$",
"$3-\\sqrt{3}$",
"I don't know"
] | images/1979.jpg | B | Let $P$ and $Q$ be the centres of two adjacent hemispheres. The faces on which these hemispheres are carved meet at an edge. Let $M$ be the midpoint of that edge. Then $M P=M Q=1$. Also $M P Q$ is a right-angled triangle since the two faces are perpendicular. By Pythagoras, $M P^{2}+M Q^{2}=P Q^{2}$, so $P Q^{2}=1+1=2$... | 2 | solid geometry | B | |
1980 | A large square of side-length $10 \mathrm{~cm}$ contains a smaller square of side-length $4 \mathrm{~cm}$, as shown in the diagram. The corresponding sides of the two squares are parallel. What percentage of the area of the large square is shaded? <image1> | [
"$25 \\%$",
"$30 \\%$",
"$40 \\%$",
"$42 \\%$",
"$45 \\%$",
"I don't know"
] | images/1980.jpg | D | The length of the big square is $10 \mathrm{~cm}$ and of the smaller is $4 \mathrm{~cm}$. The total height of the shaded regions is 6 . Hence the total area of the two trapezoids combined is $\frac{10+4}{2} \times 6=42 \mathrm{~cm}^{2}$. Since the total area is $100 \mathrm{~cm}^{2}$ this is $42 \%$. | 4 | metric geometry - area | D | |
1981 | Mary had to run to catch the train, got off two stops later and then walked to school. Which of the following speed-time graphs would best represent her journey? <image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1981.jpg | E | The order of speeds needs to be train $>$ running $>$ walk. This rules out $A$ and $B$. The train section of the trip needs to have just a single intermediate stop in the middle of the trip. However $C$ shows no intermediate stop, just a slowing, and $D$ shows two intermediate stops. This leaves $E$ as the only feasibl... | 5 | analytic geometry | E | |
1982 | The diagram shows three squares of side-length $3 \mathrm{~cm}, 5 \mathrm{~cm}$ and $8 \mathrm{~cm}$. What is the area, in $\mathrm{cm}^{2}$, of the shaded trapezium? <image1> | [
"$13$",
"$\\frac{55}{4}$",
"$\\frac{61}{4}$",
"$\\frac{65}{4}$",
"$\\frac{69}{4}$",
"I don't know"
] | images/1982.jpg | B | Let the lengths of the vertical sides of the shaded trapezium be $p$ and $q$. Using similar triangles, $\frac{p}{3}=\frac{q}{3+5}=\frac{8}{3+5+8}$. Hence $p=\frac{3}{2}$ and $q=4$. Therefore the area of the trapezium is $\frac{1}{2} \times\left(\frac{3}{2}+4\right) \times 5$ that is $\frac{55}{4} \mathrm{~cm}^{2}$. | 4 | metric geometry - area | B | |
1983 | Points $M$ and $N$ are the midpoints of two sides of the rectangle, shown in the diagram. What fraction of the rectangle's area is shaded? <image1> | [
"$\\frac{1}{6}$",
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$\\frac{1}{3}$",
"$\\frac{1}{2}$",
"I don't know"
] | images/1983.jpg | C | The diagram here shows all of the lower triangles reflected to be above the line $M N$. This makes clear that the total area of all of the shaded triangles is $\frac{1}{2} \times M N \times M L$. This area is half that of the rectangle $M N K L$, and so equals $\frac{1}{4}$ of the original rectangle. ![](https://cdn.m... | 4 | metric geometry - area | C | |
1984 | The Pentagon $P Q R S T$ is divided into four triangles with equal perimeters. The triangle $P Q R$ is equilateral. $P T U, S U T$ and $R S U$ are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon $P Q R S T$ to the perimeter of the triangle $P Q R$? <image1> | [
"$2: 1$",
"$3: 2$",
"$4: 3$",
"$5: 3$",
"$5: 2$",
"I don't know"
] | images/1984.jpg | D | Let $2 a$ be the side of equilateral triangle $P Q R$. Then $P Q R$ has a perimeter of $6 a$ and $P U$ has a length $a$. Therefore, in order for the perimeter of triangle $P T U$ to be $6 a, P T$ must be $\frac{5}{2} a$. Since the isosceles triangles are congruent $S T=P U$ and $R S=P T$. So the perimeter of the pent... | 4 | metric geometry - length | D | |
1986 | A square of side-length $30 \mathrm{~cm}$ is divided into nine smaller identical squares. The large square contains three circles with radii $5 \mathrm{~cm}$ (bottom right), $4 \mathrm{~cm}$ (top left) and $3 \mathrm{~cm}$ (top right), as shown. What is the total area of the shaded part? <image1> | [
"$400 \\mathrm{~cm}^{2}$",
"$500 \\mathrm{~cm}^{2}$",
"$(400+50 \\pi) \\mathrm{cm}^{2}$",
"$(500-25 \\pi) \\mathrm{cm}^{2}$",
"$(500+25 \\pi) \\mathrm{cm}^{2}$",
"I don't know"
] | images/1986.jpg | B | The sum of the areas of the two smaller grey circles is the same as that of the white circle since $(3 \pi)^{2}+(4 \pi)^{2}=(5 \pi)^{2}$. This means that the shaded area is the same as the area of the five of the smaller squares. Each of the smaller squares has and area of $10 \mathrm{~cm} \times 10 \mathrm{~cm}=100 \m... | 4 | metric geometry - area | B | |
1987 | The figure shows two touching semicircles of radius 1 , with parallel diameters $P Q$ and $R S$. What is the square of the distance $P S$ ? <image1> | [
"$16",
"$8+4 \\sqrt{3}$",
"$12$",
"$9$",
"$5+2 \\sqrt{3}$",
"I don't know"
] | images/1987.jpg | B | Let $T$ and $U$ be the centres of the two semicircles and $V, W$ be where the perpendiculars from $U$ and $S$ meet $P Q$. Then $T U$ has length 2, and $U V$ and $S W$ have length 1. Using Pythagoras' theorem $T V$ is $\sqrt{3}$. ![](https://cdn.mathpix.com/cropped/2023_12_27_6db5618fc6dc2044925bg-07.jpg?height=230&widt... | 4 | metric geometry - length | B | |
1989 | Lancelot has drawn a closed path on a cuboid and unfolded it into a net. Which of the nets shown could not be the net of Lancelot's cuboid? <image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1989.jpg | C | The nets given can be folded into a solid in only one way. We can continue the path along the edges that would be joined. Connections shown by dashed lines show which pieces are glued together (i.e. belong to to the same path). ![](https://cdn.mathpix.com/cropped/2023_12_27_6db5618fc6dc2044925bg-10.jpg?height=280&widt... | 2 | solid geometry | C | |
1991 | The diagram shows a map of a park. The park is divided into regions. The number inside each region gives its perimeter, in $\mathrm{km}$. What is the outer perimeter of the park? <image1> | [
"$22 \\mathrm{~km}$",
"$26 \\mathrm{~km}$",
"$28 \\mathrm{~km}$",
"$32 \\mathrm{~km}$",
"$34 \\mathrm{~km}$",
"I don't know"
] | images/1991.jpg | B | The sum of the perimeters of $F, G, H, I$ and $J$ give the length of the outside line increased by the dotted line. If we subtract the perimeters of $K, L$ and $M$, then we subtract the dotted line but we have now also subtracted the dashed line. So we add the dashed line to compensate. In other words the required peri... | 4 | metric geometry - length | B | |
1992 | Vumos wants to write the integers 1 to 9 in the nine boxes shown so that the sum of the integers in any three adjacent boxes is a multiple of 3 . In how many ways can he do this? <image1> | [
"$6 \\times 6 \\times 6 \\times 6$",
"$6 \\times 6 \\times 6$",
"$2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2$",
"$6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1$",
"$9 \\times 8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1$",
"I don't kn... | images/1992.jpg | A | Let $a, b, c, d$ be the numbers in four adjacent boxes. Then both $a+b+c$ and $b+c+d$ must be multiples of 3. Therefore $a-d$ is a multiple of 3. This applies to any entries three apart. So the numbers in the set $\{1,4,7\}$ must be listed three apart; and the same applies to $\{2,5,8\}$ and to $\{3,6,9\}$. This will a... | 4 | combinatorics | A | |
2109 | In rectangle $ ABCD$, $ AD = 1$, $ P$ is on $ \overline{AB}$, and $ \overline{DB}$ and $ \overline{DP}$ trisect $ \angle ADC$. What is the perimeter of $ \triangle BDP$?
<image1> | [
"$3 + \\frac{\\sqrt{3}}{3}$",
"$2 + \\frac{4\\sqrt{3}}{3}$",
"$2 + 2\\sqrt{2}$",
"$\\frac{3 + 3\\sqrt{5}}{2}$",
"$2 + \\frac{5\\sqrt{3}}{3}$",
"I don't know"
] | images/2109.jpg | B | null | 4 | metric geometry - length | B | |
2111 | There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?
<image1> | [
"$0$",
"$1$",
"$5!\\cdot4!\\cdot3!\\cdot2!\\cdot1!$",
"$\\frac{15!}{5!\\cdot4!\\cdot3!\\cdot2!\\cdot1!}$",
"$15!$",
"I don't know"
] | images/2111.jpg | B | null | 4 | combinatorics | B | |
2112 | The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.
<image1> | [
"$\\frac{4\\sqrt{5}}{3}$",
"$\\frac{5\\sqrt{5}}{3}$",
"$\\frac{12\\sqrt{5}}{7}$",
"$2\\sqrt{5}$",
"$\\frac{5\\sqrt{65}}{9}$",
"I don't know"
] | images/2112.jpg | B | null | 4 | metric geometry - length | B | |
2114 | Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains $ 8$ unit squares. The second ring contains $ 16$ unit squares.
<image1>
If we continue this process, the number of unit squares in the $ 100^\text{th}$ ring is | [
"$396$",
"$404$",
"$800$",
"$10,\\!000$",
"$10,\\!404$",
"I don't know"
] | images/2114.jpg | C | null | 3 | algebra | C | |
2115 | Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/2115.jpg | C | null | 2 | solid geometry | C | |
2116 | The plane is tiled by congruent squares and congruent pentagons as indicated.
<image1>
The percent of the plane that is enclosed by the pentagons is closest to | [
"50",
"52",
"54",
"56",
"58",
"I don't know"
] | images/2116.jpg | D | null | 4 | metric geometry - area | D | |
2118 | Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
<image1> | [
"$\\pi$",
"$1.5\\pi$",
"$2\\pi$",
"$3\\pi$",
"$3.5\\pi$",
"I don't know"
] | images/2118.jpg | C | null | 4 | metric geometry - area | C | |
2119 | Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct?
<image1> | [
"B = W",
"W = R",
"B = R",
"3B = 2R",
"2R = W",
"I don't know"
] | images/2119.jpg | A | null | 4 | metric geometry - area | A | |
2121 | Circles of radius $ 2$ and $ 3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.
<image1> | [
"$3\\pi$",
"$4\\pi$",
"$6\\pi$",
"$9\\pi$",
"$12\\pi$",
"I don't know"
] | images/2121.jpg | E | null | 4 | metric geometry - area | E | |
2123 | A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
<image1> | [
"$\\frac{1}{6}\\pi - \\frac{\\sqrt{3}}{4}$",
"$\\frac{\\sqrt{3}}{4} - \\frac{1}{12}\\pi$",
"$\\frac{\\sqrt{3}}{4} - \\frac{1}{24}\\pi$",
"$\\frac{\\sqrt{3}}{4} + \\frac{1}{24}\\pi$",
"$\\frac{\\sqrt{3}}{4} + \\frac{1}{12}\\pi$",
"I don't know"
] | images/2123.jpg | C | null | 4 | metric geometry - area | C | |
2125 | A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $ 3$ rows of small congruent equilateral triangles, with $ 5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle i... | [
"$1,\\!004,\\!004$",
"$1,\\!005,\\!006$",
"$1,\\!507,\\!509$",
"$3,\\!015,\\!018$",
"$6,\\!021,\\!018$",
"I don't know"
] | images/2125.jpg | C | null | 3 | algebra | C | |
2127 | Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following?
<image1> | [
"20",
"20.5",
"21",
"21.5",
"22",
"I don't know"
] | images/2127.jpg | D | null | 4 | metric geometry - length | D | |
2128 | Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircl... | [
"$\\pi-\\sqrt{3}$",
"$\\pi-\\sqrt{2}$",
"$\\frac{\\pi+\\sqrt{2}}{2}$",
"$\\frac{\\pi+\\sqrt{3}}{2}$",
"$\\frac{7}{6}\\pi-\\frac{\\sqrt{3}}{2}$",
"I don't know"
] | images/2128.jpg | E | null | 4 | metric geometry - area | E | |
2129 | In rectangle $ ABCD$, $ AB=5$ and $ BC=3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF=1$ and $ GC=2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$.
<image1> | [
"$10$",
"$\\frac{21}{2}$",
"$12$",
"$\\frac{25}{2}$",
"$15$",
"I don't know"
] | images/2129.jpg | D | null | 4 | metric geometry - area | D | |
2130 | A regular octagon $ ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ ABEF$?
<image1> | [
"$1-\\frac{\\sqrt{2}}{2}$",
"$\\frac{\\sqrt{2}}{4}$",
"$\\sqrt{2}-1$",
"$\\frac{1}{2}$",
"$\\frac{1+\\sqrt{2}}{4}$",
"I don't know"
] | images/2130.jpg | D | null | 4 | metric geometry - area | D | |
2131 | A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?
<image1> | [
"$\\frac{1}{21}$",
"$\\frac{1}{14}$",
"$\\frac{2}{21}$",
"$\\frac{1}{7}$",
"$\\frac{2}{7}$",
"I don't know"
] | images/2131.jpg | C | null | 4 | combinatorics | C | |
2134 | A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
<image1> | [
"$120$",
"$180$",
"$240$",
"$360$",
"$480$",
"I don't know"
] | images/2134.jpg | C | null | 2 | solid geometry | C | |
2135 | Points $E$ and $F$ are located on square $ABCD$ so that $\Delta BEF$ is equilateral. What is the ratio of the area of $\Delta DEF$ to that of $\Delta ABE$?
<image1> | [
"$\\frac{4}{3}$",
"$\\frac{3}{2}$",
"$\\sqrt{3}$",
"$2$",
"$1+\\sqrt{3}$",
"I don't know"
] | images/2135.jpg | D | null | 4 | metric geometry - area | D | |
2136 | Two distinct lines pass through the center of three concentric circles of radii $3$, $2$, and $1$. The area of the shaded region in the diagram is $8/13$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)
<image1> | [
"$\\frac{\\pi}8$",
"$\\frac{\\pi}7$",
"$\\frac{\\pi}6$",
"$\\frac{\\pi}5$",
"$\\frac{\\pi}4$",
"I don't know"
] | images/2136.jpg | B | null | 2 | metric geometry - angle | B | |
2137 | Square $ABCD$ has side length 2. A semicircle with diameter $AB$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $AD$ at $E$. What is the length of $CE$?
<image1> | [
"$\\frac{2+\\sqrt{5}}2$",
"$\\sqrt{5}$",
"$\\sqrt{6}$",
"$\\frac{5}{2}$",
"$5-\\sqrt{5}$",
"I don't know"
] | images/2137.jpg | D | null | 4 | metric geometry - length | D | |
2138 | An annulus is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that co... | [
"$\\pi a^2$",
"$\\pi b^2$",
"$\\pi c^2$",
"$\\pi d^2$",
"$\\pi e^2$",
"I don't know"
] | images/2138.jpg | A | null | 4 | metric geometry - area | A | |
2139 | Three circles of radius $ 1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?
<image1> | [
"$\\frac{2 + \\sqrt{6}}{3}$",
"$2$",
"$\\frac{2 + 3\\sqrt{2}}{3}$",
"$\\frac{3 + 2\\sqrt{3}}{3}$",
"$\\frac{3 + \\sqrt{3}}{2}$",
"I don't know"
] | images/2139.jpg | D | null | 4 | metric geometry - length | D | |
2140 | In right triangle $ \triangle ACE$, we have $ AC = 12$, $ CE = 16$, and $ EA = 20$. Points $ B$, $ D$, and $ F$ are located on $ \overline{AC}$, $ \overline{CE}$, and $ \overline{EA}$, respectively, so that $ AB = 3$, $ CD = 4$, and $ EF = 5$. What is the ratio of the area of $ \triangle DBF$ to that of $ \triangle ACE... | [
"$\\frac{1}{4}$",
"$\\frac{9}{25}$",
"$\\frac{3}{8}$",
"$\\frac{11}{25}$",
"$\\frac{7}{16}$",
"I don't know"
] | images/2140.jpg | E | null | 4 | metric geometry - area | E | |
2141 | In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT = 3$ and $ BT/ET = 4$, what is $ CD/BD$?
<image1> | [
"$\\frac{1}{8}$",
"$\\frac{2}{9}$",
"$\\frac{3}{10}$",
"$\\frac{4}{11}$",
"$\\frac{5}{12}$",
"I don't know"
] | images/2141.jpg | D | null | 4 | metric geometry - length | D | |
2142 | A circle of radius $ 1$ is internally tangent to two circles of radius $ 2$ at points $ A$ and $ B$, where $ AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles?
<image1> | [
"$\\frac{5}{3}\\pi - 3\\sqrt{2}$",
"$\\frac{5}{3}\\pi - 2\\sqrt{3}$",
"$\\frac{8}{3}\\pi - 3\\sqrt{3}$",
"$\\frac{8}{3}\\pi - 3\\sqrt{2}$",
"$\\frac{8}{3}\\pi - 2\\sqrt{3}$",
"I don't know"
] | images/2142.jpg | B | null | 4 | metric geometry - area | B | |
2144 | The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $ 2$?
<image1> | [
"$\\frac{1}{3}\\pi+\\frac{\\sqrt{3}}{2}$",
"$\\frac{2}{3}\\pi$",
"$\\frac{2}{3}\\pi+\\frac{\\sqrt{3}}{4}$",
"$\\frac{2}{3}\\pi+\\frac{\\sqrt{3}}{3}$",
"$\\frac{2}{3}\\pi+\\frac{\\sqrt{3}}{2}$",
"I don't know"
] | images/2144.jpg | B | null | 4 | metric geometry - length | B | |
2146 | Three one-inch squares are palced with their bases on a line. The center square is lifted out and rotated $ 45^\circ$, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $ B$ from the line on which the bases of the original s... | [
"$1$",
"$\\sqrt{2}$",
"$\\frac{3}{2}$",
"$\\sqrt{2} + \\frac{1}{2}$",
"$2$",
"I don't know"
] | images/2146.jpg | D | null | 4 | transformation geometry | D | |
2147 | Let $ \overline{AB}$ be a diameter of a circle and $ C$ be a point on $ \overline{AB}$ with $ 2 \cdot AC = BC$. Let $ D$ and $ E$ be points on the circle such that $ \overline{DC} \perp \overline{AB}$ and $ \overline{DE}$ is a second diameter. What is the ratio of the area of $ \triangle DCE$ to the area of $ \triangle... | [
"$\\frac{1}{6}$",
"$\\frac{1}{4}$",
"$\\frac{1}{3}$",
"$\\frac{1}{2}$",
"$\\frac{2}{3}$",
"I don't know"
] | images/2147.jpg | C | null | 4 | metric geometry - length | C | |
2148 | An $ 8$-foot by $ 10$-foot floor is tiled with square tiles of size $ 1$ foot by $ 1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $ 1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
<image1> | [
"$80-20\\pi$",
"$60-10\\pi$",
"$80-10\\pi$",
"$60+10\\pi$",
"$80+10\\pi$",
"I don't know"
] | images/2148.jpg | A | null | 4 | metric geometry - length | A | |
2149 | Equilateral $ \triangle ABC$ has side length $ 2$, $ M$ is the midpoint of $ \overline{AC}$, and $ C$ is the midpoint of $ \overline{BD}$. What is the area of $ \triangle CDM$?
<image1> | [
"$\\frac{\\sqrt{2}}{2}$",
"$\\frac{3}{4}$",
"$\\frac{\\sqrt{3}}{2}$",
"$1$",
"$\\sqrt{2}$",
"I don't know"
] | images/2149.jpg | C | null | 4 | metric geometry - length | C | |
2151 | Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet?
<image1> | [
"$\\text{ I, by }8\\pi$",
"$\\text{ I, by }6\\pi$",
"$\\text{ II, by }4\\pi$",
"$\\text{II, by }8\\pi$",
"$\\text{ II, by }10\\pi$",
"I don't know"
] | images/2151.jpg | C | null | 4 | metric geometry - area | C | |
2153 | A circle of radius 1 is tangent to a circle of radius 2. The sides of $ \triangle ABC$ are tangent to the circles as shown, and the sides $ \overline{AB}$ and $ \overline{AC}$ are congruent. What is the area of $ \triangle ABC$?
<image1> | [
"$\\frac{35}2$",
"$15\\sqrt{2}$",
"$\\frac{64}3$",
"$16\\sqrt{2}$",
"$24$",
"I don't know"
] | images/2153.jpg | D | null | 4 | metric geometry - area | D | |
2154 | In rectangle $ ADEH$, points $ B$ and $ C$ trisect $ \overline{AD}$, and points $ G$ and $ F$ trisect $ \overline{HE}$. In addition, $ AH = AC = 2$. What is the area of quadrilateral $ WXYZ$ shown in the figure?
<image1> | [
"$\\frac{1}{2}$",
"$\\frac{\\sqrt{2}}2$",
"$\\frac{\\sqrt{3}}2$",
"$\\frac{2\\sqrt{2}}3$",
"$\\frac{2\\sqrt{3}}3$",
"I don't know"
] | images/2154.jpg | A | null | 4 | metric geometry - area | A | |
2155 | Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE = 5$. What is $ CD$?
<image1> | [
"$13$",
"$\\frac{44}{3}$",
"$\\sqrt{221}$",
"$\\sqrt{255}$",
"$\\frac{55}{3}$",
"I don't know"
] | images/2155.jpg | B | null | 4 | metric geometry - length | B | |
2156 | A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $ 2/\pi $, as shown. What is the perimeter of this region?
<image1> | [
"$\\frac{4}\\pi$",
"$2$",
"$\\frac{8}\\pi$",
"$4$",
"$\\frac{16}{\\pi}$",
"I don't know"
] | images/2156.jpg | D | null | 4 | metric geometry - length | D | |
2157 | A square of area $40$ is inscribed in a semicircle as shown. What is the area of the semicircle?
<image1> | [
"$20\\pi$",
"$25\\pi$",
"$30\\pi$",
"$40\\pi$",
"$50\\pi$",
"I don't know"
] | images/2157.jpg | B | null | 4 | metric geometry - area | B | |
2158 | Rhombus $ ABCD$ is similar to rhombus $ BFDE$. The area of rhombus $ ABCD$ is 24, and $ \angle BAD = 60^\circ$. What is the area of rhombus $ BFDE$?
<image1> | [
"$6$",
"$4\\sqrt{3}$",
"$8$",
"$9$",
"$6\\sqrt{3}$",
"I don't know"
] | images/2158.jpg | C | null | 4 | metric geometry - area | C | |
2159 | A circle of radius 2 is centered at $ O$. Square $ OABC$ has side length 1. Sides $ \overline{AB}$ and $ \overline{CB}$ are extended past $ b$ to meet the circle at $ D$ and $ E$, respectively. What is the area of the shaded region in the figure, which is bounded by $ \overline{BD}$, $ \overline{BE}$, and the minor arc... | [
"$\\frac{\\pi}3 + 1 - \\sqrt{3}$",
"$\\frac{\\pi}2\\left( 2 - \\sqrt{3}\\right)$",
"$\\pi\\left(2 - \\sqrt{3}\\right)$",
"$\\frac{\\pi}{6} + \\frac{\\sqrt{3} - 1}{2} \\ \\indent$",
"$\\frac{\\pi}{3} - 1 + \\sqrt{3}$",
"I don't know"
] | images/2159.jpg | A | null | 4 | metric geometry - area | A | |
2160 | A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?
<image1> | [
"$15$",
"$17$",
"$\\frac{35}{2}$",
"$18$",
"$\\frac{55}{3}$",
"I don't know"
] | images/2160.jpg | D | null | 4 | metric geometry - area | D | |
2161 | Circles with centers $ O$ and $ P$ have radii 2 and 4, respectively, and are externally tangent. Points $ A$ and $ B$ are on the circle centered at $ O$, and points $ C$ and $ D$ are on the circle centered at $ P$, such that $ \overline{AD}$ and $ \overline{BC}$ are common external tangents to the circles. What is th... | [
"$18\\sqrt{3}$",
"$24\\sqrt{2}$",
"$36$",
"$24\\sqrt{3}$",
"$32\\sqrt{2}$",
"I don't know"
] | images/2161.jpg | B | null | 4 | metric geometry - area | B | |
2163 | A cube with side length $ 1$ is sliced by a plane that passes through two diagonally opposite vertices $ A$ and $ C$ and the midpoints $ B$ and $ D$ of two opposite edges not containing $ A$ and $ C$, ac shown. What is the area of quadrilateral $ ABCD$?
<image1> | [
"$\\frac{\\sqrt{6}}{2}$",
"$\\frac{5}{4}$",
"$\\sqrt{2}$",
"$\\frac{3}{2}$",
"$\\sqrt{3}$",
"I don't know"
] | images/2163.jpg | A | null | 2 | solid geometry | A | |
2164 | A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned s... | [
"$2\\sqrt{5} - \\sqrt{3}$",
"$3$",
"$\\frac{3\\sqrt{7} - \\sqrt{3}}{2}$",
"$2\\sqrt{3}$",
"$\\frac{5 + 2\\sqrt{3}}{2}$",
"I don't know"
] | images/2164.jpg | C | null | 4 | metric geometry - length | C | |
2165 | A circle of radius $ 2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
<image1> | [
"$\\frac{1}{2}$",
"$\\frac{\\pi}{6}$",
"$\\frac{2}{\\pi}$",
"$\\frac{2}{3}$",
"$\\frac{3}{\\pi}$",
"I don't know"
] | images/2165.jpg | A | null | 4 | metric geometry - area | A | |
2166 | Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD=3$, and $ DC=4$. What is the area of $ \triangle{ABC}$?
<image1> | [
"$4\\sqrt{3}$",
"$7\\sqrt{3}$",
"$21$",
"$14\\sqrt{3}$",
"$42$",
"I don't know"
] | images/2166.jpg | B | null | 4 | metric geometry - area | B | |
2168 | Four congruent rectangles are placed as shown. The area of the outer square is $ 4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
<image1> | [
"$3$",
"$\\sqrt{10}$",
"$2 + \\sqrt{2}$",
"$2\\sqrt{3}$",
"$4$",
"I don't know"
] | images/2168.jpg | A | null | 4 | metric geometry - length | A | |
2170 | Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
<image1> | [
"$3-2\\sqrt{2}$",
"$2-\\sqrt{2}$",
"$4(3-2\\sqrt{2})$",
"$\\frac{1}{2}(3-\\sqrt{2})$",
"$2\\sqrt{2}-2$",
"I don't know"
] | images/2170.jpg | C | null | 4 | metric geometry - area | C | |
2171 | A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $ 15$ and $ 25$ meters. What fraction of the yard is occupied by the flower beds?
<image1> | [
"$\\frac{1}{8}$",
"$\\frac{1}{6}$",
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$\\frac{1}{3}$",
"I don't know"
] | images/2171.jpg | C | null | 4 | metric geometry - area | C | |
2173 | As shown below, convex pentagon $ ABCDE$ has sides $ AB = 3$, $ BC = 4$, $ CD = 6$, $ DE = 3$, and $ EA = 7$. The pentagon is originally positioned in the plane with vertex $ A$ at the origin and vertex $ B$ on the positive $ x$-axis. The pentagon is then rolled clockwise to the right along the $ x$-axis. Which side wi... | [
"$\\overline{AB}$",
"$\\overline{BC}$",
"$\\overline{CD}$",
"$\\overline{DE}$",
"$\\overline{EA}$",
"I don't know"
] | images/2173.jpg | C | null | 3 | algebra | C | |
2174 | Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $ (a,0)$ to $ (3,3)$, divides the entire region into two regions of equal area. What is $ a$?
<image1> | [
"$\\frac{1}{2}$",
"$\\frac{3}{5}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"$\\frac{4}{5}$",
"I don't know"
] | images/2174.jpg | C | null | 5 | analytic geometry | C | |
2175 | Triangle $ ABC$ has a right angle at $ B$, $ AB = 1$, and $ BC = 2$. The bisector of $ \angle BAC$ meets $ \overline{BC}$ at $ D$. What is $ BD$?
<image1> | [
"$\\frac{\\sqrt{3} - 1}{2}$",
"$\\frac{\\sqrt{5} - 1}{2}$",
"$\\frac{\\sqrt{5} + 1}{2}$",
"$\\frac{\\sqrt{6} + \\sqrt{2}}{2}$",
"$2\\sqrt{3} - 1$",
"I don't know"
] | images/2175.jpg | B | null | 4 | metric geometry - length | B | |
2176 | A cubical cake with edge length $ 2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $ M$ is the midpoint of a top edge. The piece whose top is triangle $ B$ contains $ c$ cubic inches of cake and $ s$ square inches of icing. What is $ c+s$?
<image1> | [
"$\\frac{24}{5}$",
"$\\frac{32}{5}$",
"$8+\\sqrt{5}$",
"$5+\\frac{16\\sqrt{5}}{5}$",
"$10+5\\sqrt{5}$",
"I don't know"
] | images/2176.jpg | B | null | 4 | metric geometry - area | B | |
2178 | Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
<image1> | [
"$\\frac{5}{4}$",
"$\\frac{4}{3}$",
"$\\frac{3}{2}$",
"$2$",
"$3$",
"I don't know"
] | images/2178.jpg | B | null | 4 | metric geometry - length | B | |
2179 | Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?
<image1> | [
"$3 - \\frac{\\pi}{2}$",
"$\\frac{\\pi}{2}$",
"$2$",
"$\\frac{3\\pi}{4}$",
"$1+\\frac{\\pi}{2}$",
"I don't know"
] | images/2179.jpg | C | null | 4 | metric geometry - area | C | |
2180 | The area of $\triangle EBD$ is one third of the area of $3-4-5$ $ \triangle ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$?
<image1> | [
"$\\frac{4}{3}$",
"$\\sqrt{5}$",
"$\\frac{9}{4}$",
"$\\frac{4\\sqrt{3}}{3}$",
"$\\frac{5}{2}$",
"I don't know"
] | images/2180.jpg | D | null | 4 | metric geometry - length | D | |
2181 | A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?
<image1> | [
"$\\frac{\\sqrt{2} - 1}{2}$",
"$\\frac{1}{4}$",
"$\\frac{2 - \\sqrt{2}}{2}$",
"$\\frac{\\sqrt{2}}{4}$",
"$2 - \\sqrt{2}$",
"I don't know"
] | images/2181.jpg | A | null | 3 | combinatorial geometry | A | |
2183 | Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?
<image1> | [
"$\\frac{1}{6}$",
"$\\frac{1}{5}$",
"$\\frac{2}{9}$",
"$\\frac{1}{3}$",
"$\\frac{\\sqrt{2}}{4}$",
"I don't know"
] | images/2183.jpg | B | null | 4 | metric geometry - area | B | |
2184 | The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve?
<image1> | [
"$2\\pi+6$",
"$2\\pi+4\\sqrt{3}$",
"$3\\pi+4$",
"$2\\pi+3\\sqrt{3}+2$",
"$\\pi+6\\sqrt{3}$",
"I don't know"
] | images/2184.jpg | E | null | 4 | metric geometry - area | E | |
2186 | Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?
<image1> | [
"$10\\pi+4\\sqrt{3}$",
"$13\\pi-\\sqrt{3}$",
"$12\\pi+\\sqrt{3}$",
"$10\\pi+9$",
"$13\\pi$",
"I don't know"
] | images/2186.jpg | A | null | 4 | metric geometry - area | A | |
2192 | A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?
<image1> | [
"$27\\sqrt{3}-9\\pi$",
"$27\\sqrt{3}-6\\pi$",
"$54\\sqrt{3}-18\\pi$",
"$54\\sqrt{3}-12\\pi$",
"$108\\sqrt{3}-9\\pi$",
"I don't know"
] | images/2192.jpg | C | null | 4 | metric geometry - area | C | |
2193 | Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?
<image1> | [
"$\\frac{12+3\\sqrt{3}}4$",
"$\\frac{9}{2}$",
"$3+\\sqrt{3}$",
"$\\frac{6+3\\sqrt{3}}2$",
"$6$",
"I don't know"
] | images/2193.jpg | C | null | 4 | metric geometry - area | C | |
2194 | In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region?
<image1> | [
"$\\frac{1}{12}$",
"$\\frac{\\sqrt{3}}{18}$",
"$\\frac{\\sqrt{2}}{12}$",
"$\\frac{\\sqrt{3}}{12}$",
"$\\frac{1}{6}$",
"I don't know"
] | images/2194.jpg | E | null | 4 | metric geometry - area | E | |
2195 | Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$?
<image1> | [
"$\\frac{3\\sqrt{33}}5$",
"$2\\sqrt{3}$",
"$\\frac{2\\sqrt{33}}3$",
"$4$",
"$3\\sqrt{2} $",
"I don't know"
] | images/2195.jpg | A | null | 2 | solid geometry | A | |
2196 | A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to c... | [
"1:2",
"3:5",
"2:3",
"3:4",
"4:5",
"I don't know"
] | images/2196.jpg | C | null | 4 | transformation geometry | C | |
2198 | Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle ABC$?
<image1> | [
"$2\\sqrt{3}$",
"$3\\sqrt{3}$",
"$1+3\\sqrt{2}$",
"$2+2\\sqrt{3}$",
"$3+2\\sqrt{3}$",
"I don't know"
] | images/2198.jpg | B | null | 4 | metric geometry - area | B | |
2199 | In rectangle $ABCD$, $DC = 2CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$?
<image1> | [
"$\\frac{\\sqrt{3}}{6}$",
"$\\frac{\\sqrt{6}}{8}$",
"$\\frac{3\\sqrt{3}}{16}$",
"$\\frac{1}{3}$",
"$\\frac{\\sqrt{2}}{4}$",
"I don't know"
] | images/2199.jpg | A | null | 4 | metric geometry - area | A | |
2200 | Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
<image1> | [
"$\\frac{1+\\sqrt{2}}4$",
"$\\frac{\\sqrt{5}-1}2$",
"$\\frac{\\sqrt{3}+1}4$",
"$\\frac{2\\sqrt{3}}5$",
"$\\frac{\\sqrt{5}}3$",
"I don't know"
] | images/2200.jpg | B | null | 4 | metric geometry - length | B | |
2201 | A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
<image1> | [
"$\\frac{3}{2}$",
"$\\frac{1+\\sqrt{5}}{2}$",
"$\\sqrt{3}$",
"$2$",
"$\\frac{3+\\sqrt{5}}{2}$",
"I don't know"
] | images/2201.jpg | E | null | 2 | solid geometry | E | |
2203 | The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what poin... | [
"$\\text{2 o'clock}$",
"$\\text{3 o'clock}$",
"$\\text{4 o'clock}$",
"$\\text{6 o'clock}$",
"$\\text{8 o'clock}$",
"I don't know"
] | images/2203.jpg | C | null | 4 | transformation geometry | C | |
2204 | The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
<image1><image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/2204.jpg | E | null | 4 | transformation geometry | E | |
2205 | The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\frac{3}{2}$ and center $(0,\frac{3}{2})$ that lies in the first quadrant, a... | [
"$\\frac{4\\pi}{5}$",
"$\\frac{9\\pi}{8}$",
"$\\frac{4\\pi}{3}$",
"$\\frac{7\\pi}{5}$",
"$\\frac{3\\pi}{2}$",
"I don't know"
] | images/2205.jpg | B | null | 5 | analytic geometry | B | |
2206 | The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
<image1> | [
"$\\frac{75}{12}$",
"$10$",
"$12$",
"$10\\sqrt{2}$",
"$15$",
"I don't know"
] | images/2206.jpg | B | null | 2 | solid geometry | B | |
2207 | In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$?
<image1> | [
"$3$",
"$12-4\\sqrt{5}$",
"$\\frac{5+2\\sqrt{5}}{3}$",
"$1+\\sqrt{5}$",
"$\\frac{11+11\\sqrt{5}}{10}$",
"I don't know"
] | images/2207.jpg | D | null | 4 | metric geometry - length | D | |
2209 | What is the area of the shaded region of the given $8 \times 5$ rectangle?
<image1> | [
"$4\\frac{3}{5}$",
"$5$",
"$5\\frac{1}{4}$",
"$6\\frac{1}{2}$",
"$8$",
"I don't know"
] | images/2209.jpg | D | null | 4 | metric geometry - area | D | |
2210 | Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?
<image... | [
"$\\sqrt{2}$",
"$1.5$",
"$\\sqrt{\\pi}$",
"$\\sqrt{2\\pi}$",
"$\\pi$",
"I don't know"
] | images/2210.jpg | A | null | 4 | metric geometry - length | A | |
2211 | Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\fr... | [
"$\\frac{\\sqrt{13}}{16}$",
"$\\frac{\\sqrt{2}}{13}$",
"$\\frac{9}{82}$",
"$\\frac{10}{91}$",
"$\\frac{1}{9}$",
"I don't know"
] | images/2211.jpg | D | null | 4 | metric geometry - length | D | |
2215 | A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
<image1> | [
"$1+\\frac{1}{2} \\sqrt{2}$",
"$\\sqrt{3}$",
"$\\frac{7}{4}$",
"$\\frac{15}{8}$",
"$2$",
"I don't know"
] | images/2215.jpg | D | null | 4 | transformation geometry | D | |
2217 | Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest di... | [
"$\\frac{25}{27}$",
"$\\frac{26}{27}$",
"$\\frac{73}{75}$",
"$\\frac{145}{147}$",
"$\\frac{74}{75}$",
"I don't know"
] | images/2217.jpg | D | null | 4 | metric geometry - area | D | |
2220 | In the rectangular parallelpiped shown, $AB = 3, BC= 1,$ and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?
<image1> | [
"$1$",
"$\\frac{4}{3}$",
"$\\frac{3}{2}$",
"$\\frac{5}{3}$",
"$2$",
"I don't know"
] | images/2220.jpg | E | null | 2 | solid geometry | E | |
2221 | A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the side... | [
"$2(w+h)^2$",
"$\\frac{(w+h)^2}2$",
"$2w^2+4wh$",
"$2w^2$",
"$w^2h$",
"I don't know"
] | images/2221.jpg | A | null | 2 | solid geometry | A | |
2223 | The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$
<image1> | [
"$4 \\pi \\sqrt{3}$",
"$7 \\pi$",
"$\\pi(3\\sqrt{3} +2)$",
"$10 \\pi (\\sqrt{3} - 1)$",
"$\\pi(\\sqrt{3} + 6)$",
"I don't know"
] | images/2223.jpg | A | null | 4 | metric geometry - area | A | |
2224 | The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is th... | [
"$4$",
"$12 - 4\\sqrt{3}$",
"$3\\sqrt{3}$",
"$4\\sqrt{3}$",
"$16 - \\sqrt{3}$",
"I don't know"
] | images/2224.jpg | B | null | 4 | metric geometry - area | B | |
2228 | A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
<image1> | [
"$3\\pi \\sqrt{5}$",
"$4\\pi \\sqrt{3}$",
"$3 \\pi \\sqrt{7}$",
"$6\\pi \\sqrt{3}$",
"$6\\pi \\sqrt{7}$",
"I don't know"
] | images/2228.jpg | C | null | 2 | solid geometry | C | |
2229 | As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length $2$ so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?
<image1> | [
"$6\\sqrt{3}-3\\pi$",
"$\\frac{9\\sqrt{3}}{2}-2\\pi$",
"$\\frac{3\\sqrt{3}}{2}-\\frac{\\pi}{3}$",
"$3\\sqrt{3}-\\pi \\$",
"$\\frac{9\\sqrt{3}}{2}-\\pi$",
"I don't know"
] | images/2229.jpg | D | null | 4 | metric geometry - area | D | |
2230 | In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH$. Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. ... | [
"$\\frac{7}{3}$",
"$8-4\\sqrt{2}$",
"$1+\\sqrt{2}$",
"$\\frac{7}{4}\\sqrt{2}$",
"$2\\sqrt{2}$",
"I don't know"
] | images/2230.jpg | B | null | 4 | metric geometry - length | B | |
2234 | A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
<image1> | [
"$19\\frac{1}{4}$",
"$20\\frac{1}{4}$",
"$21\\frac{3}{4}$",
"$22\\frac{1}{2}$",
"$23\\frac{3}{4}$",
"I don't know"
] | images/2234.jpg | B | null | 4 | metric geometry - area | B | |
2238 | Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, which sinks to the bottom and is completely submer... | [
"1:1",
"47:43",
"2:1",
"40:13",
"4:1",
"I don't know"
] | images/2238.jpg | E | null | 2 | solid geometry | E |
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