data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A sports lottery stipulates that 7 numbers are drawn from a total of 36 numbers, ranging from 01 to 36, for a single entry, which costs 2 yuan. A person wants to select the lucky number 18 first, then choose 3 consecutive numbers from 01 to 17, 2 consecutive numbers from 19 to ... | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"index": 3600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sports team's members have unique numbers taken from the integers 1 to 100. If no member's number is the sum of the numbers of any two other members, nor is it twice the number of any other member, what is the maximum number of people this team can have?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 3601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square $ (n \\minus{} 1) \\times (n \\minus{} 1)$ is divided into $ (n \\minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly ... | math | {
"style": "rule",
"ground_truth": "$ 2^{n+1}-2$"
} | {
"index": 3602,
"solution": "\nTo solve the problem, we need to consider how we can distribute the colors such that each unit square in the \\((n-1) \\times (n-1)\\) grid has exactly two red vertices. Each unit square is defined by its four vertices, and we need each four-vertex set to have exactly two vertices co... |
DeepScaleR | [
{
"role": "user",
"content": "A square $ABCD$ with a side length of $2$ is rotated around $BC$ to form a cylinder. Find the surface area of the cylinder."
}
] | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"index": 3603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square $EFGH$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 16$ and the $x$-axis. Find the area of square $EFGH$.\n\n[asy]\nunitsize(0.8 cm);\n\nreal parab2 (real x) {\n return(x^2 - 8*x + 16);\n}\n\npair E, F, G, H;\nreal x = -2 + 2*sqrt(3);\n\nE = (4 - x... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 3604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square $EFGH$ is inscribed in the region bounded by the parabola $y = x^2 - 6x + 5$ and the $x$-axis. Find the area of square $EFGH$.\n\\[\n\\text{[asy]}\nunitsize(0.8 cm);\n\nreal parab (real x) {\n return(x^2 - 6*x + 5);\n}\n\npair E, F, G, H;\nreal x = -1 + sqrt(3);\n\nE ... | math | {
"style": "rule",
"ground_truth": "24 - 8\\sqrt{5}"
} | {
"index": 3605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square $WXYZ$ with side length 8 units is divided into four smaller squares by drawing lines from the midpoints of one side to the midpoints of the opposite sides. The top right square of each iteration is shaded. If this dividing and shading process is done 100 times, what i... | math | {
"style": "rule",
"ground_truth": "\\frac{64}{3}"
} | {
"index": 3606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square 10cm on each side has four quarter circles drawn with centers at the four corners. How many square centimeters are in the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\nunitsize (1.5 cm);\n\ndraw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle);\nfi... | math | {
"style": "rule",
"ground_truth": "100-25\\pi"
} | {
"index": 3607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square \\(ABCD\\) has a side-length of 2, and \\(M\\) is the midpoint of \\(BC\\). The circle \\(S\\) inside the quadrilateral \\(AMCD\\) touches the three sides \\(AM\\), \\(CD\\), and \\(DA\\). What is its radius?"
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\sqrt{5}"
} | {
"index": 3608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\pi}"
} | {
"index": 3609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square and a regular nonagon are coplanar and share a common side $\\overline{AD}$. What is the degree measure of the exterior angle $BAC$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{130}{1}"
} | {
"index": 3610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square and a regular octagon have equal perimeters. If the area of the square is 16, what is the area of the octagon?\nA) $8 + 4\\sqrt{2}$\nB) $4 + 2\\sqrt{2}$\nC) $16 + 8\\sqrt{2}$\nD) $4\\sqrt{2}$\nE) $8\\sqrt{2}$"
}
] | math | {
"style": "rule",
"ground_truth": "8 + 4\\sqrt{2}"
} | {
"index": 3611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square and a regular pentagon have the same perimeter. Let $C$ be the area of the circle circumscribed about the square, and $D$ the area of the circle circumscribed around the pentagon. Find $C/D$.\nA) $\\frac{25}{128}$\nB) $\\frac{25(5 + 2\\sqrt{5})}{128}$\nC) $\\frac{25(5-... | math | {
"style": "rule",
"ground_truth": "\\frac{25(5-2\\sqrt{5})}{128}"
} | {
"index": 3612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.1$ cm, $8.2$ cm and $9.7$ cm. What is the area of the square in square centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 3613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square and an equilateral triangle have the same perimeter. Let $A$ be the area of the circle circumscribed about the square and $B$ the area of the circle circumscribed around the triangle. Find $A/B$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{32}"
} | {
"index": 3614,
"solution": "1. **Assume the common perimeter**: Let the common perimeter of the square and the equilateral triangle be $P$.\n\n2. **Calculate side lengths**:\n - For the square, each side length is $\\frac{P}{4}$.\n - For the equilateral triangle, each side length is $\\frac{P}{3}$.\n\n3. **De... |
DeepScaleR | [
{
"role": "user",
"content": "A square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, of the square? [asy]\nunitsize(1mm);\ndefaultpen(linewidth(0.7pt));\n\ndraw((0,0)--(20,0)--(20,20)--(0,20)--cycle);\ndraw(Circle((5,5),5));\ndraw(Circle((15,5),... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 3615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square and four circles, each with a radius of 8 inches, are arranged as in the previous problem. What is the area, in square inches, of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"index": 3616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square array of dots with 10 rows and 10 columns is given. Each dot is coloured either blue or red. Whenever two dots of the same colour are adjacent in the same row or column, they are joined by a line segment of the same colour as the dots. If they are adjacent but of diffe... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 3617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square board with three rows and three columns contains nine cells. In how many different ways can we write the three letters A, B, and C in three different cells, so that exactly one of these three letters is written in each row?"
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 3618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square can be divided into four congruent figures as shown: If each of the congruent figures has area 1, what is the area of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 3619,
"solution": "There are four congruent figures with area 1, so the area of the square is 4."
} |
DeepScaleR | [
{
"role": "user",
"content": "A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]\n\ndraw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));\n\nfill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));\n\nfill((4,1)--(5,1)--(5,2)--(4,2)--cycle... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 3620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square field is enclosed by a wooden fence, which is made of 10-meter-long boards placed horizontally. The height of the fence is four boards. It is known that the number of boards in the fence is equal to the area of the field, expressed in hectares. Determine the dimensions... | math | {
"style": "rule",
"ground_truth": "16000"
} | {
"index": 3621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square flag features a green cross of uniform width with a yellow square in the center on a black background. The cross is symmetric with respect to each of the diagonals of the square. If the entire cross (both the green arms and the yellow center) occupies 50% of the area o... | math | {
"style": "rule",
"ground_truth": "6.25\\%"
} | {
"index": 3622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 4... | math | {
"style": "rule",
"ground_truth": "45\\%"
} | {
"index": 3623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square flag features a symmetric red cross with uniform width and two identical blue squares in the center on a white background. The entire cross, including the red arms and the two blue centers, occupies 40% of the flag's area. Determine what percent of the flag's area is o... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"index": 3624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square flag has a green cross of uniform width with a yellow square in the center on a white background. The cross is symmetric with respect to each of the diagonals of the square. If the entire cross (both the green arms and the yellow center) occupies 49% of the area of the... | math | {
"style": "rule",
"ground_truth": "25.14\\%"
} | {
"index": 3625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 3626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square floor is tiled with a large number of regular hexagonal tiles, which are either blue or white. Each blue tile is surrounded by 6 white tiles, and each white tile is surrounded by 3 white and 3 blue tiles. Determine the ratio of the number of blue tiles to the number of... | math | {
"style": "rule",
"ground_truth": "1: 2"
} | {
"index": 3627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is"
}
] | math | {
"style": "rule",
"ground_truth": "2601"
} | {
"index": 3628,
"solution": "1. **Identify the pattern of black tiles**: The problem states that the tiles on the two diagonals of the square floor are black. This means that each diagonal is filled with black tiles.\n\n2. **Determine the number of black tiles on each diagonal**: \n - If the side length of the s... |
DeepScaleR | [
{
"role": "user",
"content": "A square grid $100 \\times 100$ is tiled in two ways - only with dominoes and only with squares $2 \\times 2$. What is the least number of dominoes that are entirely inside some square $2 \\times 2$?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 3629,
"solution": "\nConsider a \\(100 \\times 100\\) square grid. We are tasked to tile this grid in two ways:\n\n1. **Tiling with Dominoes:** Dominoes are \\(1 \\times 2\\) or \\(2 \\times 1\\) tiles. This means that each domino covers exactly two squares in the grid.\n\n2. **Tiling with \\(2 \\times 2... |
DeepScaleR | [
{
"role": "user",
"content": "A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded?"
}
] | math | {
"style": "rule",
"ground_truth": "33.33\\%"
} | {
"index": 3630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has a diagonal of length $10\\sqrt{2}$ centimeters. What is the number of square centimeters in the area of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 3631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has a side length of $4$. Within this square, two equilateral triangles are placed such that one has its base along the bottom side of the square, and the other is rotated such that its vertex touches the midpoint of the top side of the square, and its base is parallel... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 3632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has a side length of $40\\sqrt{3}$ cm. Calculate the length of the diagonal of the square and the area of a circle inscribed within it."
}
] | math | {
"style": "rule",
"ground_truth": "1200\\pi"
} | {
"index": 3633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has a side length of 4 units. Points $R$ and $S$ are chosen such that $R$ is the midpoint of the bottom side of the square, and $S$ is the midpoint of the right side of the square. What fraction of the interior of the square is not shaded if the triangle formed by $R$,... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 3634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has an area of $25.$ A rectangle has the same width as the square. The length of the rectangle is double its width. What is the area of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 3635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has area $\\frac14$. What is the side length of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac12"
} | {
"index": 3636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has been divided into $2022$ rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?"
}
] | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"index": 3637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has side length 5. In how many different locations can point $X$ be placed so that the distances from $X$ to the four sides of the square are $1,2,3$, and 4?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 3638,
"solution": "Label the square as $A B C D$. Suppose that the point $X$ is 1 unit from side $A B$. Then $X$ lies on a line segment $Y Z$ that is 1 unit below side $A B$. Note that if $X$ lies on $Y Z$, then it is automatically 4 units from side $D C$. Since $X$ must be 2 units from either side $A D$... |
DeepScaleR | [
{
"role": "user",
"content": "A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "100+75\\pi"
} | {
"index": 3639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has sides of length 3 units. A second square is formed having sides that are $120\\%$ longer than the sides of the first square. This process is continued sequentially to create a total of five squares. What will be the percent increase in the perimeter from the first ... | math | {
"style": "rule",
"ground_truth": "107.4\\%"
} | {
"index": 3640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has sides of length 8, and a circle centered at one of its vertices has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "64 + 108\\pi"
} | {
"index": 3641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has two diagonals, and a convex pentagon has five diagonals. How many diagonals does a convex decagon have?"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 3642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has vertices \\( P, Q, R, S \\) labelled clockwise. An equilateral triangle is constructed with vertices \\( P, T, R \\) labelled clockwise. What is the size of angle \\( \\angle RQT \\) in degrees?"
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 3643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square has vertices at \\((-2a, -2a), (2a, -2a), (-2a, 2a), (2a, 2a)\\). The line \\( y = -\\frac{x}{2} \\) cuts this square into two congruent quadrilaterals. Calculate the perimeter of one of these quadrilaterals divided by \\( 2a \\). Express your answer in simplified radi... | math | {
"style": "rule",
"ground_truth": "4 + \\sqrt{5}"
} | {
"index": 3644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates 2, 0, and 18 in some order. Find the sum of all possible values of $A$."
}
] | math | {
"style": "rule",
"ground_truth": "1168"
} | {
"index": 3645,
"solution": "More generally, suppose three vertices of the square lie on lines $y=y_{1}, y=y_{2}, y=y_{3}$. One of these vertices must be adjacent to two others. If that vertex is on $y=y_{1}$ and the other two are on $y=y_{2}$ and $y=y_{3}$, then we can use the Pythagorean theorem to get that the ... |
DeepScaleR | [
{
"role": "user",
"content": "A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 3646,
"solution": "1. **Identify the vertices and their coordinates**: \n Given that the $y$-coordinates of the vertices of the square are $0$, $1$, $4$, and $5$, we can assume the vertices are $A=(0,0)$, $B=(x_1,1)$, $C=(x_2,5)$, and $D=(x_3,4)$ after a suitable translation.\n\n2. **Calculate the slop... |
DeepScaleR | [
{
"role": "user",
"content": "A square is completely covered by a large circle and each corner of the square touches a smaller circle of radius \\( r \\). The side length of the square is 6 units. What is the radius \\( R \\) of the large circle?"
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 3647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is constructed on one side of a regular octagon, outward. In the octagon, two diagonals intersect at point \\( B \\) (see the drawing). Find the measure of angle \\( ABC \\). \n\n(A polygon is called regular if all its sides are equal and all its angles are equal.)"
... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 3648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\\ell > 0$ such that there exists a square of side $\\ell$ contained in a cube with edge $1$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 3649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is cut along a diagonal and reassembled to form a parallelogram \\( PQRS \\). If \\( PR=90 \\mathrm{~mm} \\), what is the area of the original square, in \\( \\mathrm{mm}^{2} \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "1620 \\mathrm{~mm}^{2}"
} | {
"index": 3650,
"solution": "Suppose that the original square had side length \\( x \\mathrm{~mm} \\). We extend \\( PQ \\) and draw a line through \\( R \\) perpendicular to \\( PQ \\), meeting \\( PQ \\) extended at \\( T \\). \\( SRTQ \\) is a square, since it has three right angles at \\( S, Q, T \\) (which ma... |
DeepScaleR | [
{
"role": "user",
"content": "A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the rati... | math | {
"style": "rule",
"ground_truth": "5/2"
} | {
"index": 3651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is divided into 2016 triangles, with no vertex of any triangle lying on the sides or inside any other triangle. The sides of the square are sides of some of the triangles in the division. How many total points, which are the vertices of the triangles, are located insid... | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"index": 3653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is divided into five congruent rectangles, as shown. If the perimeter of each of these five rectangles is 36 inches, what is the perimeter of the square, in inches?\n\n[asy]\ndraw((0,0)--(0,5)--(5,5)--(5,0)--cycle);\ndraw((1,0)--(1,5));\ndraw((2,0)--(2,5));\ndraw((3,0)... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is divided into nine smaller squares of equal area. The center square is then divided into nine smaller squares of equal area and the pattern continues indefinitely. What fractional part of the figure is shaded? [asy]\nimport olympiad; size(150); defaultpen(linewidth(0... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is divided into three congruent rectangles. The middle rectangle is removed and replaced on the side of the original square to form an octagon as shown.\nWhat is the ratio of the length of the perimeter of the square to the length of the perimeter of the octagon?\nA $3... | math | {
"style": "rule",
"ground_truth": "3:5"
} | {
"index": 3656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square?"
}
] | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"index": 3657,
"solution": "1. **Assigning Dimensions Based on Given Ratios:**\n - Let the side length of the square be $s$.\n - According to the problem, the width of the rectangle is twice the side of the square. Therefore, the width of the rectangle is $2s$.\n - The ratio of the rectangle's length to its... |
DeepScaleR | [
{
"role": "user",
"content": "A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 3658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is initially divided into sixteen equal smaller squares. The center four squares are then each divided into sixteen smaller squares of equal area, and this pattern continues indefinitely. In each set of sixteen smaller squares, the four corner squares are shaded. What ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 3659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in a circle of radius 1. Find the perimeter of the square."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2}"
} | {
"index": 3660,
"solution": "The square has diagonal length 2, so side length $\\sqrt{2}$ and perimeter $4 \\sqrt{2}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in a circle. A smaller square has one side coinciding with a side of the larger square and has two vertices on the circle, as shown. What percent of the area of the larger square is the area of the smaller square?\n\n[asy]\ndraw(Circle((0,0),1.4142));\nd... | math | {
"style": "rule",
"ground_truth": "4\\%"
} | {
"index": 3661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in a circle. The number of inches in the perimeter of the square equals the number of square inches in the area of the circumscribed circle. What is the radius, in inches, of the circle? Express your answer in terms of pi."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{2}}{\\pi}"
} | {
"index": 3662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in an equilateral triangle such that each vertex of the square touches the perimeter of the triangle. One side of the square intersects and forms a smaller equilateral triangle within which we inscribe another square in the same manner, and this process co... | math | {
"style": "rule",
"ground_truth": "\\frac{3 - \\sqrt{3}}{2}"
} | {
"index": 3663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square?"
}
] | math | {
"style": "rule",
"ground_truth": "4 - 2\\sqrt{3}"
} | {
"index": 3664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in the ellipse\n\\[\\frac{x^2}{3} + \\frac{y^2}{6} = 1,\\]so that its sides are parallel to the coordinate axes. Find the area of the square.\n\n[asy]\nunitsize(1 cm);\n\ndraw(xscale(sqrt(3))*yscale(sqrt(6))*Circle((0,0),1));\ndraw((sqrt(2),sqrt(2))--(-sq... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 3665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in the ellipse\n\\[\\frac{x^2}{4} + \\frac{y^2}{8} = 1,\\]\nsuch that its sides are parallel to the coordinate axes. Find the area of this square."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{3}"
} | {
"index": 3666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in the ellipse\n\\[\\frac{x^2}{5} + \\frac{y^2}{10} = 1,\\]\nso that its sides are parallel to the coordinate axes. Find the area of the square."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"index": 3667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square is inscribed in the ellipse whose equation is $x^2 + 3y^2 = 3$. One vertex of the square is at $(0, 1)$, and one diagonal of the square lies along the y-axis. Determine the square of the length of each side of the square."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3} - 2\\sqrt{\\frac{2}{3}}"
} | {
"index": 3668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{65}"
} | {
"index": 3669,
"solution": "1. **Determine the side length of each square:**\n - The perimeter of the inner square is 20, so each side is $\\frac{20}{4} = 5$.\n - The perimeter of the outer square is 28, so each side is $\\frac{28}{4} = 7$.\n\n2. **Positioning of the inner square:**\n - The inner square is ... |
DeepScaleR | [
{
"role": "user",
"content": "A square of side length $1$ and a circle of radius $\\frac{\\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{9} - \\frac{\\sqrt{3}}{3}"
} | {
"index": 3670,
"solution": "1. **Identify the geometric setup**: We have a square with side length $1$ and a circle with radius $\\frac{\\sqrt{3}}{3}$ sharing the same center. We need to find the area inside the circle but outside the square.\n\n2. **Calculate the diagonal of the square**: The diagonal of the squ... |
DeepScaleR | [
{
"role": "user",
"content": "A square piece of paper has a side length of 1. It is folded such that vertex $C$ meets edge $\\overline{AD}$ at point $C'$, and edge $\\overline{BC}$ intersects edge $\\overline{AB}$ at point $E$. Given $C'D = \\frac{1}{4}$, find the perimeter of triangle $\\bigtriangleup AEC'... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"index": 3671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\\circ}$ (see the figu... | math | {
"style": "rule",
"ground_truth": "871"
} | {
"index": 3672,
"solution": "In the final pyramid, let $ABCD$ be the smaller square and let $A'B'C'D'$ be the larger square such that $AA'$, etc. are edges.\n It is obvious from the diagram that $\\angle A'AB = \\angle A'AD = 105^\\circ$.\n Let $AB$ and $AD$ be the positive $x$ and $y$ axes in a 3-d coordinate sys... |
DeepScaleR | [
{
"role": "user",
"content": "A square piece of paper has sides of length $120$. From each corner, a wedge is cut such that each of the two cuts for the wedge starts at a distance $10$ from the corner, and they meet on the diagonal at an angle of $45^{\\circ}$. After the cuts, the paper is folded up along t... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"index": 3673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square piece of paper is modified by cutting out one smaller right triangle from each corner, each having one leg along a side of the square facing outside. The remaining form is a rectangle. If each triangle has legs of lengths $4$ units and $3$ units, and the original squar... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 3675,
"solution": "1. **Understanding the problem**: We start with a square piece of paper with each side measuring 4 inches. The paper is folded in half vertically, which means the fold divides the square into two equal rectangles each with dimensions $4 \\times 2$ inches.\n\n2. **Cutting the paper**: A... |
DeepScaleR | [
{
"role": "user",
"content": "A square pyramid has a base edge of 32 inches and an altitude of 1 foot. A square pyramid whose altitude is one-fourth of the original altitude is cut away at the apex of the original pyramid. The volume of the remaining frustum is what fractional part of the volume of the orig... | math | {
"style": "rule",
"ground_truth": "\\frac{63}{64}"
} | {
"index": 3676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$. A plane passes through the midpoints of $AE$, $BC$, and $CD$. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 3677,
"solution": "[asy]import three; import math; pointpen = black; pathpen = black+linewidth(0.7); currentprojection = perspective(2.5,-12,4); triple A=(-2,2,0), B=(2,2,0), C=(2,-2,0), D=(-2,-2,0), E=(0,0,2*2^.5), P=(A+E)/2, Q=(B+C)/2, R=(C+D)/2, Y=(-3/2,-3/2,2^.5/2),X=(3/2,3/2,2^.5/2), H=(4,2,0), I=(-... |
DeepScaleR | [
{
"role": "user",
"content": "A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\\overline{AE}$, $\\overline{BC}$, and $\\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$."... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 3678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square sheet contains 1000 points, with any three points, including the vertices of the square, not being collinear. Connect some of these points and the vertices of the square with line segments to divide the entire square into smaller triangles (using the connected line seg... | math | {
"style": "rule",
"ground_truth": "2002"
} | {
"index": 3679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$ . In what ratio does the fold line divide side $BC$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "5/3"
} | {
"index": 3680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square sheet of paper has an area of $12 \\text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Gi... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"index": 3681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square sheet of paper has area $6 \\text{ cm}^2$. The front is white and the back is black. When the sheet is folded so that point $A$ rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is $A$ from its original posi... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 3682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square sheet of paper that measures $18$ cm on a side has corners labeled $A$ , $B$ , $C$ , and $D$ in clockwise order. Point $B$ is folded over to a point $E$ on $\\overline{AD}$ with $DE=6$ cm and the paper is creased. When the paper is unfolded, the crease i... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 3683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square sheet of paper with sides of length $10$ cm is initially folded in half horizontally. The folded paper is then folded diagonally corner to corner, forming a triangular shape. If this shape is then cut along the diagonal fold, what is the ratio of the perimeter of one o... | math | {
"style": "rule",
"ground_truth": "\\frac{15 + \\sqrt{125}}{40}"
} | {
"index": 3684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with a perimeter of 36 is inscribed in a square with a perimeter of 40. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?\nA) $\\sqrt{101}$\nB) $9\\sqrt{2}$\nC) $8\\sqrt{2}$\nD) $\\sqrt{90}$\nE) $10$"
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2}"
} | {
"index": 3685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with a side length of $1$ is divided into one triangle and three trapezoids by joining the center of the square to points on each side. These points divide each side into segments such that the length from a vertex to the point is $\\frac{1}{4}$ and from the point to t... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 3686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with a side length of 10 centimeters is rotated about its horizontal line of symmetry. Calculate the volume of the resulting cylinder in cubic centimeters and express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "250\\pi"
} | {
"index": 3687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with a side length of 100 cm is drawn on a board. Alexei intersected it with two lines parallel to one pair of its sides. After that, Danil intersected the square with two lines parallel to the other pair of sides. As a result, the square was divided into 9 rectangles,... | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"index": 3688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with a side length of 2 rotates around one of its sides, which is the axis of rotation. What is the volume of the cylinder obtained from this rotation?"
}
] | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"index": 3689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with a side length of one unit has one of its vertices separated from the other three by a line \\( e \\). The products of the distances of opposite vertex pairs from \\( e \\) are equal. What is the distance of the center of the square from \\( e \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with an area of 40 is inscribed in a semicircle. If another square is inscribed in a full circle with the same radius, what is the area of this square?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 3691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with an area of one square unit is inscribed in an isosceles triangle such that one side of the square lies on the base of the triangle. Find the area of the triangle, given that the centers of mass of the triangle and the square coincide (the center of mass of the tri... | math | {
"style": "rule",
"ground_truth": "9/4"
} | {
"index": 3692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with an integer side length was cut into 2020 smaller squares. It is known that the areas of 2019 of these squares are 1, while the area of the 2020th square is not equal to 1. Find all possible values that the area of the 2020th square can take. Provide the smallest o... | math | {
"style": "rule",
"ground_truth": "112225"
} | {
"index": 3693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a 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 th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3694,
"solution": "1. **Identify the side lengths of the squares**: Given that the area of the smaller square is $4$, its side length is $\\sqrt{4} = 2$. For the larger square with area $5$, the side length is $\\sqrt{5}$.\n\n2. **Set up the relationship between $a$ and $b$**: Each vertex of the smaller ... |
DeepScaleR | [
{
"role": "user",
"content": "A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 3695,
"solution": "1. **Understanding the Problem:**\n - We need to find the smallest integer side length of a square that can be divided into 10 smaller squares, where at least 8 of these squares have an area of 1 (i.e., side length of 1).\n\n2. **Eliminating the Smallest Option:**\n - **Option (A) ... |
DeepScaleR | [
{
"role": "user",
"content": "A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin wi... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"index": 3696,
"solution": "To solve this problem, we need to calculate the probability that a randomly placed coin will cover part of the black region on the square. We start by determining the total possible region where the center of the coin can land and then calculate the area of the regions where the coin w... |
DeepScaleR | [
{
"role": "user",
"content": "A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of lengt... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{35}"
} | {
"index": 3697,
"solution": "#### Analyzing the first right triangle:\nConsider a right triangle $ABC$ with sides $3$, $4$, and $5$, where $5$ is the hypotenuse. Let a square be inscribed such that one vertex of the square coincides with the right-angle vertex $C$ of the triangle. Let the side length of the square... |
DeepScaleR | [
{
"role": "user",
"content": "A square with side length $x$ is inscribed in a right triangle with sides of length $6$, $8$, and $10$ so that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in the same triangle, but one side of ... | math | {
"style": "rule",
"ground_truth": "\\frac{111}{175}"
} | {
"index": 3698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A square with side length 1 is rotated about one vertex by an angle of $\\alpha,$ where $0^\\circ < \\alpha < 90^\\circ$ and $\\cos \\alpha = \\frac{4}{5}.$ Find the area of the shaded region that is common to both squares.\n\n[asy]\nunitsize(3 cm);\n\npair A, B, C, D, Bp, Cp,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3699,
"solution": ""
} |
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